{"id":10708,"date":"2017-08-25t21:19:01","date_gmt":"2017-08-26t04:19:01","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=10708"},"modified":"2017-08-25t21:19:01","modified_gmt":"2017-08-26t04:19:01","slug":"ap-calculus-review-exponential-functions","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-exponential-functions\/","title":{"rendered":"ap calculus review: exponential functions"},"content":{"rendered":"<p>exponential functions play a key role in a wide array of applications including population growth.  these important functions show up on both the ap calculus ab and bc exams.  so here&#8217;s what you should know about them for the test.<\/p>\n<h2>exponential functions &#8212; definitions<\/h2>\n<p>an <strong>exponential function<\/strong> is one that involves a constant positive base to a variable exponent.  the most basic exponential is: <em>f<\/em>(<em>x<\/em>) = <em>a<sup>x<\/sup><\/em>, where <em>a<\/em> &gt; 0 is a constant.<\/p>\n<p>other variations include coefficients that scale the graph horizontally or vertically.  adding or subtracting a constant shifts the graph up or down.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_general1.gif\" alt=\"y = b a^(kx) + c, general exponential\" width=\"109\" height=\"20\" class=\"aligncenter size-full wp-image-10797\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_general1.gif 109w, \/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_general1-30x6.gif 30w\" sizes=\"(max-width: 109px) 100vw, 109px\" \/><\/p>\n<p>furthermore, there is a certain constant called <em>e<\/em> (<strong>euler&#8217;s constant<\/strong>) that is so useful as a base that we call <em>e<sup>x<\/sup><\/em> the <strong>natural exponential<\/strong> function.  <\/p>\n<figure id=\"attachment_8335\" aria-describedby=\"caption-attachment-8335\" style=\"width: 458px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/exponential.jpg\" alt=\"exponential functions\" width=\"458\" height=\"408\" class=\"size-full wp-image-8335\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2016\/12\/exponential.jpg 458w, \/\/www.catharsisit.com\/hs\/files\/2016\/12\/exponential-300x267.jpg 300w\" sizes=\"(max-width: 458px) 100vw, 458px\" \/><figcaption id=\"caption-attachment-8335\" class=\"wp-caption-text\">an exponential function.  here, the base is <em>e<\/em>, and the graph has been shifted down by 4 units.<\/figcaption><\/figure>\n<h3>facts and properties<\/h3>\n<p>the exponential function <em>f<\/em>(<em>x<\/em>) = <em>a<sup>x<\/sup><\/em> satisfies the following properties.<\/p>\n<ul>\n<li>the domain is all real numbers, (-&infin;, &infin;).<\/li>\n<li>the range is all positive real numbers, (0, &infin;).<\/li>\n<li>the <em>y<\/em>-intercept is (0, 1), but there is no <em>x<\/em>-intercept.\n<li>if <em>a<\/em> &gt; 1, then <em>f<\/em> increases on its domain.<\/li>\n<li>if 0 &lt; <em>a<\/em> &lt; 1, then <em>f<\/em> decreases on its domain.<\/li>\n<li>the graph is concave up on its entire domain.<\/li>\n<li>the line <em>y<\/em> = 0 is a <a href=\"\/\/www.catharsisit.com\/hs\/ap\/find-horizontal-asymptotes\/\">horizontal asymptote<\/a>.<\/li>\n<\/ul>\n<h2>differentiating exponential functions<\/h2>\n<p>the natural exponential has the remarkable property that it is its own derivative.  that property alone is what makes the function <em>e<sup>x<\/sup><\/em> so essential to almost all branches of science.<\/p>\n<p>when the base is something other than <em>e<\/em>, then the derivative formula involves a multiple of the <em>natural logarithm<\/em> of the base.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential-derivatives.gif\" alt=\"exponential derivatives\" width=\"157\" height=\"169\" class=\"aligncenter size-full wp-image-10795\" \/><\/p>\n<p>check out the following article for more details about logarithms and exponent properties: <a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-properties-exponents-logarithms\/\">ap calculus review: properties of exponents and logarithms<\/a>.<\/p>\n<h4>example 1<\/h4>\n<p>find the absolute minimum and maximum values of <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_example_function.gif\" alt=\"example exponential function\" width=\"104\" height=\"21\" class=\"alignnone size-full wp-image-10804\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_example_function.gif 104w, \/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_example_function-30x6.gif 30w\" sizes=\"(max-width: 104px) 100vw, 104px\" \/><\/p>\n<h4>solution<\/h4>\n<p>the function <em>g<\/em>(<em>x<\/em>) is technically not an exponential function, however it does <em>involve<\/em> an exponential.<\/p>\n<p>let&#8217;s take the first derivative.  don&#8217;t forget product rule!<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_example_function_derivative.gif\" alt=\"derivative of example\" width=\"241\" height=\"55\" class=\"aligncenter size-full wp-image-10805\" \/><\/p>\n<p>find the critical numbers by setting <em>g<\/em>&nbsp;&#039; equal to 0 and solving.  note that because the exponential function itself can never be 0, the only factor that matters here is 1 &#8211; 6<em>x<\/em><sup>2<\/sup>.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/part_of_solution.gif\" alt=\"solving for x = 1\/sqrt(6) and -1\/sqrt(60\" width=\"215\" height=\"114\" class=\"aligncenter size-full wp-image-10806\" \/><\/p>\n<p>finally, plug the two critical numbers into the original function to determine minimum and maximum.  a sketch of the graph also helps to prove that the absolute min and max do occur at those points.<\/p>\n<ul>\n<li><em>g<\/em>(-0.4082) = -0.2476 (minimum value)<\/li>\n<li><em>g<\/em>(0.4082) = 0.2476 (maximum value)<\/li>\n<\/ul>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_example_graph.png\" alt=\"graph of example exponential.\" width=\"300\" height=\"300\" class=\"aligncenter size-full wp-image-10807\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_example_graph.png 300w, \/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_example_graph-150x150.png 150w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2>integrating exponential functions<\/h2>\n<p>because the derivative of <em>e<sup>x<\/sup><\/em> is equal to <em>e<sup>x<\/sup><\/em>, you can expect its antiderivative to be the exact same thing.  (but don&#8217;t forget the &#8220;+ <em>c<\/em>&#8220;!)<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/01\/exponential_antiderivatives.gif\" alt=\"exponential antiderivatives\" width=\"397\" height=\"88\" class=\"aligncenter size-full wp-image-8740\" \/><\/p>\n<h4>example 2<\/h4>\n<p>let <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/e_2x.gif\" alt=\"e to the 2x\" width=\"80\" height=\"20\" class=\"alignnone size-full wp-image-10808\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/e_2x.gif 80w, \/\/www.catharsisit.com\/hs\/files\/2017\/07\/e_2x-30x8.gif 30w\" sizes=\"(max-width: 80px) 100vw, 80px\" \/>.  find the value of <em>c<\/em> guaranteed by the mean value theorem for integrals so that <em>f<\/em>(<em>c<\/em>) is equal to the average value of <em>f<\/em> on the interval [1, 3].<\/p>\n<h4>solution<\/h4>\n<p>we must first find the average value of the function on the given interval.  this is a job for an integral.  note that there will be a substitution: <em>u<\/em> = 2<em>x<\/em>.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/average_value_exponential.gif\" alt=\"computing the average value of an exponential function\" width=\"281\" height=\"182\" class=\"aligncenter size-full wp-image-10809\" \/><\/p>\n<p>next, set <em>f<\/em>(<em>c<\/em>) = 99.01 and solve for <em>c<\/em>.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/solution_example_2_mvti.gif\" alt=\"solution to mean value theorem for integrals example\" width=\"196\" height=\"93\" class=\"aligncenter size-full wp-image-10810\" \/><\/p>\n<h2>related topics<\/h2>\n<p>the exponential also shows up in a number of applications on the ap calculus exams.  <\/p>\n<h3>exponential growth models<\/h3>\n<p>any situation in which the rate of growth is proportional to the amount present lends itself directly to an exponential model.<\/p>\n<p>the differential equation <em>y<\/em>&nbsp;&#039; = <em>ky<\/em>, where <em>k<\/em> is a constant, has the general solution, <em>y<\/em> = <em>ae<sup>kx<\/sup><\/em>.  here, the value of the constant <em>a<\/em> is equal to the <em>initial population<\/em>, <em>y<\/em>(0).<\/p>\n<h4>example 3<\/h4>\n<p>a certain bacteria culture has 1000 cells initially.  <\/p>\n<p>the amount <em>x<\/em>, in grams, of a radioactive isotope decays according to the equation <em>dx<\/em>\/<em>dt<\/em> = <em>-0.03x<\/em>, where time <em>t<\/em> is measured in days.  determine the <em>half-life<\/em> of the isotope to the nearest day. (note: the <strong>half-life<\/strong> of a substance is the amount of time it takes for exactly one-half of the substance to decay.)<\/p>\n<h4>solution<\/h4>\n<p>first, we can build the exponential model (<em>ae<sup>kx<\/sup><\/em>) based on the given information.  here, the initial amount is <em>a<\/em> = 1000, and the decay constant is <em>k<\/em> = -0.03.  so, the appropriate exponential model would be:<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_model.gif\" alt=\"x = 1000 e^(-0.03t)\" width=\"142\" height=\"20\" class=\"aligncenter size-full wp-image-10811\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_model.gif 142w, \/\/www.catharsisit.com\/hs\/files\/2017\/07\/exponential_model-30x4.gif 30w\" sizes=\"(max-width: 142px) 100vw, 142px\" \/><\/p>\n<p>next, to find the half-life, we set <em>x<\/em>(<em>t<\/em>) = 500 (because that&#8217;s half of 1000), and solve for <em>t<\/em>.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/solution_example_3.gif\" alt=\"solution to half-life problem\" width=\"244\" height=\"151\" class=\"aligncenter size-full wp-image-10812\" \/><\/p>\n<p>thus the half-life is about 21 days.<\/p>\n<h3>the logistics growth model<\/h3>\n<p>the <strong>logistics growth model<\/strong> is an equation the models population growth up to a constant <strong>carrying capacity<\/strong> <em>m<\/em>.<\/p>\n<figure id=\"attachment_10612\" aria-describedby=\"caption-attachment-10612\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/06\/logistics_plot.png\" alt=\"logistics curve\" width=\"300\" height=\"300\" class=\"size-full wp-image-10612\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/06\/logistics_plot.png 300w, \/\/www.catharsisit.com\/hs\/files\/2017\/06\/logistics_plot-150x150.png 150w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-10612\" class=\"wp-caption-text\">the logistics growth model shows how populations might grow under limited resources.<\/figcaption><\/figure>\n<p>the natural exponential <em>e<sup>x<\/sup><\/em> makes an appearance as part of the solution to  the logistics differential equation.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/06\/logistics_solution.gif\" alt=\"logistics function\" width=\"119\" height=\"39\" class=\"aligncenter size-full wp-image-10619\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/06\/logistics_solution.gif 119w, \/\/www.catharsisit.com\/hs\/files\/2017\/06\/logistics_solution-30x10.gif 30w\" sizes=\"(max-width: 119px) 100vw, 119px\" \/><\/p>\n<p>for more information and examples, check out: <a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-logistics-growth-model\/\">ap calculus bc review: logistics growth model<\/a>.<\/p>\n<h3>the maclaurin series for <em>e<sup>x<\/sup><\/em><\/h3>\n<p>taylor and maclaurin series only show up on the ap calculus bc exam.  you should probably memorize the <strong>maclaurin series<\/strong> for the natural exponential function.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/maclaurin-exponential.gif\" alt=\"maclaurin series for e^x\" width=\"330\" height=\"50\" class=\"aligncenter size-full wp-image-10801\" \/><\/p>\n<h4>example 4<\/h4>\n<p>find a power series expansion for <img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/x_e_x_sq.gif\" alt=\"x times e to the x-squared\" width=\"30\" height=\"17\" class=\"alignnone size-full wp-image-10802\" \/>.<\/p>\n<h4>solution<\/h4>\n<p>we just have to work with the maclaurin series for the natural exponential.  i&#8217;ve color coded certain parts to make it a little easier to follow.<\/p>\n<p><img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/maclaurin-example.gif\" alt=\"solution to maclaurin series example \" width=\"278\" height=\"216\" class=\"aligncenter size-full wp-image-10803\" \/><\/p>\n<h2>summary<\/h2>\n<ul>\n<li>exponential functions have the basic form <em>f<\/em>(<em>x<\/em>) = <em>a<sup>x<\/sup><\/em>, where <em>a<\/em> &gt; 0 is a constant.  but the graph can be scaled or shifted by including appropriate coefficients and constant terms.<\/li>\n<li>the derivative of <em>e<sup>x<\/sup><\/em> is equal to <em>e<sup>x<\/sup><\/em>.  the derivative of <em>a<sup>x<\/sup><\/em> is <em>a<sup>x<\/sup><\/em> ln <em>a<\/em>.<\/li>\n<li>the antiderivative (integral) of <em>e<sup>x<\/sup><\/em> is equal to <em>e<sup>x<\/sup><\/em> + <em>c<\/em>.  the antiderivative of <em>a<sup>x<\/sup><\/em> is <em>a<sup>x<\/sup><\/em> \/ (ln <em>a<\/em>) + <em>c<\/em>.<\/li>\n<li>the general solution of the differential equation <em>y<\/em>&nbsp;&#039; = <em>ky<\/em> is the exponential function, <em>y<\/em> = <em>ae<sup>kx<\/sup><\/em>. <\/li>\n<li>the solution to the logistics growth model equation also involves exponentials.<\/li>\n<li>the maclaurin series (power series) of the natural exponential function is:<br \/>\n<img decoding=\"async\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/07\/maclaurin-exponential.gif\" alt=\"maclaurin series for e^x\" width=\"330\" height=\"50\" class=\"aligncenter size-full wp-image-10801\" \/>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>exponential functions show up on both the ap calculus ab and bc exams. here&#8217;s what you should know about them for the test!<\/p>\n","protected":false},"author":223,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[240],"tags":[241],"ppma_author":[24932],"acf":[],"yoast_head":"<!-- this site is optimized with the yoast seo premium plugin v21.7 (yoast seo v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>ap calculus 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