{"id":9399,"date":"2017-04-07t11:15:12","date_gmt":"2017-04-07t18:15:12","guid":{"rendered":"\/\/www.catharsisit.com\/hs\/?p=9399"},"modified":"2017-04-05t21:15:44","modified_gmt":"2017-04-06t04:15:44","slug":"ap-calculus-review-theorems-definitions","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-theorems-definitions\/","title":{"rendered":"ap calculus review: reasoning with theorems and definitions"},"content":{"rendered":"<p>many people believe that mathematics is about number-crunching, but much more importantly, math is about reasoning. for example, when you solve a word problem, you are using your reasoning skills to put together the given information in just the right way.<\/p>\n<p>in a way, ap calculus is all about reasoning. you have to interpret each problem and correctly apply the appropriate methods (limits, derivatives, integrals, etc.) to solve it. however sometimes we have to take it one step further and reason with theorems and definitions as well, gluing our thoughts together with mathematical <strong>logic<\/strong>.<\/p>\n<p>why is this important? well using nothing more than a handful of assumptions and plenty of definitions, theorems, and logic, euclid developed the entire subject of geometry from the ground up! if that&#8217;s not a reason to respect the power of definitions and theorems, then nothing else is.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter wp-image-9408 size-full\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/1_euclid.jpg\" alt=\"ap calculus review theorems\" width=\"512\" height=\"632\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/1_euclid.jpg 512w, \/\/www.catharsisit.com\/hs\/files\/2017\/03\/1_euclid-243x300.jpg 243w, \/\/www.catharsisit.com\/hs\/files\/2017\/03\/1_euclid-24x30.jpg 24w\" sizes=\"(max-width: 512px) 100vw, 512px\" \/><\/p>\n<h2>what are definitions and theorems?<\/h2>\n<p>in mathematics, every term must be defined in some way. a <strong>definition<\/strong> of a mathematical object is formal description of the essential properties that make that object what it is. for instance,<\/p>\n<ul>\n<li><em>definition:<\/em> a <strong>triangle<\/strong> is a three-sided polygon.<\/li>\n<\/ul>\n<p>it&#8217;s very important to understand the definitions of our mathematical terms so that we can employ just the right tool in each specific case. so if you see a three-sided polygon in a problem, then you know that it&#8217;s a triangle by definition. then you may use a property or formula related to triangles as part of your reasoning steps.<\/p>\n<p>we also rely on general statements of truth called <em>theorems<\/em> in order to reason about a specific situation. speaking of triangles, perhaps one of the most famous (and useful) theorems of all time is the <a href=\"https:\/\/www.mathsisfun.com\/pythagoras.html\" target=\"_blank\" rel=\"noopener noreferrer\">pythagorean theorem<\/a>. (by the way, this theorem shows up in book 1 of euclid&#8217;s elements, over 2000 years ago!)<\/p>\n<figure id=\"attachment_9404\" aria-describedby=\"caption-attachment-9404\" style=\"width: 460px\" class=\"wp-caption aligncenter\"><img decoding=\"async\" class=\" wp-image-9404\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/2000px-pythagoraas.svg_-600x792.png\" alt=\"diagram for pythagoras theorem\" width=\"460\" height=\"607\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/2000px-pythagoraas.svg_-600x792.png 600w, \/\/www.catharsisit.com\/hs\/files\/2017\/03\/2000px-pythagoraas.svg_-227x300.png 227w, \/\/www.catharsisit.com\/hs\/files\/2017\/03\/2000px-pythagoraas.svg_-768x1014.png 768w, \/\/www.catharsisit.com\/hs\/files\/2017\/03\/2000px-pythagoraas.svg_-1164x1536.png 1164w, \/\/www.catharsisit.com\/hs\/files\/2017\/03\/2000px-pythagoraas.svg_-1552x2048.png 1552w, \/\/www.catharsisit.com\/hs\/files\/2017\/03\/2000px-pythagoraas.svg_.png 2000w\" sizes=\"(max-width: 460px) 100vw, 460px\" \/><figcaption id=\"caption-attachment-9404\" class=\"wp-caption-text\">diagram for pythagoras theorem by <a href=\"https:\/\/commons.wikimedia.org\/wiki\/file:pythagoraas.svg\" target=\"_blank\" rel=\"noopener noreferrer\">drini (pedro sanchez)<\/a><\/figcaption><\/figure>\n<h2>reasoning with definitions<\/h2>\n<p>on the ap calculus exams, you must know and be able to apply the definitions of calculus. let&#8217;s see what that means in an example problem.<\/p>\n<h3>example 1<\/h3>\n<p>if <img decoding=\"async\" class=\"alignnone size-full wp-image-9409\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/piecewise_function.gif\" alt=\"piecewise function for example\" width=\"213\" height=\"55\" \/>, which of the following is true?<\/p>\n<p>(a) <em>f<\/em>(<em>x<\/em>) is continuous and differentiable at <em>x<\/em> = 3<\/p>\n<p>(b) <em>f<\/em>(<em>x<\/em>) is continuous but not differentiable at <em>x<\/em> = 3<\/p>\n<p>(c) <em>f<\/em>(<em>x<\/em>) is neither continuous nor differentiable at <em>x<\/em> = 3<\/p>\n<p>(d) <em>f<\/em>(<em>x<\/em>) is differentiable but not continuous at <em>x<\/em> = 3<\/p>\n<p>in order to properly address this question, we must know the definitions of continuous and differentiable.<\/p>\n<ul>\n<li>a function <em>f<\/em> is <strong>continuous<\/strong> at a point <em>x<\/em> = <em>a<\/em> if <img decoding=\"async\" class=\"alignnone size-full wp-image-8617\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/01\/limit_def_continuity.jpg\" alt=\"the limit as x approaches a is equal to f(a)\" width=\"138\" height=\"33\" \/><\/li>\n<li>a function <em>f<\/em> is <strong>differentiable<\/strong> at a point <em>x<\/em> = <em>a<\/em> if the derivative <em>f<\/em>\u00a0&#8216;(<em>a<\/em>) exists.<\/li>\n<\/ul>\n<h4>example solution \u2014 continuity<\/h4>\n<p>first let&#8217;s determine if the function is continuous at <em>x<\/em> = 3. because <em>f<\/em> is defined <em>piece-wise<\/em>, we must compute both the left and right hand limits.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9410\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/lefthandlimit.gif\" alt=\"lefthand limit\" width=\"359\" height=\"29\" \/><\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9411\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/righthandlimit.gif\" alt=\"right hand limit\" width=\"376\" height=\"29\" \/><\/p>\n<p>now because the left and right hand limits agree, we know that the two-sided limit as <em>x<\/em> \u2192 3 exists and equals 0. next, check the function value at <em>x<\/em> = 3.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9412\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/functionvalue.gif\" alt=\"evaluating the piecewise function at x = 3\" width=\"179\" height=\"20\" \/><\/p>\n<p>therefore, since the limiting value equals the function value (both are 0), the function <em>f<\/em> is continuous at <em>x<\/em> = 3 <em>by definition<\/em>. (for more about this topic, check out <a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-exam-review-limits-continuity\/\">ap calculus exam review: limits and continuity<\/a>.)<\/p>\n<h4>example solution \u2014 differentiability<\/h4>\n<p>moving on to differentiability, now we must check whether <em>f<\/em>\u00a0&#8216;(<em>3<\/em>) exists. again, because <em>f<\/em> is defined piece-wise, we must be careful at the point where the function changes behavior. first find the derivative of each piece.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9413\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/derivative_of_piecewise.gif\" alt=\"derivative of piecewise function\" width=\"202\" height=\"55\" \/><\/p>\n<p>note, there is no typo here \u2014 the derivative of the first piece can only be found when <em>x<\/em> &lt; 3. in fact it takes more analysis to figure out what happens <em>at<\/em> <em>x<\/em> = 3.<\/p>\n<p>because the derivative itself is actually a certain kind of limit (by definition!), we&#8217;ll have to see what the limiting values for <em>f<\/em>\u00a0&#8216; are as <em>x<\/em> \u2192 3. as before, examine each piece separately.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9414\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/derivative_value.gif\" alt=\"derivatives from the left and right\" width=\"118\" height=\"47\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/derivative_value.gif 118w, \/\/www.catharsisit.com\/hs\/files\/2017\/03\/derivative_value-30x12.gif 30w\" sizes=\"(max-width: 118px) 100vw, 118px\" \/><\/p>\n<p>this time there is a mismatch. because the left and right derivatives do not agree (18 \u2260 -9), the derivative does not exist at <em>x<\/em> = 3. thus <em>by definition<\/em>, <em>f<\/em> is not differentiable at <em>x<\/em> = 3.<\/p>\n<p>in summary, <em>f<\/em> is continuous, but not differentiable at <em>x<\/em> = 3. choice (b) is correct.<\/p>\n<h2>reasoning with theorems<\/h2>\n<p>remember, a theorem is a true mathematical statement. typically theorems are general facts that can apply to lots of different situations. here is a small list of important theorems in calculus.<\/p>\n<ul>\n<li>intermediate value theorem<\/li>\n<li>extreme value theorem<\/li>\n<li><a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-review-mean-value-theorem\/\">mean value theorem<\/a> for derivatives<\/li>\n<li>rolle&#8217;s theorem<\/li>\n<li><a href=\"\/\/www.catharsisit.com\/hs\/ap\/ap-calculus-exam-review-fundamental-theorem-calculus\/\">fundamental theorem of calculus<\/a> (two parts)<\/li>\n<li>mean value theorem for integrals<\/li>\n<\/ul>\n<h3>a theorem by any other name&#8230;<\/h3>\n<p>there are many other results and formulas in calculus that may not have the title of &#8220;theorem&#8221; but are nevertheless important theorems. every one of your derivative and antidifferentiation rules is actually a theorem. here is a partial list of other theorems that may not be explicitly identified as theorems in your textbook.<\/p>\n<ul>\n<li>differentiability implies continuity<\/li>\n<li>the first derivative rule for increase and decrease<\/li>\n<li>the second derivative rule for concavity<\/li>\n<li>first and second derivative rules for relative extrema<\/li>\n<li>product rule, quotient rule, chain rule, etc.<\/li>\n<li><a href=\"\/\/www.catharsisit.com\/hs\/ap\/what-is-lhospitals-rule\/\">l&#8217;hospital&#8217;s rule<\/a><\/li>\n<li>additivity and linearity of the definite integral<\/li>\n<li>techniques of antidifferentiation such as substitution, integration by parts, etc.<\/li>\n<li>various tests for convergence of series<\/li>\n<\/ul>\n<p>now let&#8217;s see if we can use the right theorems to crack the next example.<\/p>\n<h3>example 2<\/h3>\n<p>find <img decoding=\"async\" class=\"alignnone size-full wp-image-9415\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/derivative_of_integral.gif\" alt=\"example - derivative of an accumulation function\" width=\"112\" height=\"46\" srcset=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/derivative_of_integral.gif 112w, \/\/www.catharsisit.com\/hs\/files\/2017\/03\/derivative_of_integral-30x12.gif 30w\" sizes=\"(max-width: 112px) 100vw, 112px\" \/>.<\/p>\n<p>notice that this is a derivative of an integral. that means we may be able to apply the fundamental theorem of calculus. there are two parts to the theorem, but the one we need is:<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-8717\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/01\/ftc2.jpg\" alt=\"second fundamental theorem of calculus\" width=\"171\" height=\"49\" \/><\/p>\n<p>however, before we can apply this theorem, we must change the form of the integral. the theorem requires that the lower limit of integration must be a constant. by using the rule for switching the order of integration (another <em>theorem<\/em>!), we may write:<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9416\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/ftc_problem_solution_parta.gif\" alt=\"changing the order of integration in an ftc problem\" width=\"300\" height=\"116\" \/><\/p>\n<p>next, because the upper limit of integration is not a simple variable, <em>x<\/em>, we must use yet another theorem: the chain rule. here, the &#8220;inside function&#8221; is <em>u<\/em> = <em>x<\/em><sup>3<\/sup>.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter size-full wp-image-9417\" src=\"\/\/www.catharsisit.com\/hs\/files\/2017\/03\/ftc_problem_solution_partb.gif\" alt=\"ftc problem using chain rule solution\" width=\"327\" height=\"93\" \/><\/p>\n<p>it&#8217;s interesting to note in this case that no other method could have led to the solution. it is impossible to write down an antiderivative for the function, sin <em>t<\/em><sup>2<\/sup>. fortunately the fundamental theorem of calculus in the form we used it avoids the antidifferentiation step altogether.<\/p>\n<h2>conclusion<\/h2>\n<p>definitions and theorems form the backbone of mathematical reasoning. knowing your definitions means knowing which tools can apply in each situation. and by understanding the theorems, you can avoid doing a lot of unnecessary or difficult work.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>many people believe that mathematics is about number-crunching, but much more importantly, math is about reasoning. click here to find out more!<\/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 review: reasoning with theorems and definitions - magoosh blog | high school<\/title>\n<meta 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