{"id":334,"date":"2017-08-28T04:30:39","date_gmt":"2017-08-28T04:30:39","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=334"},"modified":"2017-09-25T16:41:26","modified_gmt":"2017-09-25T16:41:26","slug":"differential-calculus-ii","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/differential-calculus-ii\/","title":{"rendered":"Differential Calculus II"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/differential-calculus\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/calculus\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/integral-calculus\">Next\u00a0Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Differential Calculus II<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will cover the definition of a derivative and apply it to the evaluation of slopes and tangent\u00a0lines of curves, instantaneous rates of changes of functions, relative maxima and minima, and methods to compute the\u00a0derivative of several types of functions.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>The <em><strong>limit\u00a0<\/strong><\/em>of a function is the value the function approaches as its\u00a0variable approaches a number to the left and right of that\u00a0number. The function does not need to be defined at that number,\u00a0but must be defined for values slightly less and slightly more.\u00a0Limits can be determined (if they exist) by plugging in the exact\u00a0value and values on either side of the value itself, by graphing\u00a0the function, or by mathematical manipulation, such as factoring.<\/li>\n<li>A <strong><em>one-sided limit\u00a0<\/em><\/strong>approaches a value to one side of the variable but not the other.\u00a0If a function is defined for numbers slightly higher than\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image003.gif\" width=\"7\" height=\"7\" name=\"graphics2\" align=\"BOTTOM\" border=\"0\" \/>\u00a0but not for numbers slightly lower, the function has a right-hand\u00a0limit, and we say that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image006.gif\" width=\"57\" height=\"23\" name=\"graphics3\" align=\"MIDDLE\" border=\"0\" \/>\u00a0exists.<\/li>\n<li><strong><em>Infinite limits\u00a0<\/em><\/strong>are limits that approach\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p1_html_65cbb539.gif\" width=\"25\" height=\"16\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>\u00a0as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image012.gif\" width=\"7\" height=\"7\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>\u00a0approaches\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image015.gif\" width=\"7\" height=\"7\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0This is represented graphically with a vertical asymptote at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image018.gif\" width=\"32\" height=\"7\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/li>\n<li>A <strong><em>limit at infinity\u00a0<\/em><\/strong>is the value of a function as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image020.gif\" width=\"7\" height=\"7\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>\u00a0approaches\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p1_html_65cbb539.gif\" width=\"25\" height=\"16\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0For example, if\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image025.gif\" width=\"80\" height=\"20\" name=\"graphics10\" align=\"MIDDLE\" border=\"0\" \/>,\u00a0the limit at infinity is represented with a horizontal asymptote\u00a0at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image028.gif\" width=\"35\" height=\"14\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>.<\/li>\n<li>A function is <strong><em>continuous\u00a0<\/em><\/strong>over an interval if there are no breaks in the graph. In other\u00a0words, for all points\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p1_html_m588f3dc7.gif\" width=\"37\" height=\"27\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/>\u00a0in the interval (except the endpoints if the interval is open),\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p1_clip_image033.gif\" width=\"102\" height=\"20\" name=\"graphics13\" align=\"MIDDLE\" border=\"0\" \/>,\u00a0and terms both to the left and right of the equal sign exist.<\/li>\n<li>The <strong><em>Intermediate Value Theorem\u00a0<\/em><\/strong>states that, if a function is continuous over a closed interval,\u00a0every value of the function between the endpoints exists at least\u00a0once, and there is a value of the variable within the open range\u00a0that defines the function\u2019s value.<\/li>\n<\/ul>\n<section>\n<h3>Differential Calculus II<\/h3>\n<p>Up to this point, we have differentiated functions defined as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image003.gif\" width=\"56\" height=\"14\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0In other words,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image006.gif\" width=\"9\" height=\"10\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is defined <em><span style=\"text-decoration: none;\">explicitly <\/span><\/em><span style=\"text-decoration: none;\">in\u00a0terms of the variable\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image009.gif\" width=\"7\" height=\"7\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0There are situations where we define\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image012.gif\" width=\"46\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0in terms of a third variable, or <\/span><em><span style=\"text-decoration: none;\">parameter<\/span><\/em>,\u00a0such as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image015.gif\" width=\"4\" height=\"9\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0In order to find\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image018.gif\" width=\"17\" height=\"34\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we apply <abbr title=\"A differentiation technique used when two variables x and y are each defined in terms of a third variable, t.\">parametric\u00a0differentiation<\/abbr>. Similarly, we may have an equation (not\u00a0necessarily a function) that relates\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image021.gif\" width=\"46\" height=\"14\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0but is not in the form\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image024.gif\" width=\"56\" height=\"14\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0An example is the equation of a circle with a radius\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image027.gif\" width=\"6\" height=\"7\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0centered about the origin:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image030.gif\" width=\"74\" height=\"17\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0We find\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image033.gif\" width=\"17\" height=\"34\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0through a similar technique known as <abbr title=\" a differentiation technique used when an equation exists that relates two variables a, but not in the simple form a \">implicit\u00a0differentiation<\/abbr>.<\/p>\n<p>The rule for parametric differentiation is simple:<\/p>\n<p>If\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image036.gif\" width=\"129\" height=\"14\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0we can find\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image039.gif\" width=\"17\" height=\"34\" name=\"graphics15\" align=\"MIDDLE\" border=\"0\" \/>\u00a0by solving the ratio\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image042.gif\" width=\"57\" height=\"74\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image045.gif\" width=\"41\" height=\"34\" name=\"graphics17\" align=\"MIDDLE\" border=\"0\" \/>.<\/p>\n<p>This is actually a consequence of the chain rule.<\/p>\n<p>For example, suppose <em>x<\/em> = <em>t<\/em><sup>3<\/sup> + 4<em>t\u00a0<\/em>\u2013 1, and <em>y<\/em> = 6 \u2013 <em>t<\/em><sup>2<\/sup>. Use\u00a0parametric differentiation to find\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image051.gif\" width=\"17\" height=\"34\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>If we were to solve this using only the tools we had in the\u00a0last section, we would have to find a formula for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image054.gif\" width=\"4\" height=\"9\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/>\u00a0in terms of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image057.gif\" width=\"7\" height=\"7\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/>\u00a0(which may be next to impossible) and substitute that formula into\u00a0the equation for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image060.gif\" width=\"9\" height=\"10\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0so we would be left with an expression for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image063.gif\" width=\"28\" height=\"14\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\nHowever, parametric differentiation makes this much easier:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image066.gif\" width=\"305\" height=\"35\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>If, as in this case, the formula for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image069.gif\" width=\"17\" height=\"34\" name=\"graphics24\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0cannot easily be defined in terms of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image072.gif\" width=\"7\" height=\"7\" name=\"graphics25\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0you can create a table of values for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image075.gif\" width=\"96\" height=\"34\" name=\"graphics26\" align=\"ABSMIDDLE\" border=\"0\" \/>.<br \/>\nThis should provide information on the relationship of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image078.gif\" width=\"17\" height=\"34\" name=\"graphics27\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image081.gif\" width=\"46\" height=\"14\" name=\"graphics28\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Some problems simply provide the\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image084.gif\" width=\"4\" height=\"9\" name=\"graphics29\" align=\"BOTTOM\" border=\"0\" \/>\u00a0value.<\/p>\n<p>Given the equation of a circle,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image087.gif\" width=\"74\" height=\"17\" name=\"graphics30\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0with constant radius\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image090.gif\" width=\"6\" height=\"7\" name=\"graphics31\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0use implicit differentiation to solve for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image093.gif\" width=\"17\" height=\"34\" name=\"graphics32\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Without implicit differentiation, we can solve for <em>y<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image096.gif\" width=\"92\" height=\"21\" name=\"graphics33\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>We have two separate functions, one for the top semicircle, and\u00a0the other for the bottom semicircle. We can then use the chain\u00a0rule to solve for both values of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image099.gif\" width=\"17\" height=\"34\" name=\"graphics34\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Now we will examine how to use implicit differentiation. First,\u00a0since both sides of the equation are equal, their derivatives are\u00a0equal. Also, recall that the derivative of the left side of the\u00a0equation is the same as the derivative of each of the two terms.\u00a0To find\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image102.gif\" width=\"17\" height=\"34\" name=\"graphics35\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we will take the derivative\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image105.gif\" width=\"17\" height=\"34\" name=\"graphics36\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0of all three terms.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image108.gif\" width=\"169\" height=\"34\" name=\"graphics37\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image111.gif\" width=\"93\" height=\"34\" name=\"graphics38\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0(The second term uses the <span style=\"text-decoration: none;\">chain\u00a0rule since <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is indeed dependent upon <em>x<\/em>, and the third term is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image114.gif\" width=\"6\" height=\"11\" name=\"graphics39\" align=\"BOTTOM\" border=\"0\" \/>\u00a0because\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image117.gif\" width=\"6\" height=\"7\" name=\"graphics40\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is a constant with respect to <em>x<\/em>). <\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image120.gif\" width=\"160\" height=\"37\" name=\"graphics41\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>We actually have two solutions for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image123.gif\" width=\"17\" height=\"34\" name=\"graphics42\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0depending on whether we are interested in the upper or lower\u00a0semicircle.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image126.gif\" width=\"212\" height=\"40\" name=\"graphics43\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The first equation describes the slope of the upper semicircle\u00a0of a circle of radius\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p11_clip_image129.gif\" width=\"6\" height=\"7\" name=\"graphics44\" align=\"BOTTOM\" border=\"0\" \/>\u00a0centered about the origin, while the second equation describes the\u00a0slope of the lower semicircle.<\/p>\n<h3>What are related rates of change?<\/h3>\n<p>Using the tools of implicit differentiation, we are able to use\u00a0differential calculus for applications known as <abbr title=\" given the rate of change of certain quantities, find the rate of change of other related quantities\">related\u00a0rates of change<\/abbr>. We will solve two classic related rates\u00a0problems: the <em>sliding ladder<\/em> and the <em>leaking container<\/em>.<\/p>\n<h4>The Sliding Ladder<\/h4>\n<p>A 10-foot ladder is leaning at an angle against a wall, with\u00a0the foot of the ladder on the floor. The foot of the ladder is\u00a0sliding across the floor at a rate of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image003.gif\" width=\"6\" height=\"11\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>\u00a0feet per second. When the foot of the ladder is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image006.gif\" width=\"6\" height=\"11\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>\u00a0feet from the wall, how fast is the top of the ladder sliding down\u00a0the wall?<\/p>\n<p>The wall, floor, and ladder form a right triangle, so we will\u00a0use the Pythagorean Theorem to solve. The equation is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image009.gif\" width=\"74\" height=\"17\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image012.gif\" width=\"7\" height=\"7\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is the distance from the wall to the foot of the ladder,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image015.gif\" width=\"9\" height=\"10\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the height from the floor to the top of the ladder, and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image018.gif\" width=\"7\" height=\"7\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is the length of the ladder, which we know is 10 feet. We are\u00a0asked to find the rate at which the top of the ladder is sliding\u00a0down the wall, which according to our choice of variables is\u00a0equivalent to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image021.gif\" width=\"17\" height=\"34\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Be careful when you approach problems such as this, as we could\u00a0have been asked to find the rate at which the bottom of the ladder\u00a0is sliding across the floor. In that case we would be looking for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image024.gif\" width=\"17\" height=\"34\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Since we are interested in changing rates over time, we take\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image027.gif\" width=\"14\" height=\"34\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0of all terms.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image030.gif\" width=\"195\" height=\"34\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image033.gif\" width=\"113\" height=\"34\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>At the beginning of this problem, we identified that we are\u00a0looking for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image036.gif\" width=\"17\" height=\"34\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0so we will use this equation, and isolate that particular term.\u00a0Solving for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image039.gif\" width=\"17\" height=\"34\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image042.gif\" width=\"77\" height=\"37\" name=\"graphics16\" align=\"MIDDLE\" border=\"0\" \/>.\u00a0We are given\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image045.gif\" width=\"30\" height=\"11\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image048.gif\" width=\"42\" height=\"34\" name=\"graphics18\" align=\"MIDDLE\" border=\"0\" \/>\u00a0feet per second. We still need to find the value of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image051.gif\" width=\"9\" height=\"10\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0That can be determined by using the Pythagorean Theorem again.\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image054.gif\" width=\"102\" height=\"17\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0which gives us\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image057.gif\" width=\"33\" height=\"14\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0feet as the only positive answer. Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image060.gif\" width=\"166\" height=\"34\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0feet per second. Note that the answer is negative, since the\u00a0ladder is sliding in the\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p12_clip_image063.gif\" width=\"19\" height=\"10\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0direction.<\/p>\n<h3>The Leaking Container<\/h3>\n<p>A paper cup in the shape of a right circular cone has a height\u00a0of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image003.gif\" width=\"6\" height=\"11\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>\u00a0inches and a radius of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image006.gif\" width=\"6\" height=\"11\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>\u00a0inches at its rim. It is filled with water, but develops a leak\u00a0out of the bottom, at a rate of 0.1 cubic inch per minute. At what\u00a0rate is the height of the water dropping when the cup is filled to\u00a0a height of 3 inches?<\/p>\n<p>In order to solve this problem, you need to know that the\u00a0volume of a right circular cone is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image012.gif\" width=\"68\" height=\"34\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and, according to the law of similar triangles, the ratio\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image015.gif\" width=\"9\" height=\"30\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is constant at all heights. From this, we eliminate\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image018.gif\" width=\"6\" height=\"7\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>\u00a0in the volume equation.<\/p>\n<p>We are given\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image021.gif\" width=\"21\" height=\"34\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and need to solve for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image024.gif\" width=\"21\" height=\"40\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0or the rate at which the height of the water is changing with\u00a0respect to time. The first sentence tells us\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image027.gif\" width=\"37\" height=\"34\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0so\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image030.gif\" width=\"34\" height=\"34\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image033.gif\" width=\"140\" height=\"43\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Taking\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image036.gif\" width=\"14\" height=\"34\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0of both sides, we find<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image039.gif\" width=\"86\" height=\"36\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0or\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image042.gif\" width=\"86\" height=\"35\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>We are given\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image045.gif\" width=\"66\" height=\"34\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image048.gif\" width=\"31\" height=\"11\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p13_clip_image051.gif\" width=\"173\" height=\"38\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0inches per minute.<\/p>\n<h3>Derivatives of Higher Order<\/h3>\n<p>The derivative\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image003.gif\" width=\"33\" height=\"14\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is known as a <abbr title=\"f '(x) or the slope of a line tangent to a graph. \">first\u00a0derivative<\/abbr> of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image006.gif\" width=\"10\" height=\"14\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0or the <em>first derived function<\/em>. A derivative of a derivative\u00a0is known as a <abbr title=\"the derivative of a first derivative \">second\u00a0derivative function<\/abbr>. One way a second derivative is denoted is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image009.gif\" width=\"37\" height=\"14\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0which is read, \u201c<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image012.gif\" width=\"10\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0<em>double prime<\/em>.\u201d We actually computed a second\u00a0derivative earlier, when we calculated the acceleration in the\u00a0example of a moving particle.<\/p>\n<p>Recall that we found the velocity\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image015.gif\" width=\"23\" height=\"14\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>\u00a0by taking the derivative of the position\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image018.gif\" width=\"22\" height=\"14\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image021.gif\" width=\"101\" height=\"34\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and then found the acceleration <em>g<\/em> by taking the derivative\u00a0of the velocity,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image027.gif\" width=\"43\" height=\"34\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We see that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image030.gif\" width=\"54\" height=\"14\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>Another way to write the second derivative function is<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image033.gif\" width=\"152\" height=\"40\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Typically, third derivative functions also use primes, such as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image036.gif\" width=\"39\" height=\"14\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Beyond that, we typically put the order in parentheses, like this:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image039.gif\" width=\"161\" height=\"17\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Alternatively, we can use the notation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image042.gif\" width=\"26\" height=\"35\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0which does not use parentheses. Power functions may eventually\u00a0reach a point where higher-order derivatives are all zero. Other\u00a0functions, like trigonometric, logarithmic, and exponential\u00a0functions, can have infinite orders of derivatives.<\/p>\n<p>Let\u2019s try an example. Compute all higher order\u00a0derivatives of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image045.gif\" width=\"161\" height=\"17\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0(until\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image048.gif\" width=\"68\" height=\"17\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>).<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image051.gif\" width=\"142\" height=\"91\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Let\u2019s try another example. Let\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image054.gif\" width=\"36\" height=\"34\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Compute\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image057.gif\" width=\"25\" height=\"37\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and extrapolate an equation for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image060.gif\" width=\"26\" height=\"35\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>To solve, rewrite\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image063.gif\" width=\"44\" height=\"17\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image066.gif\" width=\"162\" height=\"170\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Note that the higher order derivatives alternate between + and\u00a0\u2013 . You may also notice another pattern \u2014 the\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image069.gif\" width=\"14\" height=\"14\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/>\u00a0order derivative has\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image072.gif\" width=\"11\" height=\"11\" name=\"graphics25\" align=\"BOTTOM\" border=\"0\" \/>\u00a0in the numerator and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image075.gif\" width=\"23\" height=\"14\" name=\"graphics26\" align=\"BOTTOM\" border=\"0\" \/>\u00a0in the denominator. The alternating signs can be mathematically\u00a0represented as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image078.gif\" width=\"30\" height=\"15\" name=\"graphics27\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Therefore, the\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image081.gif\" width=\"14\" height=\"14\" name=\"graphics28\" align=\"BOTTOM\" border=\"0\" \/>\u00a0derivative of <em>y<\/em> is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image087.gif\" width=\"90\" height=\"35\" name=\"graphics29\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Try one final example. Given\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image090.gif\" width=\"92\" height=\"17\" name=\"graphics30\" align=\"TEXTTOP\" border=\"0\" \/>,\u00a0compute\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image093.gif\" width=\"25\" height=\"37\" name=\"graphics31\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>We are being asked to find the second derivative of <em>y<\/em>,\u00a0with respect to <em>x<\/em>. We need to use implicit differentiation,\u00a0the chain rule, and the quotient rule to solve this problem.<\/p>\n<p>First, find\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image096.gif\" width=\"17\" height=\"34\" name=\"graphics32\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0by taking\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image099.gif\" width=\"17\" height=\"34\" name=\"graphics33\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0of each term.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image102.gif\" width=\"98\" height=\"34\" name=\"graphics34\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Next, solve for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image105.gif\" width=\"17\" height=\"34\" name=\"graphics35\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p5_html_3c34a5ff.gif\" width=\"231\" height=\"44\" name=\"graphics36\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p5_html_78c88e1e.gif\" width=\"399\" height=\"85\" name=\"graphics37\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Next, substitute the value of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image114.gif\" width=\"17\" height=\"34\" name=\"graphics38\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0into the above equation.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image117.gif\" width=\"523\" height=\"92\" name=\"graphics39\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Finally, notice that due to some careful algebra, we have\u00a0managed to write this last quantity in such a way that it contains\u00a0the term\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image120.gif\" width=\"60\" height=\"17\" name=\"graphics40\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We know from the original problem statement that this term has a\u00a0specific numerical value of 16 for that expression. When solving\u00a0implicit differentiation problems on your own, look carefully for\u00a0opportunities to use initial information like this. We now\u00a0substitute\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image123.gif\" width=\"13\" height=\"11\" name=\"graphics41\" align=\"BOTTOM\" border=\"0\" \/>\u00a0for the value of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_image126.gif\" width=\"61\" height=\"17\" name=\"graphics42\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p14_clip_extra.gif\" width=\"133\" height=\"40\" name=\"graphics43\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3>How do we find the derivative of trigonometric functions,\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"non_block_image no_margin\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image003.gif\" width=\"109\" height=\"18\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/h3>\n<p>Before we proceed, we will define other key derivatives without\u00a0proof. We are going to assume the reader is familiar with natural\u00a0logarithms and exponential functions, as well as their properties.<\/p>\n<p>If we know the derivative of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image006.gif\" width=\"90\" height=\"11\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0we can find the derivatives of other trigonometric functions. For\u00a0example, it is easy to prove\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image009.gif\" width=\"115\" height=\"34\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Simply write\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image012.gif\" width=\"78\" height=\"34\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0differentiate using the quotient rule, and utilize the\u00a0trigonometric identity\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image015.gif\" width=\"109\" height=\"14\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>The following are important derivatives to know:<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image018.gif\" width=\"226\" height=\"121\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<p>Do not forget that the chain rule can apply to any function. For example, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image021.gif\" width=\"123\" height=\"34\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Derivatives for inverse trigonometric functions (e.g. <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image024.gif\" width=\"38\" height=\"14\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/>) and\u00a0hyperbolic trigonometric functions (e.g. <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image027.gif\" width=\"38\" height=\"11\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>) are beyond the scope of this\u00a0module, but can be found in many calculus textbooks, math formula books, and are often provided as references when\u00a0taking tests.<\/p>\n<p>Two interesting rules apply in differential calculus when a function is continuous and\u00a0differentiable on an interval. The first is <abbr title=\" Let s be a function such that it is continuous on the closed interval s. it is differentiable on the open interval s. s Then there is at least one number c in s such that s . \">Rolle\u2019s Theorem<\/abbr>.<\/p>\n<div class=\"callout\">\n<h4>Rolle&#8217;s Theorem<\/h4>\n<p>Let <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image030.gif\" width=\"10\" height=\"14\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/> be a\u00a0function such that:<\/p>\n<ul>\n<li class=\"notebox_text\">it is continuous on the closed interval <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p6_html_m588f3dc7.gif\" width=\"37\" height=\"27\" name=\"graphics13\" align=\"absmiddle\" border=\"0\" \/>,<\/li>\n<li class=\"notebox_text\">it is differentiable on the open interval <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image036.gif\" width=\"35\" height=\"14\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>, and<\/li>\n<li class=\"notebox_text\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image039.gif\" width=\"101\" height=\"14\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<\/ul>\n<p class=\"notebox_text\" align=\"LEFT\">Then there is at least one number <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image042.gif\" width=\"6\" height=\"7\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/> in (<em>a<\/em>, <em>b<\/em>) such that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image048.gif\" width=\"57\" height=\"14\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<p>Of course, this is obvious if <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image051.gif\" width=\"54\" height=\"14\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/> for all\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image054.gif\" width=\"7\" height=\"7\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/> in <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image057.gif\" width=\"35\" height=\"14\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Since <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image060.gif\" width=\"58\" height=\"14\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/> for all <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image063.gif\" width=\"35\" height=\"14\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>, any number <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image066.gif\" width=\"6\" height=\"7\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/> in\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image069.gif\" width=\"35\" height=\"14\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0will satisfy <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image072.gif\" width=\"57\" height=\"14\" name=\"graphics26\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>We will illustrate Rolle\u2019s Theorem with the graph <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image075.gif\" width=\"84\" height=\"14\" name=\"graphics27\" align=\"ABSMIDDLE\" border=\"0\" \/> ( <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image078.gif\" width=\"7\" height=\"7\" name=\"graphics28\" align=\"BOTTOM\" border=\"0\" \/> in\u00a0degrees), on the interval <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image105.gif\" width=\"78\" height=\"15\" name=\"graphics37\" align=\"ABSMIDDLE\" border=\"0\" \/> and the interval <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image123.gif\" width=\"86\" height=\"15\" name=\"graphics43\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This function is both continuous and differentiable for all <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image087.gif\" width=\"7\" height=\"7\" name=\"graphics31\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20001.JPG\" alt=\"rolle's theorem\" width=\"354\" height=\"333\" name=\"graphics32\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"left\">In the first graph, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image096.gif\" width=\"101\" height=\"14\" name=\"graphics34\" align=\"ABSMIDDLE\" border=\"0\" \/> in the interval for <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image099.gif\" width=\"54\" height=\"12\" name=\"graphics35\" align=\"BOTTOM\" border=\"0\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image102.gif\" width=\"52\" height=\"12\" name=\"graphics36\" align=\"BOTTOM\" border=\"0\" \/>. According to Rolle\u2019s Theorem, there must be at least one point in the interval <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image105.gif\" width=\"78\" height=\"15\" name=\"graphics37\" align=\"ABSMIDDLE\" border=\"0\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image108.gif\" width=\"58\" height=\"14\" name=\"graphics38\" align=\"ABSMIDDLE\" border=\"0\" \/>. That point is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image111.gif\" width=\"44\" height=\"12\" name=\"graphics39\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20002.JPG\" alt=\"rolle's theorem\" width=\"354\" height=\"333\" name=\"graphics33\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>In the second graph, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image114.gif\" width=\"101\" height=\"14\" name=\"graphics40\" align=\"ABSMIDDLE\" border=\"0\" \/> in the interval\u00a0for <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image117.gif\" width=\"53\" height=\"12\" name=\"graphics41\" align=\"BOTTOM\" border=\"0\" \/>\u00a0and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image120.gif\" width=\"52\" height=\"12\" name=\"graphics42\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0According to Rolle\u2019s Theorem, there must be at least one point in the interval <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image123.gif\" width=\"86\" height=\"15\" name=\"graphics43\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image126.gif\" width=\"58\" height=\"14\" name=\"graphics44\" align=\"ABSMIDDLE\" border=\"0\" \/>. That point is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image129.gif\" width=\"52\" height=\"12\" name=\"graphics45\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>N<span style=\"text-decoration: none;\">ow we can make a couple of observations about this graph. First, the line tangent to the point at which <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image132.gif\" width=\"58\" height=\"14\" name=\"graphics46\" align=\"ABSMIDDLE\" border=\"0\" \/> is always parallel to the <\/span><em><span style=\"text-decoration: none;\">x-<\/span><\/em><span style=\"text-decoration: none;\">axis. Second, the points at which <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p15_clip_image135.gif\" width=\"58\" height=\"14\" name=\"graphics47\" align=\"ABSMIDDLE\" border=\"0\" \/> appear to define some type of maximum or minimum of the curve in an interval. We will be discussing this second observation in great detail later<\/span>.<\/p>\n<h3>Differential Calculus II<\/h3>\n<p>Let\u2019s try an example. The position of a ball thrown up in\u00a0the air from the ground is described by the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image003.gif\" width=\"103\" height=\"34\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,<br \/>\nwhere\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image006.gif\" width=\"11\" height=\"12\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the upward velocity at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image009.gif\" width=\"29\" height=\"11\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image012.gif\" width=\"8\" height=\"10\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is the downward acceleration due to gravity<br \/>\n(<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image015.gif\" width=\"53\" height=\"16\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0are constant). At some time\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image018.gif\" width=\"33\" height=\"11\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0the ball will strike the ground. Use Rolle\u2019s Theorem to\u00a0prove that, at some time\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image021.gif\" width=\"72\" height=\"13\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0the velocity of the ball will be exactly 0.<\/p>\n<p>You know from experience that when you throw a ball in the air,\u00a0the position\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image024.gif\" width=\"22\" height=\"14\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and velocity\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image027.gif\" width=\"67\" height=\"14\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0of the ball do not abruptly change at any time. Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image030.gif\" width=\"22\" height=\"14\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is continuous and differentiable. Also, we are given\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image033.gif\" width=\"133\" height=\"14\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Since all three conditions for Rolle\u2019s Theorem are met, we\u00a0know there is some time\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image036.gif\" width=\"4\" height=\"9\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/>\u00a0in the interval\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image039.gif\" width=\"32\" height=\"14\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image042.gif\" width=\"91\" height=\"14\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This time is when the ball reaches its peak height before coming\u00a0back down.<\/p>\n<p>Rolle\u2019s Theorem is used to prove a very important rule in\u00a0calculus known as the <abbr title=\" Let c be continuous on the closed interval c and differentiable on the open interval c. Then there exists at least one number c in the open interval c such that c. \">Mean\u00a0Value Theorem<\/abbr>.<\/p>\n<div class=\"callout\">\n<h4>The Mean Value Theorem<\/h4>\n<p>Let <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image045.gif\" width=\"10\" height=\"14\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/> be continuous on the closed\u00a0interval\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p7_html_m588f3dc7.gif\" width=\"37\" height=\"27\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/>\u00a0and\u00a0differentiable on the open interval <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image051.gif\" width=\"35\" height=\"14\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>. Then there exists at least\u00a0one number\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image054.gif\" width=\"6\" height=\"7\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/> in the\u00a0open interval <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image057.gif\" width=\"35\" height=\"14\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/> such that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image060.gif\" width=\"129\" height=\"34\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<p>You may recall that we saw an equation similar to this when we\u00a0interpreted the derivative in terms of a tangent line. It is the\u00a0formula for the slope of the secant line connecting the points\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image063.gif\" width=\"148\" height=\"23\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0on the\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image066.gif\" width=\"29\" height=\"14\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/>\u00a0curve. We also know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image069.gif\" width=\"32\" height=\"14\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the slope of the curve at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image072.gif\" width=\"30\" height=\"7\" name=\"graphics26\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Therefore, this theorem simply states that, for any secant line\u00a0drawn on a curve, there is at least one point in between at which\u00a0the tangent line is parallel to the secant line.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20003.JPG\" alt=\"Mean Value Theorem illustration\" width=\"266\" height=\"250\" name=\"graphics27\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Since this curve meets the continuity and differentiability\u00a0rules of the Mean Value Theorem, the point\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image075.gif\" width=\"6\" height=\"7\" name=\"graphics28\" align=\"BOTTOM\" border=\"0\" \/>\u00a0must exist somewhere in the\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image078.gif\" width=\"35\" height=\"14\" name=\"graphics29\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0interval.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Given the function\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image081.gif\" width=\"57\" height=\"34\" name=\"graphics30\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0over the interval\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image084.gif\" width=\"43\" height=\"14\" name=\"graphics31\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0what is the value of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image087.gif\" width=\"6\" height=\"7\" name=\"graphics32\" align=\"BOTTOM\" border=\"0\" \/>\u00a0that satisfies the Mean Value Theorem?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image090.gif\" width=\"43\" height=\"19\" name=\"graphics33\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image093.gif\" width=\"54\" height=\"19\" name=\"graphics34\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image096.gif\" width=\"31\" height=\"11\" name=\"graphics35\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li>None of the above<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D. If we were to try to find\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image099.gif\" width=\"6\" height=\"7\" name=\"graphics36\" align=\"BOTTOM\" border=\"0\" \/>\u00a0whereby\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image102.gif\" width=\"129\" height=\"34\" name=\"graphics37\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we would first compute\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image105.gif\" width=\"151\" height=\"34\" name=\"graphics38\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image108.gif\" width=\"103\" height=\"34\" name=\"graphics39\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image111.gif\" width=\"229\" height=\"50\" name=\"graphics40\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We compute\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image114.gif\" width=\"78\" height=\"35\" name=\"graphics41\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0so we need to find\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image117.gif\" width=\"6\" height=\"7\" name=\"graphics42\" align=\"BOTTOM\" border=\"0\" \/>\u00a0such that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image120.gif\" width=\"54\" height=\"35\" name=\"graphics43\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This is true when\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image123.gif\" width=\"47\" height=\"14\" name=\"graphics44\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0which has no real solutions.<\/p>\n<p>You might think that this example implies that the Mean Value\u00a0Theorem is incorrect. However, this is not the case. Remember\u00a0that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image126.gif\" width=\"29\" height=\"14\" name=\"graphics45\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0must be <em>continuous and differentiable<\/em> in the interval\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image129.gif\" width=\"43\" height=\"14\" name=\"graphics46\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image132.gif\" width=\"29\" height=\"14\" name=\"graphics47\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is neither continuous nor differentiable at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p16_clip_image135.gif\" width=\"31\" height=\"11\" name=\"graphics48\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0which is in the open interval. Therefore, the Mean Value Theorem\u00a0does not apply here.<\/p>\n<\/div>\n<\/section>\n<h3>What is L\u2019H\u00f4pital\u2019s rule?<\/h3>\n<p>When discussing how to determine limits in the previous module,\u00a0we mentioned that there was a way to determine the limits of\u00a0quotients in the form\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image003.gif\" width=\"32\" height=\"37\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0which reduce to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image006.gif\" width=\"48\" height=\"34\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This is known as <abbr title=\"A differentiation technique to determine the limits of quotients in the form c that reduce to c. If f and g are functions that are differentiable for in an interval around a (except possibly at a), and both c and c, then if c, where L is a real number or c, then c. If f and g are functions that are differentiable for in an interval around a (except possibly at a), and both c and c, then if c, where c is a real number or c, then c. \">L\u2019H\u00f4pital\u2019s\u00a0rule<\/abbr>. There are two versions of this rule; the first is for\u00a0the situation in which we find the limit of the quotient as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image009.gif\" width=\"9\" height=\"34\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and the second for the situation in which we find the limit of the\u00a0quotient as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image012.gif\" width=\"13\" height=\"30\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p><strong>L\u2019H\u00f4pital\u2019s rule for the indeterminate form <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image015.gif\" width=\"9\" height=\"34\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/><span style=\"text-decoration: none;\">: <\/span><\/strong><\/p>\n<p>If <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image018.gif\" width=\"50\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/> are\u00a0functions that are differentiable for in an interval around <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image021.gif\" width=\"7\" height=\"7\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>\u00a0(except possibly at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image024.gif\" width=\"7\" height=\"7\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/>), and both <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image027.gif\" width=\"78\" height=\"20\" name=\"graphics11\" align=\"MIDDLE\" border=\"0\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image030.gif\" width=\"77\" height=\"20\" name=\"graphics12\" align=\"MIDDLE\" border=\"0\" \/>, then if <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image033.gif\" width=\"87\" height=\"37\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image036.gif\" width=\"8\" height=\"11\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/> is a real number or , <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p8_html_m11a5a54a.gif\" width=\"52\" height=\"17\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/> then\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image042.gif\" width=\"83\" height=\"37\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p><strong>L\u2019H\u00f4pital\u2019s rule for the indeterminate form <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image045.gif\" width=\"13\" height=\"30\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/><span style=\"text-decoration: none;\">: <\/span><\/strong><\/p>\n<p>If <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image048.gif\" width=\"50\" height=\"14\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/> are\u00a0functions that are differentiable for in an interval around <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image051.gif\" width=\"7\" height=\"7\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/>\u00a0(except possibly at <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image054.gif\" width=\"7\" height=\"7\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/>), and both <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image057.gif\" width=\"91\" height=\"20\" name=\"graphics21\" align=\"MIDDLE\" border=\"0\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image060.gif\" width=\"90\" height=\"20\" name=\"graphics22\" align=\"MIDDLE\" border=\"0\" \/>, then if <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image063.gif\" width=\"87\" height=\"37\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image066.gif\" width=\"8\" height=\"11\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/> is a real number or , <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p8_html_m11a5a54a.gif\" width=\"52\" height=\"17\" name=\"graphics25\" align=\"BOTTOM\" border=\"0\" \/> then\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image072.gif\" width=\"83\" height=\"37\" name=\"graphics26\" align=\"ABSMIDDLE\" border=\"0\" \/><\/p>\n<\/div>\n<p>When written out formally, this rule sounds much more\u00a0complicated than it really is. L\u2019H\u00f4pital\u2019s rule\u00a0is a method used to compute the limit of a function in which the\u00a0variable is in the numerator and denominator. If plugging the\u00a0value of the limit that <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image075.gif\" width=\"7\" height=\"7\" name=\"graphics27\" align=\"BOTTOM\" border=\"0\" \/>\u00a0approaches into the function leads to either\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image078.gif\" width=\"48\" height=\"34\" name=\"graphics28\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0then just take the derivative of the numerator and the derivative\u00a0of the denominator. Check out this new ratio. If you still get,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image081.gif\" width=\"48\" height=\"34\" name=\"graphics29\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0take the derivative again. Finally, whatever number you get as the\u00a0limit, is the limit for the original function.<\/p>\n<div class=\"callout\">\n<h4>Be Aware!<\/h4>\n<p>You can only use L\u2019H\u00f4pital\u2019s rule whenever the limit is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image084.gif\" width=\"48\" height=\"34\" name=\"graphics30\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Applying this rule to any finite, nonzero limit will produce an erroneous result.<\/p>\n<\/div>\n<p>Let\u2019s try an example. Use L\u2019H\u00f4pital\u2019s\u00a0rule to prove the trigonometric limits that you were previously\u00a0asked to memorize:<\/p>\n<p>(A)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image087.gif\" width=\"76\" height=\"34\" name=\"graphics31\" align=\"ABSMIDDLE\" border=\"0\" \/>;\u00a0and<\/p>\n<p>(B)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p8_html_78bc1dad.gif\" width=\"105\" height=\"41\" name=\"graphics32\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>(A) First, note that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image093.gif\" width=\"55\" height=\"34\" name=\"graphics33\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0so we know we can apply this rule.<\/p>\n<p>The derivative of the numerator is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image096.gif\" width=\"31\" height=\"11\" name=\"graphics34\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0and the derivative of the denominator is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image099.gif\" width=\"4\" height=\"11\" name=\"graphics35\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Therefore,<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image102.gif\" width=\"196\" height=\"34\" name=\"graphics36\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>(B) First, note that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p8_html_782075c4.gif\" width=\"131\" height=\"41\" name=\"graphics37\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0so we know that we can apply this rule again. The derivative of\u00a0the numerator is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p8_html_893ce93.gif\" width=\"128\" height=\"27\" name=\"graphics38\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0and the derivative of the denominator is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p17_clip_image111.gif\" width=\"4\" height=\"11\" name=\"graphics39\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Therefore,<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p8_html_m7a2c9258.gif\" width=\"208\" height=\"41\" name=\"graphics40\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3>Differential Calculus II<\/h3>\n<p>Let\u2019s try a second example. Find\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image003.gif\" width=\"94\" height=\"35\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>By plugging in\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image006.gif\" width=\"29\" height=\"11\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0you should find that both the numerator and denominator are 0 (<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image012.gif\" width=\"43\" height=\"11\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/>).\u00a0Therefore, take the derivative of the numerator and denominator.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image015.gif\" width=\"182\" height=\"48\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Again, plugging in\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image018.gif\" width=\"29\" height=\"11\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>\u00a0leaves us with\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image021.gif\" width=\"9\" height=\"34\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>. Therefore, take the derivatives again.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image024.gif\" width=\"190\" height=\"49\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image027.gif\" width=\"139\" height=\"35\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>Let\u2019s try one more example. Find\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image030.gif\" width=\"86\" height=\"35\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>First, note that the numerator and denominator both approach\u00a0infinity as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image033.gif\" width=\"7\" height=\"7\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/>\u00a0approaches infinity, so we have the indeterminate form\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image036.gif\" width=\"13\" height=\"30\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0If we apply L\u2019H\u00f4pital\u2019s rule, we find that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image039.gif\" width=\"214\" height=\"49\" name=\"graphics15\" align=\"MIDDLE\" border=\"0\" \/>.\u00a0Note that the second term in the denominator approaches 0 at the\u00a0limit, but the numerator and denominator still results in\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image045.gif\" width=\"13\" height=\"30\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>This means it is necessary to take derivatives again. You\u00a0should see a pattern: the numerator will always be\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image048.gif\" width=\"12\" height=\"12\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0the first term of the denominator will always have a power\u00a0function that will go down by one, and the second term in the\u00a0denominator will always approach zero as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image051.gif\" width=\"41\" height=\"9\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Rather than take\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image054.gif\" width=\"22\" height=\"11\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/>\u00a0derivatives before the first term in the denominator fails to\u00a0approach\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image057.gif\" width=\"11\" height=\"7\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0you should see that the numerator will eventu<span style=\"text-decoration: none;\">ally\u00a0<\/span><em><span style=\"text-decoration: none;\">win,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">so\u00a0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image060.gif\" width=\"115\" height=\"35\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0The end result is that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image063.gif\" width=\"12\" height=\"12\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/>\u00a0blows up more rapidly as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image066.gif\" width=\"41\" height=\"9\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/>\u00a0than the\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image069.gif\" width=\"12\" height=\"12\" name=\"graphics25\" align=\"BOTTOM\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p18_clip_image072.gif\" width=\"22\" height=\"11\" name=\"graphics26\" align=\"BOTTOM\" border=\"0\" \/>\u00a0functions combined.<\/p>\n<h3>What is curve sketching?<\/h3>\n<p>In previous sections, we have applied differential calculus to\u00a0determine slopes of tangent lines and rates of change of\u00a0functions. We will now introduce other useful applications. First,\u00a0we will use differentiation as a tool for <abbr title=\"The process of using the first derivative and second derivative to graph a function or relation.. \">curve\u00a0sketching<\/abbr>.<\/p>\n<p><span style=\"text-decoration: none;\">From what we know about\u00a0derivatives, we can look at the graph of a function and roughly\u00a0determine its slope at all points. Wherever the curve\u2019s\u00a0<\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-value\u00a0increases with increasing\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image003.gif\" width=\"7\" height=\"7\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0we know its slope (hence its derivative) is positive. The steeper\u00a0the climb, the larger the value of the derivative in that\u00a0interval. <\/span><\/p>\n<p>Likewise, a curve that decreases in an interval will have a\u00a0negative derivative. At a point where a curve changes direction,\u00a0the derivative is zero. We will illustrate this point in the\u00a0following example.<\/p>\n<p>&nbsp;<\/p>\n<p>For what values of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image006.gif\" width=\"7\" height=\"7\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is the function\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image009.gif\" width=\"115\" height=\"17\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0increasing and decreasing? Determine this by (A) computing\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image012.gif\" width=\"33\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and (B) sketching the curve\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image015.gif\" width=\"29\" height=\"14\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>(A)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image018.gif\" width=\"90\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0The curve\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image021.gif\" width=\"29\" height=\"14\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0increases in value whenever\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image024.gif\" width=\"123\" height=\"11\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Likewise, the curve decreases for all\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image027.gif\" width=\"42\" height=\"11\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>(B) The curve for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image030.gif\" width=\"29\" height=\"14\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is depicted below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20004.JPG\" alt=\" curve for f(x)\" width=\"266\" height=\"250\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The curve clearly shows a parabola that decreases in value\u00a0until\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image033.gif\" width=\"42\" height=\"11\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0after which it increases. Note also that, at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image036.gif\" width=\"42\" height=\"11\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image039.gif\" width=\"58\" height=\"14\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Earlier we saw that the line tangent to the curve at this point is\u00a0horizontal.<\/p>\n<p>For this curve, the point\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image042.gif\" width=\"42\" height=\"11\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0at which\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image045.gif\" width=\"58\" height=\"14\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0is an example of a <abbr title=\" a number MathType in an interval MathType of a function MathType, where MathType, or MathType does not exist. The function MathType must be defined and continuous on MathType. Relative maxima and minima are examples of critical points. \">critical\u00a0point<\/abbr>. This point, at which the curve changes from decreasing\u00a0to increasing, is known as a <abbr title=\"a critical point c in an interval c of a function c, where c for every c. If the function changes in the interval, it represents the point where the curve changes from decreasing to increasing. \">local\u00a0minimum<\/abbr>. The local minimum is the point in the given interval\u00a0at which the curve is at its smallest value. When a function\u00a0changes from decreasing to increasing (or its derivative increases\u00a0continuously from negative to positive), we say that the function\u00a0is <abbr title=\"a property of the graph of the functionMathType, where the derivative MathType changes signs; if MathType increases continuously from negative to positive, the function is concave up. If MathType decreases continuously from positive to negative, the function is concave down.\">concave\u00a0up<\/abbr>,\u00a0or has positive concavity.<\/p>\n<p>&nbsp;<\/p>\n<p>Now let\u2019s examine the parabola\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image048.gif\" width=\"123\" height=\"17\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20005.JPG\" alt=\" Parabola g(x) =-x2+3x+4\" width=\"266\" height=\"250\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Its derivative\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image051.gif\" width=\"97\" height=\"14\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0It also has a critical point at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image054.gif\" width=\"42\" height=\"11\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0However, it changes from increasing to decreasing at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image057.gif\" width=\"42\" height=\"11\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/>\u00a0(i.e. its derivative changes from + to \u2013). The point\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image060.gif\" width=\"42\" height=\"11\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/>\u00a0in the given interval, is known as a <abbr title=\"a critical point c in an interval c of a function c, where c for every c. If the function changes in the interval, it represents the point where the curve changes from increasing to decreasing. \">local\u00a0maximum<\/abbr>. Also, in this interval, the function is <abbr title=\"a property of the graph of the functionMathType, where the derivative MathType changes signs; if MathType increases continuously from negative to positive, the function is concave up. If MathType decreases continuously from positive to negative, the function is concave down. \">concave\u00a0do<span style=\"text-decoration: none;\">wn<\/span><\/abbr><strong><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/strong><\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>Some books refer to <abbr title=\" Where a function has positive concavity, it has negative convexity, and vice versa. \">convexity<\/abbr> rather than concavity. Where a function has positive concavity, it has negative convexity, and vice\u00a0versa. Some books also use the terms a <strong>relative maximum and minimum<\/strong>, rather than local\u00a0maximum and minimum.<\/p>\n<\/div>\n<p>There is a third type of critical point, which is any point\u00a0where a function is not differentiable. Thus,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image063.gif\" width=\"31\" height=\"11\" name=\"graphics25\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is a critical point for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p10_html_m2f71b6c4.gif\" width=\"69\" height=\"27\" name=\"graphics26\" align=\"absmiddle\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p19_clip_image069.gif\" width=\"63\" height=\"17\" name=\"graphics27\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<h3>Differential Calculus II<\/h3>\n<p>For\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image003.gif\" width=\"129\" height=\"17\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0use the first derivative test to find:<\/p>\n<p>(A) over which interval the function is increasing and\u00a0decreasing; and<\/p>\n<p>(B) all local maxima and minima.<\/p>\n<p>(A)\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image006.gif\" width=\"284\" height=\"17\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image009.gif\" width=\"29\" height=\"14\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is increasing whenever the product\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image012.gif\" width=\"108\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This occurs when either both terms are positive, or both terms are\u00a0negative. Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image015.gif\" width=\"318\" height=\"11\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0The first condition is met whenever\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image018.gif\" width=\"31\" height=\"11\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0and the second condition is met whenever\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image021.gif\" width=\"40\" height=\"11\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Another way to write this is that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image024.gif\" width=\"29\" height=\"14\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is increasing in the ranges\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image027.gif\" width=\"136\" height=\"14\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0It should be obvious (or you can run through the entire argument)\u00a0that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image030.gif\" width=\"29\" height=\"14\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is decreasing in the open range\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image033.gif\" width=\"45\" height=\"14\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>(B) The local extrema (the local maximum and minimum) are\u00a0located at the points at which\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image036.gif\" width=\"58\" height=\"14\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0which are the points\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image039.gif\" width=\"41\" height=\"11\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0In the interval around\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image042.gif\" width=\"40\" height=\"11\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image045.gif\" width=\"33\" height=\"14\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0changes from positive to negative, which means\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image048.gif\" width=\"40\" height=\"11\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is a local maximum. In the interval around\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image051.gif\" width=\"31\" height=\"11\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image054.gif\" width=\"33\" height=\"14\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0changes from negative to positive, so\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image057.gif\" width=\"31\" height=\"11\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is a local minimum.<\/p>\n<p>A graph of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image060.gif\" width=\"29\" height=\"14\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0in the interval\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p11_html_m4a72678c.gif\" width=\"45\" height=\"27\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is depicted below.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20006.JPG\" alt=\"Graph of the interval [5, -5]\" width=\"266\" height=\"250\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>We can see the local maximum and minimum and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image066.gif\" width=\"40\" height=\"11\" name=\"graphics25\" align=\"BOTTOM\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image069.gif\" width=\"31\" height=\"11\" name=\"graphics26\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0respectively. However, it is clear that the local maximum is not\u00a0the largest value of the function in this interval, and the local\u00a0minimum is not the smallest value. In the interval\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/s3_p11_html_m4a72678c.gif\" width=\"45\" height=\"27\" name=\"graphics27\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0the <abbr title=\" a point MathTypewhere MathTypefor every x in the domain of the function MathType \">absolute\u00a0maximum<\/abbr> is at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image075.gif\" width=\"31\" height=\"11\" name=\"graphics28\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0and the <abbr title=\"\u2014 a point MathType where MathType for every x in the domain of the function MathType \">absolute\u00a0minimum<\/abbr> is at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p20_clip_image078.gif\" width=\"40\" height=\"11\" name=\"graphics29\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0The first derivative test only identifies relative extrema, that\u00a0is, the points at which the function shifts direction. Additional\u00a0analysis is required to determine absolute extrema.<\/p>\n<h3>What are first and second derivative tests?<\/h3>\n<p>As we have learned, the first derivative of a function can be\u00a0used to determine concavity (i.e. over what interval the function\u00a0increases and decreases), and the location of relative extrema. We\u00a0will call this analysis the <abbr title=\"f '(x) or the slope of a line tangent to a graph. \">first\u00a0derivative test<\/abbr>. We will now show that the <abbr title=\" the derivative of a first derivative \">second\u00a0derivative test<\/abbr> can be used to determine relative\u00a0extrema and more.<\/p>\n<p>Let\u2019s look again at our two parabolas,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image003.gif\" width=\"115\" height=\"17\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image006.gif\" width=\"123\" height=\"17\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We have already computed their derivatives,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image009.gif\" width=\"90\" height=\"14\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image012.gif\" width=\"97\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Below,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image015.gif\" width=\"29\" height=\"14\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image018.gif\" width=\"28\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0are sketched again.<\/p>\n<table>\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20007.JPG\" alt=\" f(x)\" width=\"266\" height=\"250\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20008.JPG\" width=\"266\" height=\"250\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image021.gif\" width=\"29\" height=\"14\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image024.gif\" width=\"28\" height=\"14\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Look carefully at their second derivatives,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image027.gif\" width=\"62\" height=\"14\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image030.gif\" width=\"69\" height=\"14\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Note that the second derivative is always positive for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image033.gif\" width=\"10\" height=\"14\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and always negative for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image036.gif\" width=\"8\" height=\"10\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This is illustrated below by graphs of the derivatives for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image039.gif\" width=\"10\" height=\"14\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image042.gif\" width=\"8\" height=\"10\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>:<\/p>\n<table>\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20009.JPG\" alt=\" f'(x)\" width=\"266\" height=\"250\" name=\"graphics19\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20010.JPG\" alt=\"g'(x)\" width=\"266\" height=\"250\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image045.gif\" width=\"33\" height=\"14\" name=\"graphics21\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image048.gif\" width=\"32\" height=\"14\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The second derivative is nothing more than the slope of the\u00a0first derivatives. In the vicinity of a local minimum, the slope\u00a0of the derivative (i.e. the second derivative) is positive; in the\u00a0vicinity of a local maximum, the slope of the derivative (i.e. the\u00a0second derivative) is negative. The same principle holds for any\u00a0differentiable function where the second derivative exists.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>If <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image051.gif\" width=\"10\" height=\"14\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/> is differentiable\u00a0on an open interval around <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image054.gif\" width=\"32\" height=\"7\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image057.gif\" width=\"37\" height=\"14\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/> exists, then:<\/p>\n<p class=\"notebox_text\" align=\"LEFT\">(i) <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image060.gif\" width=\"29\" height=\"14\" name=\"graphics26\" align=\"ABSMIDDLE\" border=\"0\" \/> is a local\u00a0minimum (and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image063.gif\" width=\"10\" height=\"14\" name=\"graphics27\" align=\"ABSMIDDLE\" border=\"0\" \/> is concave up) if <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image066.gif\" width=\"58\" height=\"14\" name=\"graphics28\" align=\"ABSMIDDLE\" border=\"0\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image069.gif\" width=\"61\" height=\"14\" name=\"graphics29\" align=\"ABSMIDDLE\" border=\"0\" \/>;<\/p>\n<p class=\"notebox_text\" align=\"LEFT\">(ii) <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image072.gif\" width=\"29\" height=\"14\" name=\"graphics30\" align=\"ABSMIDDLE\" border=\"0\" \/> is a local\u00a0maximum (and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image075.gif\" width=\"10\" height=\"14\" name=\"graphics31\" align=\"ABSMIDDLE\" border=\"0\" \/> is concave down) if <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image078.gif\" width=\"58\" height=\"14\" name=\"graphics32\" align=\"ABSMIDDLE\" border=\"0\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image081.gif\" width=\"61\" height=\"14\" name=\"graphics33\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<p>At this point, you may be wondering what happens when\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image084.gif\" width=\"62\" height=\"14\" name=\"graphics34\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Let\u2019s look at two examples of this situation, <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image087.gif\" width=\"59\" height=\"17\" name=\"graphics35\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image090.gif\" width=\"58\" height=\"17\" name=\"graphics36\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<table>\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20011.JPG\" alt=\" f(x)=x3\" width=\"266\" height=\"250\" name=\"graphics37\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20012.JPG\" alt=\" f(x)=x4\" width=\"266\" height=\"250\" name=\"graphics38\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image093.gif\" width=\"29\" height=\"14\" name=\"graphics39\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p21_clip_image096.gif\" width=\"28\" height=\"14\" name=\"graphics40\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>What is a point of inflection?<\/h3>\n<p>It is easy to show\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image003.gif\" width=\"61\" height=\"14\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image006.gif\" width=\"58\" height=\"14\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0However, the point\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image009.gif\" width=\"30\" height=\"11\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is neither a maximum nor a minimum for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image012.gif\" width=\"30\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and it is actually a relative minimum for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image015.gif\" width=\"28\" height=\"14\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0This tells us that the second derivative test does not always\u00a0work. If the second derivative is zero, you may need to rely on\u00a0the first derivative test to determine relative extrema and\u00a0concavity.<\/p>\n<p>For\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image017.gif\" width=\"30\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0the point at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image019.gif\" width=\"30\" height=\"11\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is known as a <abbr title=\"any point along a curve at which the concavity changes directions from down to up or from up to down \">point\u00a0of inflection<\/abbr>. A point of inflection is any point along a\u00a0curve at which the concavity changes directions from down to up or\u00a0from up to down. In other words, the point\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image022.gif\" width=\"29\" height=\"8\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is a point of inflection if\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image025.gif\" width=\"61\" height=\"14\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0when\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image028.gif\" width=\"29\" height=\"8\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image031.gif\" width=\"61\" height=\"14\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0if\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image034.gif\" width=\"29\" height=\"8\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0or if\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image036.gif\" width=\"61\" height=\"14\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0when\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image038.gif\" width=\"29\" height=\"9\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image040.gif\" width=\"61\" height=\"14\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0if\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image042.gif\" width=\"29\" height=\"9\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0The function does not have to be differentiable at this point,\u00a0however. For example, we have seen that the function\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image045.gif\" width=\"67\" height=\"18\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is not differentiable at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image047.gif\" width=\"32\" height=\"13\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/>. However, you can prove that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image049.gif\" width=\"30\" height=\"14\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0satisfies the condition for a point of inflection at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image051.gif\" width=\"32\" height=\"13\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p>With what we know about maxima and minima, we are able to use\u00a0the first and second derivative tests to solve\u00a0<abbr title=\" the mathematical procedures involved in making something as fully perfect, functional, or effective as possible or maximizing or minimizing the ouput of the function or situation. \">optimization<\/abbr> <span style=\"text-decoration: none;\">pro<\/span>blems.\u00a0We will demonstrate this with a couple of classic examples.<\/p>\n<p>A rectangular field which runs to the bank of a straight river\u00a0needs to be fenced off. There is a total of 2,000 yards of fencing\u00a0material that can be used. Assuming the river forms one side, what\u00a0is the maximum area that can be fenced in? Assume the field will\u00a0have a length\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image057.gif\" width=\"9\" height=\"9\" name=\"graphics23\" align=\"BOTTOM\" border=\"0\" \/>\u00a0and width\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image060.gif\" width=\"11\" height=\"12\" name=\"graphics24\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0The total area is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image063.gif\" width=\"41\" height=\"14\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Let\u2019s assume the river runs along the width. That means that\u00a0the total amount of fencing material used is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image066.gif\" width=\"113\" height=\"14\" name=\"graphics26\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0yards. Solving for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image069.gif\" width=\"107\" height=\"14\" name=\"graphics27\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We then substitute\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image072.gif\" width=\"9\" height=\"10\" name=\"graphics28\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0into the equation of the area, and the result is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image075.gif\" width=\"219\" height=\"17\" name=\"graphics29\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0An extreme point occurs where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image078.gif\" width=\"57\" height=\"14\" name=\"graphics30\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Therefore, <em>A<\/em>&#8216;(<em>x<\/em>) = 2,000 \u2013 4<em>x<\/em> = 0 or <em>x\u00a0<\/em>= 500 yards, <em>y<\/em> = 2,000 \u2013 1,000 = 1,000 yards, and the\u00a0area is<\/p>\n<p align=\"CENTER\"><em>A<\/em> = (500)(1,000) = 500,000 square yards.<\/p>\n<p>Is this the maximum or minimum area? The answer can be found by\u00a0computing\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image090.gif\" width=\"36\" height=\"14\" name=\"graphics31\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image093.gif\" width=\"92\" height=\"14\" name=\"graphics32\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Since the second derivative is negative, this area is the\u00a0maximum.<\/p>\n<p>Let\u2019s try ano<span style=\"text-decoration: none;\">ther\u00a0example. What is the area of the largest rectangle that can be\u00a0inscribed within the parabola\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image096.gif\" width=\"60\" height=\"17\" name=\"graphics33\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0as illustrated below? The <\/span><em><span style=\"text-decoration: none;\">x-<\/span><\/em><span style=\"text-decoration: none;\">axis\u00a0defines the length of one side, and two corners lie on the\u00a0parabola. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20013.JPG\" alt=\"Parabola y=4-x2\" width=\"266\" height=\"250\" name=\"graphics34\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">Assume the corners of the\u00a0rectangle on the <\/span><em><span style=\"text-decoration: none;\">x-<\/span><\/em><span style=\"text-decoration: none;\">axis\u00a0is at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image099.gif\" width=\"35\" height=\"11\" name=\"graphics35\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image102.gif\" width=\"45\" height=\"11\" name=\"graphics36\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Therefore, the rectangle touches the parabola at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image105.gif\" width=\"65\" height=\"18\" name=\"graphics37\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0which is also the length of the rectangle. Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image108.gif\" width=\"188\" height=\"18\" name=\"graphics38\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0The extreme value for <\/span><em><span style=\"text-decoration: none;\">A\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image111.gif\" width=\"96\" height=\"18\" name=\"graphics39\" align=\"TEXTTOP\" border=\"0\" \/>,\u00a0or\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image114.gif\" width=\"101\" height=\"41\" name=\"graphics40\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0(we defined\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image117.gif\" width=\"11\" height=\"11\" name=\"graphics41\" align=\"BOTTOM\" border=\"0\" \/>\u00a0as positive). Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image120.gif\" width=\"83\" height=\"34\" name=\"graphics42\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image123.gif\" width=\"61\" height=\"11\" name=\"graphics43\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Since\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image126.gif\" width=\"94\" height=\"15\" name=\"graphics44\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p22_clip_image129.gif\" width=\"36\" height=\"15\" name=\"graphics45\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we know this area is the maximum.<\/span><\/p>\n<h3>What is Newton\u2019s method for approximating the roots of\u00a0function?<\/h3>\n<p>Another interesting application of derivatives is <strong>Newton\u2019s\u00a0method for approximating the roots of a function.<\/strong> The\u00a0<em>roots<\/em> of a function are those points where the function\u00a0crosses the <em>x<\/em>-axis, or where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image003.gif\" width=\"32\" height=\"14\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0They are also known as the <em>zeros<\/em> of a function. As an\u00a0example, the function\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image006.gif\" width=\"127\" height=\"17\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0has four roots:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image009.gif\" width=\"122\" height=\"13\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Below, we plot\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image012.gif\" width=\"30\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0in a range where all its roots are visible, as well as the<br \/>\nvicinity of the root\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image015.gif\" width=\"28\" height=\"11\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<table>\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20Mod%208.3%20Art%20014.JPG\" alt=\"graph of the function f(x) =x4-10x+9\" width=\"266\" height=\"250\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/Math%20%20Mod%208.3%20Art%20015.JPG\" alt=\"vicinity of the root x=1\" width=\"266\" height=\"250\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the second graph, we also include the tangent line at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image018.gif\" width=\"42\" height=\"11\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Let\u2019s say\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image020.gif\" width=\"42\" height=\"11\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is our initial guess of the root around\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image023.gif\" width=\"30\" height=\"11\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0That initial guess is not accurate, but notice that the tangent\u00a0line intersects the <em>x<\/em>-axis at a point much closer to the\u00a0root (at about\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image029.gif\" width=\"50\" height=\"11\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/>).\u00a0Now, suppose you take the tangent line at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image032.gif\" width=\"50\" height=\"11\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0That tangent line will intersect the <em>x<\/em>-axis at a point even\u00a0closer to the root (at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image037.gif\" width=\"58\" height=\"11\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/>).\u00a0After a few iterations, you will come very close to the actual\u00a0root.<\/p>\n<p>Given\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image039.gif\" width=\"30\" height=\"14\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0try\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image042.gif\" width=\"11\" height=\"11\" name=\"graphics17\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0as our initial guess for a root. The tangent line hits the curve\u00a0at the point\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image045.gif\" width=\"61\" height=\"16\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0, has a slope\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image048.gif\" width=\"67\" height=\"15\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and we\u2019ll say it crosses the <em>x<\/em>-axis at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image053.gif\" width=\"35\" height=\"15\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0With these two points and the slope, we can solve for\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image056.gif\" width=\"10\" height=\"11\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image059.gif\" width=\"180\" height=\"38\" name=\"graphics22\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Therefore,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image062.gif\" width=\"231\" height=\"38\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0With\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image064.gif\" width=\"10\" height=\"11\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/>\u00a0as our new guess, we can repeat the process.<\/p>\n<div class=\"callout\">\n<h4>Important Tidbit<\/h4>\n<p>It should be clear that, for the <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image067.gif\" width=\"45\" height=\"16\" name=\"graphics25\" align=\"ABSMIDDLE\" border=\"0\" \/> iteration:<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image070.gif\" width=\"110\" height=\"38\" name=\"graphics26\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<p>Let\u2019s try an example. Assuming an initial guess of <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image073.gif\" width=\"4\" height=\"11\" name=\"graphics27\" align=\"BOTTOM\" border=\"0\" \/>,\u00a0use Newton\u2019s method to approximate <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image076.gif\" width=\"18\" height=\"18\" name=\"graphics28\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>The value\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image079.gif\" width=\"43\" height=\"18\" name=\"graphics29\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is the positive root of the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image082.gif\" width=\"83\" height=\"17\" name=\"graphics30\" align=\"BOTTOM\" border=\"0\" \/>.\u00a0Since\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image085.gif\" width=\"67\" height=\"14\" name=\"graphics31\" align=\"BOTTOM\" border=\"0\" \/>, we have\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image088.gif\" width=\"110\" height=\"41\" name=\"graphics32\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0We start with\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image091.gif\" width=\"34\" height=\"15\" name=\"graphics33\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0You may want to use a calculator for these calculations.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image094.gif\" width=\"219\" height=\"177\" name=\"graphics34\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The value\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image097.gif\" width=\"11\" height=\"11\" name=\"graphics35\" align=\"BOTTOM\" border=\"0\" \/>\u00a0is as close to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p23_clip_image099.gif\" width=\"18\" height=\"18\" name=\"graphics36\" align=\"BOTTOM\" border=\"0\" \/>\u00a0as a 10-digit calculator can calculate.<\/p>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>The <em><strong>derivative\u00a0<\/strong><\/em>of a function\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p24_clip_image003.gif\" width=\"30\" height=\"14\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is expressed as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p24_clip_image006.gif\" width=\"187\" height=\"34\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The derivative\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p24_clip_image009.gif\" width=\"39\" height=\"15\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0can be interpreted as the <strong><em>slope of the tangent line\u00a0<\/em><\/strong>of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p24_clip_image011.gif\" width=\"30\" height=\"14\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p24_clip_image014.gif\" width=\"35\" height=\"11\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0or the <strong><em>instantaneous rate of change<\/em><\/strong> of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p24_clip_image016.gif\" width=\"30\" height=\"14\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0at\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p24_clip_image019.gif\" width=\"11\" height=\"11\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <em><strong>first\u00a0derivative<\/strong><\/em> is useful in calculations involving the\u00a0<strong><em>relative <\/em><\/strong><em><strong>maxima\u00a0<\/strong><\/em>and <em><strong>minima<\/strong><\/em> of functions, the <strong><em>intervals\u00a0of increase and decrease<\/em><\/strong> of a function, the <em><strong>roots\u00a0<\/strong><\/em>of functions, <strong><em>optimization <\/em><\/strong>problems,\u00a0and <em><strong>related rate <\/strong><\/em>problems.<\/li>\n<li>The <strong><em>second\u00a0derivative<\/em><\/strong> of a function is helpful in determining\u00a0<strong><em>concavity<\/em>,<\/strong> the type of <em><strong>relative\u00a0extrema<\/strong><\/em> within an interval, and <em><strong>inflection\u00a0points<\/strong><\/em>.<\/li>\n<li>The <em><strong>chain rule<\/strong>,\u00a0<strong>implicit differentiation<\/strong><\/em>, and\u00a0<strong><em>parametric differentiation<\/em><\/strong> are\u00a0important principles in finding derivatives of <em><strong>composite\u00a0functions<\/strong><\/em> and <em><strong>implicit equations<\/strong><\/em>.<\/li>\n<li><strong><em>L\u2019H\u00f4pital\u2019s rule\u00a0<\/em><\/strong>is used to determine the limits of functions of the indeterminate\u00a0form\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/8\/images\/s2_p24_clip_image022.gif\" width=\"49\" height=\"34\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/differential-calculus\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/calculus\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/integral-calculus\">Next\u00a0Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next\u00a0Lesson \u27a1 Differential Calculus II Objective In this lesson, you will cover the definition of a derivative and apply it to the evaluation of slopes and tangent\u00a0lines of curves, instantaneous rates of changes of functions, relative maxima and minima, and methods to compute the\u00a0derivative of several types of functions. Previously Covered: The [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-334","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/334","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=334"}],"version-history":[{"count":24,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/334\/revisions"}],"predecessor-version":[{"id":830,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/334\/revisions\/830"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=334"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}