{"id":10708,"date":"2017-08-25T21:19:01","date_gmt":"2017-08-26T04:19:01","guid":{"rendered":"https:\/\/magoosh.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":"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.com\/hs\/files\/2017\/07\/exponential_general1.gif 109w, https:\/\/magoosh.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=\"https:\/\/magoosh.com\/hs\/files\/2016\/12\/exponential.jpg\" alt=\"Exponential functions\" width=\"458\" height=\"408\" class=\"size-full wp-image-8335\" srcset=\"https:\/\/magoosh.com\/hs\/files\/2016\/12\/exponential.jpg 458w, https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.com\/hs\/files\/2017\/07\/exponential_example_function.gif 104w, https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.com\/hs\/files\/2017\/07\/exponential_example_graph.png 300w, https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.com\/hs\/files\/2017\/07\/e_2x.gif 80w, https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.com\/hs\/files\/2017\/07\/exponential_model.gif 142w, https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.com\/hs\/files\/2017\/06\/logistics_plot.png\" alt=\"Logistics curve\" width=\"300\" height=\"300\" class=\"size-full wp-image-10612\" srcset=\"https:\/\/magoosh.com\/hs\/files\/2017\/06\/logistics_plot.png 300w, https:\/\/magoosh.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=\"https:\/\/magoosh.com\/hs\/files\/2017\/06\/Logistics_solution.gif\" alt=\"Logistics function\" width=\"119\" height=\"39\" class=\"aligncenter size-full wp-image-10619\" srcset=\"https:\/\/magoosh.com\/hs\/files\/2017\/06\/Logistics_solution.gif 119w, https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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=\"https:\/\/magoosh.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],"class_list":["post-10708","post","type-post","status-publish","format-standard","hentry","category-ap","tag-ap-calculus"],"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 Review: Exponential Functions - Magoosh Blog | High School<\/title>\n<meta name=\"description\" content=\"Exponential functions show up on both the AP Calculus AB and BC exams. Here&#039;s what you should know about them for the test!\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"AP Calculus Review: Exponential Functions\" \/>\n<meta property=\"og:description\" content=\"Exponential functions show up on both the AP Calculus AB and BC exams. Here&#039;s what you should know about them for the test!\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog | High School\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/MagooshSat\/\" \/>\n<meta property=\"article:published_time\" content=\"2017-08-26T04:19:01+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/magoosh.com\/hs\/files\/2017\/07\/exponential_general1.gif\" \/>\n<meta name=\"author\" content=\"Shaun Ault\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@ShaunAultMath\" \/>\n<meta name=\"twitter:site\" content=\"@MagooshSAT_ACT\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Shaun Ault\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/\"},\"author\":{\"name\":\"Shaun Ault\",\"@id\":\"https:\/\/magoosh.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f\"},\"headline\":\"AP Calculus Review: Exponential Functions\",\"datePublished\":\"2017-08-26T04:19:01+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/\"},\"wordCount\":944,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/hs\/#organization\"},\"keywords\":[\"AP Calculus\"],\"articleSection\":[\"AP\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/\",\"url\":\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/\",\"name\":\"AP Calculus Review: Exponential Functions - Magoosh Blog | High School\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/hs\/#website\"},\"datePublished\":\"2017-08-26T04:19:01+00:00\",\"description\":\"Exponential functions show up on both the AP Calculus AB and BC exams. Here's what you should know about them for the test!\",\"breadcrumb\":{\"@id\":\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/magoosh.com\/hs\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"AP Calculus Review: Exponential Functions\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/magoosh.com\/hs\/#website\",\"url\":\"https:\/\/magoosh.com\/hs\/\",\"name\":\"Magoosh Blog | High School\",\"description\":\"ACT, SAT, College Admissions, Life\",\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/hs\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/magoosh.com\/hs\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/magoosh.com\/hs\/#organization\",\"name\":\"Magoosh\",\"url\":\"https:\/\/magoosh.com\/hs\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/magoosh.com\/hs\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/magoosh.com\/hs\/files\/2019\/02\/Magoosh-logo-purple-60h.png\",\"contentUrl\":\"https:\/\/magoosh.com\/hs\/files\/2019\/02\/Magoosh-logo-purple-60h.png\",\"width\":265,\"height\":60,\"caption\":\"Magoosh\"},\"image\":{\"@id\":\"https:\/\/magoosh.com\/hs\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.facebook.com\/MagooshSat\/\",\"https:\/\/twitter.com\/MagooshSAT_ACT\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/magoosh.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f\",\"name\":\"Shaun Ault\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/magoosh.com\/hs\/#\/schema\/person\/image\/d3984d52deb82187299202f51fb828ce\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f10cdb687137bc0ad4e885404588101b7cd4aa01ae2be48abda61f14fa3715e2?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f10cdb687137bc0ad4e885404588101b7cd4aa01ae2be48abda61f14fa3715e2?s=96&d=mm&r=g\",\"caption\":\"Shaun Ault\"},\"description\":\"Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!\",\"sameAs\":[\"http:\/\/valdosta.academia.edu\/ShaunAult\",\"https:\/\/twitter.com\/ShaunAultMath\"],\"url\":\"https:\/\/magoosh.com\/hs\/author\/shaunault\/\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"AP Calculus Review: Exponential Functions - Magoosh Blog | High School","description":"Exponential functions show up on both the AP Calculus AB and BC exams. Here's what you should know about them for the test!","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/","og_locale":"en_US","og_type":"article","og_title":"AP Calculus Review: Exponential Functions","og_description":"Exponential functions show up on both the AP Calculus AB and BC exams. Here's what you should know about them for the test!","og_url":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/","og_site_name":"Magoosh Blog | High School","article_publisher":"https:\/\/www.facebook.com\/MagooshSat\/","article_published_time":"2017-08-26T04:19:01+00:00","og_image":[{"url":"https:\/\/magoosh.com\/hs\/files\/2017\/07\/exponential_general1.gif"}],"author":"Shaun Ault","twitter_card":"summary_large_image","twitter_creator":"@ShaunAultMath","twitter_site":"@MagooshSAT_ACT","twitter_misc":{"Written by":"Shaun Ault","Est. reading time":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/#article","isPartOf":{"@id":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/"},"author":{"name":"Shaun Ault","@id":"https:\/\/magoosh.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f"},"headline":"AP Calculus Review: Exponential Functions","datePublished":"2017-08-26T04:19:01+00:00","mainEntityOfPage":{"@id":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/"},"wordCount":944,"commentCount":0,"publisher":{"@id":"https:\/\/magoosh.com\/hs\/#organization"},"keywords":["AP Calculus"],"articleSection":["AP"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/","url":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/","name":"AP Calculus Review: Exponential Functions - Magoosh Blog | High School","isPartOf":{"@id":"https:\/\/magoosh.com\/hs\/#website"},"datePublished":"2017-08-26T04:19:01+00:00","description":"Exponential functions show up on both the AP Calculus AB and BC exams. Here's what you should know about them for the test!","breadcrumb":{"@id":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/magoosh.com\/hs\/ap\/ap-calculus-review-exponential-functions\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/magoosh.com\/hs\/"},{"@type":"ListItem","position":2,"name":"AP Calculus Review: Exponential Functions"}]},{"@type":"WebSite","@id":"https:\/\/magoosh.com\/hs\/#website","url":"https:\/\/magoosh.com\/hs\/","name":"Magoosh Blog | High School","description":"ACT, SAT, College Admissions, Life","publisher":{"@id":"https:\/\/magoosh.com\/hs\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/magoosh.com\/hs\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/magoosh.com\/hs\/#organization","name":"Magoosh","url":"https:\/\/magoosh.com\/hs\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/hs\/#\/schema\/logo\/image\/","url":"https:\/\/magoosh.com\/hs\/files\/2019\/02\/Magoosh-logo-purple-60h.png","contentUrl":"https:\/\/magoosh.com\/hs\/files\/2019\/02\/Magoosh-logo-purple-60h.png","width":265,"height":60,"caption":"Magoosh"},"image":{"@id":"https:\/\/magoosh.com\/hs\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/MagooshSat\/","https:\/\/twitter.com\/MagooshSAT_ACT"]},{"@type":"Person","@id":"https:\/\/magoosh.com\/hs\/#\/schema\/person\/f01e70874cef77d6f6392c12c43f6b6f","name":"Shaun Ault","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/hs\/#\/schema\/person\/image\/d3984d52deb82187299202f51fb828ce","url":"https:\/\/secure.gravatar.com\/avatar\/f10cdb687137bc0ad4e885404588101b7cd4aa01ae2be48abda61f14fa3715e2?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f10cdb687137bc0ad4e885404588101b7cd4aa01ae2be48abda61f14fa3715e2?s=96&d=mm&r=g","caption":"Shaun Ault"},"description":"Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition. Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass. Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!","sameAs":["http:\/\/valdosta.academia.edu\/ShaunAult","https:\/\/twitter.com\/ShaunAultMath"],"url":"https:\/\/magoosh.com\/hs\/author\/shaunault\/"}]}},"authors":[{"term_id":24932,"user_id":223,"is_guest":0,"slug":"shaunault","display_name":"Shaun Ault","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/f10cdb687137bc0ad4e885404588101b7cd4aa01ae2be48abda61f14fa3715e2?s=96&d=mm&r=g","user_url":"http:\/\/valdosta.academia.edu\/ShaunAult","last_name":"Ault","first_name":"Shaun","description":"Shaun earned his Ph. D. in mathematics from The Ohio State University in 2008 (Go Bucks!!). He received his BA in Mathematics with a minor in computer science from Oberlin College in 2002. In addition, Shaun earned a B. Mus. from the Oberlin Conservatory in the same year, with a major in music composition.  Shaun still loves music -- almost as much as math! -- and he (thinks he) can play piano, guitar, and bass.  Shaun has taught and tutored students in mathematics for about a decade, and hopes his experience can help you to succeed!"}],"_links":{"self":[{"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/posts\/10708","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/users\/223"}],"replies":[{"embeddable":true,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/comments?post=10708"}],"version-history":[{"count":0,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/posts\/10708\/revisions"}],"wp:attachment":[{"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/media?parent=10708"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/categories?post=10708"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/tags?post=10708"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/magoosh.com\/hs\/wp-json\/wp\/v2\/ppma_author?post=10708"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}