{"id":3597,"date":"2024-06-28T17:12:44","date_gmt":"2024-06-28T17:12:44","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3597"},"modified":"2024-08-05T02:05:26","modified_gmt":"2024-08-05T02:05:26","slug":"analytical-applications-of-derivatives-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/analytical-applications-of-derivatives-cheat-sheet\/","title":{"raw":"Analytical Applications of Derivatives: Cheat Sheet","rendered":"Analytical Applications of Derivatives: Cheat Sheet"},"content":{"raw":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Analytical+Applications+of+Derivatives.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\r\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\r\n<h2>Essential Concepts<\/h2>\r\n<p><strong>Related Rates<\/strong><\/p>\r\n<ul id=\"fs-id1165043098599\">\r\n\t<li>To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time.<\/li>\r\n\t<li>In terms of the quantities, state the information given and the rate to be found.<\/li>\r\n\t<li>Find an equation relating the quantities.<\/li>\r\n\t<li>Use differentiation, applying the chain rule as necessary, to find an equation that relates the rates.<\/li>\r\n\t<li>Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates.<\/li>\r\n<\/ul>\r\n<p><strong>Linear Approximations and Differentials<\/strong><\/p>\r\n<ul id=\"fs-id1165043309846\">\r\n\t<li>A differentiable function [latex]y=f(x)[\/latex] can be approximated at [latex]a[\/latex] by the linear function<br \/>\r\n<div id=\"fs-id1165042638768\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/div>\r\n<\/li>\r\n\t<li>For a function [latex]y=f(x)[\/latex], if [latex]x[\/latex] changes from [latex]a[\/latex] to [latex]a+dx[\/latex], then<br \/>\r\n<div id=\"fs-id1165043309024\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]dy=f^{\\prime}(x) \\, dx[\/latex]<\/div>\r\n<p>is an approximation for the change in [latex]y[\/latex]. The actual change in [latex]y[\/latex] is<\/p>\r\n<div id=\"fs-id1165042514151\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\Delta y=f(a+dx)-f(a)[\/latex]<\/div>\r\n<\/li>\r\n\t<li>A measurement error [latex]dx[\/latex] can lead to an error in a calculated quantity [latex]f(x)[\/latex]. The error in the calculated quantity is known as the <em>propagated error<\/em>. The propagated error can be estimated by<br \/>\r\n<div id=\"fs-id1165042582705\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]dy\\approx f^{\\prime}(x) \\, dx[\/latex]<\/div>\r\n<\/li>\r\n\t<li style=\"text-align: left;\">To estimate the relative error of a particular quantity [latex]q[\/latex], we estimate [latex]\\dfrac{\\Delta q}{q}[\/latex]<\/li>\r\n<\/ul>\r\n<p><strong>Maxima and Minima<\/strong><\/p>\r\n<ul id=\"fs-id1165040729423\">\r\n\t<li>A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.<\/li>\r\n\t<li>If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.<\/li>\r\n\t<li>A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.<\/li>\r\n<\/ul>\r\n<p><strong>The Mean Value Theorem<\/strong><\/p>\r\n<ul id=\"fs-id1165042651664\">\r\n\t<li>If [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex] and [latex]f(a)=0=f(b)[\/latex], then there exists a point [latex]c \\in (a,b)[\/latex] such that [latex]f^{\\prime}(c)=0[\/latex]. This is Rolle\u2019s theorem.<\/li>\r\n\t<li>If [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], then there exists a point [latex]c \\in (a,b)[\/latex] such that<br \/>\r\n<div id=\"fs-id1165042711722\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\dfrac{f(b)-f(a)}{b-a}[\/latex].<\/div>\r\n<p>This is the Mean Value Theorem.<\/p>\r\n<\/li>\r\n\t<li>If [latex]f^{\\prime}(x)=0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is constant over [latex]I[\/latex].<\/li>\r\n\t<li>If two differentiable functions [latex]f[\/latex] and [latex]g[\/latex] satisfy [latex]f^{\\prime}(x)=g^{\\prime}(x)[\/latex] over [latex]I[\/latex], then [latex]f(x)=g(x)+C[\/latex] for some constant [latex]C[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime}(x)&gt;0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is increasing over [latex]I[\/latex]. If [latex]f^{\\prime}(x)&lt;0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is decreasing over [latex]I[\/latex].<\/li>\r\n<\/ul>\r\n<p><strong>Derivatives and the Shape of a Graph<\/strong><\/p>\r\n<ul id=\"fs-id1165043163899\">\r\n\t<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)&gt;0[\/latex] for [latex]x &lt; c[\/latex] and [latex]f^{\\prime}(x)&lt;0[\/latex] for [latex]x&gt;c[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\r\n\t<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)&lt;0[\/latex] for [latex]x &lt; c[\/latex] and [latex]f^{\\prime}(x)&gt;0[\/latex] for [latex]x&gt;c[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)&gt;0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)&lt;0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)=0[\/latex], then evaluate [latex]f^{\\prime}(x)[\/latex] at a test point [latex]x[\/latex] to the left of [latex]c[\/latex] and a test point [latex]x[\/latex] to the right of [latex]c[\/latex], to determine whether [latex]f[\/latex] has a local extremum at [latex]c[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime \\prime}(x)&gt;0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime \\prime}(x)&lt;0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)&gt;0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)&lt;0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\r\n\t<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)=0[\/latex], then evaluate [latex]f^{\\prime}(x)[\/latex] at a test point [latex]x[\/latex] to the left of [latex]c[\/latex] and a test point [latex]x[\/latex] to the right of [latex]c[\/latex], to determine whether [latex]f[\/latex] has a local extremum at [latex]c[\/latex].<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1165042390258\">\r\n\t<li><strong>Linear approximation<\/strong><br \/>\r\n[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/li>\r\n\t<li><strong>A differential<\/strong><br \/>\r\n[latex]dy=f^{\\prime}(x) \\, dx[\/latex].<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165042059050\" class=\"definition\">\r\n<dt>absolute extremum<\/dt>\r\n<dd id=\"fs-id1165042059056\">if [latex]f[\/latex] has an absolute maximum or absolute minimum at [latex]c[\/latex], we say [latex]f[\/latex] has an absolute extremum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281356\" class=\"definition\">\r\n<dt>absolute maximum<\/dt>\r\n<dd id=\"fs-id1165042281361\">if [latex]f(c)\\ge f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], we say [latex]f[\/latex] has an absolute maximum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281408\" class=\"definition\">\r\n<dt>absolute minimum<\/dt>\r\n<dd id=\"fs-id1165042281414\">if [latex]f(c)\\le f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], we say [latex]f[\/latex] has an absolute minimum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043427379\" class=\"definition\">\r\n<dt>concave down<\/dt>\r\n<dd id=\"fs-id1165043427385\">if [latex]f[\/latex] is differentiable over an interval [latex]I[\/latex] and [latex]f^{\\prime}[\/latex] is decreasing over [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043396223\" class=\"definition\">\r\n<dt>concave up<\/dt>\r\n<dd id=\"fs-id1165043396229\">if [latex]f[\/latex] is differentiable over an interval [latex]I[\/latex] and [latex]f^{\\prime}[\/latex] is increasing over [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043396264\" class=\"definition\">\r\n<dt>concavity<\/dt>\r\n<dd id=\"fs-id1165043396269\">the upward or downward curve of the graph of a function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043396274\" class=\"definition\">\r\n<dt>concavity test<\/dt>\r\n<dd id=\"fs-id1165043396279\">suppose [latex]f[\/latex] is twice differentiable over an interval [latex]I[\/latex]; if [latex]f^{\\prime \\prime}&gt;0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex]; if [latex]f^{\\prime \\prime}&lt;0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>critical point<\/dt>\r\n<dd id=\"fs-id1165042281467\">if [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined, we say that [latex]c[\/latex] is a critical point of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>differential<\/dt>\r\n<dd id=\"fs-id1165043315303\">the differential [latex]dx[\/latex] is an independent variable that can be assigned any nonzero real number; the differential [latex]dy[\/latex] is defined to be [latex]dy=f^{\\prime}(x) \\, dx[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043199989\" class=\"definition\">\r\n<dt>differential form<\/dt>\r\n<dd id=\"fs-id1165043422532\">given a differentiable function [latex]y=f^{\\prime}(x)[\/latex], the equation [latex]dy=f^{\\prime}(x) \\, dx[\/latex] is the differential form of the derivative of [latex]y[\/latex] with respect to [latex]x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281514\" class=\"definition\">\r\n<dt>extreme value theorem<\/dt>\r\n<dd id=\"fs-id1165042281519\">if [latex]f[\/latex] is a continuous function over a finite, closed interval, then [latex]f[\/latex] has an absolute maximum and an absolute minimum<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281532\" class=\"definition\">\r\n<dt>Fermat\u2019s theorem<\/dt>\r\n<dd id=\"fs-id1165042281538\">if [latex]f[\/latex] has a local extremum at [latex]c[\/latex], then [latex]c[\/latex] is a critical point of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043281605\" class=\"definition\">\r\n<dt>first derivative test<\/dt>\r\n<dd id=\"fs-id1165043281610\">let [latex]f[\/latex] be a continuous function over an interval [latex]I[\/latex] containing a critical point [latex]c[\/latex] such that [latex]f[\/latex] is differentiable over [latex]I[\/latex] except possibly at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] changes sign from positive to negative as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] changes sign from negative to positive as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] does not change sign as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] does not have a local extremum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042364543\" class=\"definition\">\r\n<dt>inflection point<\/dt>\r\n<dd id=\"fs-id1165042364548\">if [latex]f[\/latex] is continuous at [latex]c[\/latex] and [latex]f[\/latex] changes concavity at [latex]c[\/latex], the point [latex](c,f(c))[\/latex] is an inflection point of [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>linear approximation<\/dt>\r\n<dd id=\"fs-id1165043380421\">the linear function [latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex] is the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042281561\" class=\"definition\">\r\n<dt>local extremum<\/dt>\r\n<dd id=\"fs-id1165042281566\">if [latex]f[\/latex] has a local maximum or local minimum at [latex]c[\/latex], we say [latex]f[\/latex] has a local extremum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042071370\" class=\"definition\">\r\n<dt>local maximum<\/dt>\r\n<dd id=\"fs-id1165042071375\">if there exists an interval [latex]I[\/latex] such that [latex]f(c)\\ge f(x)[\/latex] for all [latex]x\\in I[\/latex], we say [latex]f[\/latex] has a local maximum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042071428\" class=\"definition\">\r\n<dt>local minimum<\/dt>\r\n<dd id=\"fs-id1165042071434\">if there exists an interval [latex]I[\/latex] such that [latex]f(c)\\le f(x)[\/latex] for all [latex]x\\in I[\/latex], we say [latex]f[\/latex] has a local minimum at [latex]c[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042595459\" class=\"definition\">\r\n<dt>mean value theorem<\/dt>\r\n<dd id=\"fs-id1165042595464\">\r\n<p>if [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], then there exists [latex]c \\in (a,b)[\/latex] such that<\/p>\r\n<div id=\"fs-id1165043183378\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\dfrac{f(b)-f(a)}{b-a}[\/latex]<\/div>\r\n<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>percentage error<\/dt>\r\n<dd id=\"fs-id1165042478964\">the relative error expressed as a percentage<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042478968\" class=\"definition\">\r\n<dt>propagated error<\/dt>\r\n<dd id=\"fs-id1165042321686\">the error that results in a calculated quantity [latex]f(x)[\/latex] resulting from a measurement error [latex]dx[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165042370920\" class=\"definition\">\r\n<dt>relative error<\/dt>\r\n<dd id=\"fs-id1165042370925\">given an absolute error [latex]\\Delta q[\/latex] for a particular quantity, [latex]\\dfrac{\\Delta q}{q}[\/latex] is the relative error.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043120519\" class=\"definition\">\r\n<dt>related rates<\/dt>\r\n<dd id=\"fs-id1165043120524\">are rates of change associated with two or more related quantities that are changing over time<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165043183432\" class=\"definition\">\r\n<dt>rolle\u2019s theorem<\/dt>\r\n<dd id=\"fs-id1165043183437\">if [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], and if [latex]f(a)=f(b)[\/latex], then there exists [latex]c \\in (a,b)[\/latex] such that [latex]f^{\\prime}(c)=0[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>second derivative test<\/dt>\r\n<dd id=\"fs-id1165043131584\">suppose [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}[\/latex] is continuous over an interval containing [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)&gt;0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)&lt;0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)=0[\/latex], then the test is inconclusive<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>tangent line approximation (linearization)<\/dt>\r\n<dd id=\"fs-id1165043393042\">since the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex] is defined using the equation of the tangent line, the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex] is also known as the tangent line approximation to [latex]f[\/latex] at [latex]x=a[\/latex]<\/dd>\r\n<\/dl>\r\n<h2>Study Tips<\/h2>\r\n<p><strong>Related-Rates Problem-Solving<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying the related quantities in word problems.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Memorize common geometric formulas (area, volume, Pythagorean theorem).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Review chain rule applications, as they're crucial in related rates.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Always check if your answer makes physical sense in the context of the problem.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Be careful with positive\/negative rates (e.g., increasing vs. decreasing quantities).<\/li>\r\n<\/ul>\r\n<p><strong>Linear Approximation of a Function at a Point<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Visualize the tangent line and the function to understand the approximation geometrically.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Compare approximations with actual function values to gauge accuracy.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]f(a)[\/latex] is the [latex]y[\/latex]-intercept and [latex]f'(a)[\/latex] is the slope of the approximation line.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use linear approximation to estimate values of complex functions or irrational numbers.<\/li>\r\n<\/ul>\r\n<p><strong>Differentials and Amount of Error<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice computing differentials for various functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Understand the connection between differentials and the slope of the tangent line.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">In error estimation problems, identify the measured quantity and the calculated quantity.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that relative error is often more meaningful than absolute error.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice converting between absolute, relative, and percentage errors.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When estimating errors, consider both positive and negative variations in measurements.<\/li>\r\n<\/ul>\r\n<p><strong>Absolute Extrema<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying absolute extrema on various function graphs.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Understand the difference between the extremum ([latex]y[\/latex]-value) and its location ([latex]x[\/latex]-value).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Recognize scenarios where the Extreme Value Theorem applies or doesn't apply.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to the domain of the function, especially its endpoints and continuity.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Learn to distinguish between functions that have both, one, or no absolute extrema.<\/li>\r\n<\/ul>\r\n<p><strong>Local Extrema and Critical Points<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying local extrema on various function graphs.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Learn to distinguish between local and absolute extrema.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When finding critical points, check both for [latex]f'(x) = 0[\/latex] and where [latex]f'(x)[\/latex] is undefined.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that critical points are only candidates for local extrema, not guarantees.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to endpoints of closed intervals, as they may contain extrema but are not considered local extrema.<\/li>\r\n<\/ul>\r\n<p><strong>Locating Absolute Extrema<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Always check both endpoints of the interval, even if they're not critical points.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When finding critical points, check for both [latex]f'(x) = 0[\/latex] and where [latex]f'(x)[\/latex] is undefined.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Don't forget to evaluate the function at each critical point within the interval.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Organize your work: list all candidate points (endpoints and critical points) with their function values.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that a function can have multiple points with the same extreme value.<\/li>\r\n<\/ul>\r\n<p><strong>Rolle\u2019s Theorem<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Verify all conditions of Rolle's Theorem before applying it.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying functions that satisfy or violate the conditions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use Rolle's Theorem to prove the existence of zeros of a derivative, not to find them explicitly.<\/li>\r\n<\/ul>\r\n<p><strong>The Mean Value Theorem and Its Meaning<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Visualize the theorem geometrically: think of a secant line and parallel tangent line(s).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Connect the Mean Value Theorem with the concept of average rate of change.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use the theorem to solve problems involving motion, particularly relating average and instantaneous velocities.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that the theorem doesn't tell you how to find [latex]c[\/latex], just that it exists.<\/li>\r\n<\/ul>\r\n<p><strong>Corollaries of the Mean Value Theorem<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Understand the relationship between these corollaries and the Mean Value Theorem.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice applying each corollary to different types of functions and scenarios.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For Corollary 1, remember that it applies to an entire interval, not just a single point.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For Corollary 2, think about how this relates to the concept of antiderivatives.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For Corollary 3, visualize how the sign of the derivative affects the graph of the function.<\/li>\r\n<\/ul>\r\n<p><strong>The First Derivative Test<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice finding critical points for various function types<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize the relationship between [latex]f'(x)[\/latex] and the graph of [latex]f(x)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that not all critical points are local extrema<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use a number line to organize information about [latex]f'(x)[\/latex] signs<\/li>\r\n<\/ul>\r\n<p><strong>Concavity and Points of Inflection<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice finding [latex]f''(x)[\/latex] for various function types<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize how [latex]f''(x)[\/latex] relates to the \"bending\" of the graph<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]f''(x) = 0[\/latex] doesn't guarantee an inflection point<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use a number line to organize information about [latex]f''(x)[\/latex] signs<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Compare concavity analysis with first derivative analysis for a complete picture<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Sketch graphs to confirm your analytical results<\/li>\r\n<\/ul>\r\n<p><strong>The Second Derivative Test<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Use this test in conjunction with other analytical tools (e.g., concavity analysis)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Be prepared to use the First Derivative Test when this test is inconclusive<\/li>\r\n<\/ul>","rendered":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+Analytical+Applications+of+Derivatives.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>Related Rates<\/strong><\/p>\n<ul id=\"fs-id1165043098599\">\n<li>To solve a related rates problem, first draw a picture that illustrates the relationship between the two or more related quantities that are changing with respect to time.<\/li>\n<li>In terms of the quantities, state the information given and the rate to be found.<\/li>\n<li>Find an equation relating the quantities.<\/li>\n<li>Use differentiation, applying the chain rule as necessary, to find an equation that relates the rates.<\/li>\n<li>Be sure not to substitute a variable quantity for one of the variables until after finding an equation relating the rates.<\/li>\n<\/ul>\n<p><strong>Linear Approximations and Differentials<\/strong><\/p>\n<ul id=\"fs-id1165043309846\">\n<li>A differentiable function [latex]y=f(x)[\/latex] can be approximated at [latex]a[\/latex] by the linear function\n<div id=\"fs-id1165042638768\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/div>\n<\/li>\n<li>For a function [latex]y=f(x)[\/latex], if [latex]x[\/latex] changes from [latex]a[\/latex] to [latex]a+dx[\/latex], then\n<div id=\"fs-id1165043309024\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]dy=f^{\\prime}(x) \\, dx[\/latex]<\/div>\n<p>is an approximation for the change in [latex]y[\/latex]. The actual change in [latex]y[\/latex] is<\/p>\n<div id=\"fs-id1165042514151\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\Delta y=f(a+dx)-f(a)[\/latex]<\/div>\n<\/li>\n<li>A measurement error [latex]dx[\/latex] can lead to an error in a calculated quantity [latex]f(x)[\/latex]. The error in the calculated quantity is known as the <em>propagated error<\/em>. The propagated error can be estimated by\n<div id=\"fs-id1165042582705\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]dy\\approx f^{\\prime}(x) \\, dx[\/latex]<\/div>\n<\/li>\n<li style=\"text-align: left;\">To estimate the relative error of a particular quantity [latex]q[\/latex], we estimate [latex]\\dfrac{\\Delta q}{q}[\/latex]<\/li>\n<\/ul>\n<p><strong>Maxima and Minima<\/strong><\/p>\n<ul id=\"fs-id1165040729423\">\n<li>A function may have both an absolute maximum and an absolute minimum, have just one absolute extremum, or have no absolute maximum or absolute minimum.<\/li>\n<li>If a function has a local extremum, the point at which it occurs must be a critical point. However, a function need not have a local extremum at a critical point.<\/li>\n<li>A continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. Each extremum occurs at a critical point or an endpoint.<\/li>\n<\/ul>\n<p><strong>The Mean Value Theorem<\/strong><\/p>\n<ul id=\"fs-id1165042651664\">\n<li>If [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex] and [latex]f(a)=0=f(b)[\/latex], then there exists a point [latex]c \\in (a,b)[\/latex] such that [latex]f^{\\prime}(c)=0[\/latex]. This is Rolle\u2019s theorem.<\/li>\n<li>If [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], then there exists a point [latex]c \\in (a,b)[\/latex] such that\n<div id=\"fs-id1165042711722\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\dfrac{f(b)-f(a)}{b-a}[\/latex].<\/div>\n<p>This is the Mean Value Theorem.<\/p>\n<\/li>\n<li>If [latex]f^{\\prime}(x)=0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is constant over [latex]I[\/latex].<\/li>\n<li>If two differentiable functions [latex]f[\/latex] and [latex]g[\/latex] satisfy [latex]f^{\\prime}(x)=g^{\\prime}(x)[\/latex] over [latex]I[\/latex], then [latex]f(x)=g(x)+C[\/latex] for some constant [latex]C[\/latex].<\/li>\n<li>If [latex]f^{\\prime}(x)>0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is increasing over [latex]I[\/latex]. If [latex]f^{\\prime}(x)<0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is decreasing over [latex]I[\/latex].<\/li>\n<\/ul>\n<p><strong>Derivatives and the Shape of a Graph<\/strong><\/p>\n<ul id=\"fs-id1165043163899\">\n<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)>0[\/latex] for [latex]x < c[\/latex] and [latex]f^{\\prime}(x)<0[\/latex] for [latex]x>c[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\n<li>If [latex]c[\/latex] is a critical point of [latex]f[\/latex] and [latex]f^{\\prime}(x)<0[\/latex] for [latex]x < c[\/latex] and [latex]f^{\\prime}(x)>0[\/latex] for [latex]x>c[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(x)>0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(x)<0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex].<\/li>\n<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)>0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)<0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)=0[\/latex], then evaluate [latex]f^{\\prime}(x)[\/latex] at a test point [latex]x[\/latex] to the left of [latex]c[\/latex] and a test point [latex]x[\/latex] to the right of [latex]c[\/latex], to determine whether [latex]f[\/latex] has a local extremum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(x)>0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex].<\/li>\n<li>If [latex]f^{\\prime \\prime}(x)<0[\/latex] over an interval [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex].<\/li>\n<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)>0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)<0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex].<\/li>\n<li>If [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}(c)=0[\/latex], then evaluate [latex]f^{\\prime}(x)[\/latex] at a test point [latex]x[\/latex] to the left of [latex]c[\/latex] and a test point [latex]x[\/latex] to the right of [latex]c[\/latex], to determine whether [latex]f[\/latex] has a local extremum at [latex]c[\/latex].<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1165042390258\">\n<li><strong>Linear approximation<\/strong><br \/>\n[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/li>\n<li><strong>A differential<\/strong><br \/>\n[latex]dy=f^{\\prime}(x) \\, dx[\/latex].<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165042059050\" class=\"definition\">\n<dt>absolute extremum<\/dt>\n<dd id=\"fs-id1165042059056\">if [latex]f[\/latex] has an absolute maximum or absolute minimum at [latex]c[\/latex], we say [latex]f[\/latex] has an absolute extremum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281356\" class=\"definition\">\n<dt>absolute maximum<\/dt>\n<dd id=\"fs-id1165042281361\">if [latex]f(c)\\ge f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], we say [latex]f[\/latex] has an absolute maximum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281408\" class=\"definition\">\n<dt>absolute minimum<\/dt>\n<dd id=\"fs-id1165042281414\">if [latex]f(c)\\le f(x)[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex], we say [latex]f[\/latex] has an absolute minimum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043427379\" class=\"definition\">\n<dt>concave down<\/dt>\n<dd id=\"fs-id1165043427385\">if [latex]f[\/latex] is differentiable over an interval [latex]I[\/latex] and [latex]f^{\\prime}[\/latex] is decreasing over [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043396223\" class=\"definition\">\n<dt>concave up<\/dt>\n<dd id=\"fs-id1165043396229\">if [latex]f[\/latex] is differentiable over an interval [latex]I[\/latex] and [latex]f^{\\prime}[\/latex] is increasing over [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043396264\" class=\"definition\">\n<dt>concavity<\/dt>\n<dd id=\"fs-id1165043396269\">the upward or downward curve of the graph of a function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043396274\" class=\"definition\">\n<dt>concavity test<\/dt>\n<dd id=\"fs-id1165043396279\">suppose [latex]f[\/latex] is twice differentiable over an interval [latex]I[\/latex]; if [latex]f^{\\prime \\prime}>0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is concave up over [latex]I[\/latex]; if [latex]f^{\\prime \\prime}<0[\/latex] over [latex]I[\/latex], then [latex]f[\/latex] is concave down over [latex]I[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>critical point<\/dt>\n<dd id=\"fs-id1165042281467\">if [latex]f^{\\prime}(c)=0[\/latex] or [latex]f^{\\prime}(c)[\/latex] is undefined, we say that [latex]c[\/latex] is a critical point of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>differential<\/dt>\n<dd id=\"fs-id1165043315303\">the differential [latex]dx[\/latex] is an independent variable that can be assigned any nonzero real number; the differential [latex]dy[\/latex] is defined to be [latex]dy=f^{\\prime}(x) \\, dx[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043199989\" class=\"definition\">\n<dt>differential form<\/dt>\n<dd id=\"fs-id1165043422532\">given a differentiable function [latex]y=f^{\\prime}(x)[\/latex], the equation [latex]dy=f^{\\prime}(x) \\, dx[\/latex] is the differential form of the derivative of [latex]y[\/latex] with respect to [latex]x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281514\" class=\"definition\">\n<dt>extreme value theorem<\/dt>\n<dd id=\"fs-id1165042281519\">if [latex]f[\/latex] is a continuous function over a finite, closed interval, then [latex]f[\/latex] has an absolute maximum and an absolute minimum<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281532\" class=\"definition\">\n<dt>Fermat\u2019s theorem<\/dt>\n<dd id=\"fs-id1165042281538\">if [latex]f[\/latex] has a local extremum at [latex]c[\/latex], then [latex]c[\/latex] is a critical point of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043281605\" class=\"definition\">\n<dt>first derivative test<\/dt>\n<dd id=\"fs-id1165043281610\">let [latex]f[\/latex] be a continuous function over an interval [latex]I[\/latex] containing a critical point [latex]c[\/latex] such that [latex]f[\/latex] is differentiable over [latex]I[\/latex] except possibly at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] changes sign from positive to negative as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] changes sign from negative to positive as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex]; if [latex]f^{\\prime}[\/latex] does not change sign as [latex]x[\/latex] increases through [latex]c[\/latex], then [latex]f[\/latex] does not have a local extremum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042364543\" class=\"definition\">\n<dt>inflection point<\/dt>\n<dd id=\"fs-id1165042364548\">if [latex]f[\/latex] is continuous at [latex]c[\/latex] and [latex]f[\/latex] changes concavity at [latex]c[\/latex], the point [latex](c,f(c))[\/latex] is an inflection point of [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>linear approximation<\/dt>\n<dd id=\"fs-id1165043380421\">the linear function [latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex] is the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042281561\" class=\"definition\">\n<dt>local extremum<\/dt>\n<dd id=\"fs-id1165042281566\">if [latex]f[\/latex] has a local maximum or local minimum at [latex]c[\/latex], we say [latex]f[\/latex] has a local extremum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042071370\" class=\"definition\">\n<dt>local maximum<\/dt>\n<dd id=\"fs-id1165042071375\">if there exists an interval [latex]I[\/latex] such that [latex]f(c)\\ge f(x)[\/latex] for all [latex]x\\in I[\/latex], we say [latex]f[\/latex] has a local maximum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042071428\" class=\"definition\">\n<dt>local minimum<\/dt>\n<dd id=\"fs-id1165042071434\">if there exists an interval [latex]I[\/latex] such that [latex]f(c)\\le f(x)[\/latex] for all [latex]x\\in I[\/latex], we say [latex]f[\/latex] has a local minimum at [latex]c[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042595459\" class=\"definition\">\n<dt>mean value theorem<\/dt>\n<dd id=\"fs-id1165042595464\">\n<p>if [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], then there exists [latex]c \\in (a,b)[\/latex] such that<\/p>\n<div id=\"fs-id1165043183378\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]f^{\\prime}(c)=\\dfrac{f(b)-f(a)}{b-a}[\/latex]<\/div>\n<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>percentage error<\/dt>\n<dd id=\"fs-id1165042478964\">the relative error expressed as a percentage<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042478968\" class=\"definition\">\n<dt>propagated error<\/dt>\n<dd id=\"fs-id1165042321686\">the error that results in a calculated quantity [latex]f(x)[\/latex] resulting from a measurement error [latex]dx[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165042370920\" class=\"definition\">\n<dt>relative error<\/dt>\n<dd id=\"fs-id1165042370925\">given an absolute error [latex]\\Delta q[\/latex] for a particular quantity, [latex]\\dfrac{\\Delta q}{q}[\/latex] is the relative error.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043120519\" class=\"definition\">\n<dt>related rates<\/dt>\n<dd id=\"fs-id1165043120524\">are rates of change associated with two or more related quantities that are changing over time<\/dd>\n<\/dl>\n<dl id=\"fs-id1165043183432\" class=\"definition\">\n<dt>rolle\u2019s theorem<\/dt>\n<dd id=\"fs-id1165043183437\">if [latex]f[\/latex] is continuous over [latex][a,b][\/latex] and differentiable over [latex](a,b)[\/latex], and if [latex]f(a)=f(b)[\/latex], then there exists [latex]c \\in (a,b)[\/latex] such that [latex]f^{\\prime}(c)=0[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>second derivative test<\/dt>\n<dd id=\"fs-id1165043131584\">suppose [latex]f^{\\prime}(c)=0[\/latex] and [latex]f^{\\prime \\prime}[\/latex] is continuous over an interval containing [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)>0[\/latex], then [latex]f[\/latex] has a local minimum at [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)<0[\/latex], then [latex]f[\/latex] has a local maximum at [latex]c[\/latex]; if [latex]f^{\\prime \\prime}(c)=0[\/latex], then the test is inconclusive<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>tangent line approximation (linearization)<\/dt>\n<dd id=\"fs-id1165043393042\">since the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex] is defined using the equation of the tangent line, the linear approximation of [latex]f[\/latex] at [latex]x=a[\/latex] is also known as the tangent line approximation to [latex]f[\/latex] at [latex]x=a[\/latex]<\/dd>\n<\/dl>\n<h2>Study Tips<\/h2>\n<p><strong>Related-Rates Problem-Solving<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying the related quantities in word problems.<\/li>\n<li class=\"whitespace-normal break-words\">Memorize common geometric formulas (area, volume, Pythagorean theorem).<\/li>\n<li class=\"whitespace-normal break-words\">Review chain rule applications, as they&#8217;re crucial in related rates.<\/li>\n<li class=\"whitespace-normal break-words\">Always check if your answer makes physical sense in the context of the problem.<\/li>\n<li class=\"whitespace-normal break-words\">Be careful with positive\/negative rates (e.g., increasing vs. decreasing quantities).<\/li>\n<\/ul>\n<p><strong>Linear Approximation of a Function at a Point<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Visualize the tangent line and the function to understand the approximation geometrically.<\/li>\n<li class=\"whitespace-normal break-words\">Compare approximations with actual function values to gauge accuracy.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]f(a)[\/latex] is the [latex]y[\/latex]-intercept and [latex]f'(a)[\/latex] is the slope of the approximation line.<\/li>\n<li class=\"whitespace-normal break-words\">Use linear approximation to estimate values of complex functions or irrational numbers.<\/li>\n<\/ul>\n<p><strong>Differentials and Amount of Error<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice computing differentials for various functions.<\/li>\n<li class=\"whitespace-normal break-words\">Understand the connection between differentials and the slope of the tangent line.<\/li>\n<li class=\"whitespace-normal break-words\">In error estimation problems, identify the measured quantity and the calculated quantity.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that relative error is often more meaningful than absolute error.<\/li>\n<li class=\"whitespace-normal break-words\">Practice converting between absolute, relative, and percentage errors.<\/li>\n<li class=\"whitespace-normal break-words\">When estimating errors, consider both positive and negative variations in measurements.<\/li>\n<\/ul>\n<p><strong>Absolute Extrema<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying absolute extrema on various function graphs.<\/li>\n<li class=\"whitespace-normal break-words\">Understand the difference between the extremum ([latex]y[\/latex]-value) and its location ([latex]x[\/latex]-value).<\/li>\n<li class=\"whitespace-normal break-words\">Recognize scenarios where the Extreme Value Theorem applies or doesn&#8217;t apply.<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to the domain of the function, especially its endpoints and continuity.<\/li>\n<li class=\"whitespace-normal break-words\">Learn to distinguish between functions that have both, one, or no absolute extrema.<\/li>\n<\/ul>\n<p><strong>Local Extrema and Critical Points<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying local extrema on various function graphs.<\/li>\n<li class=\"whitespace-normal break-words\">Learn to distinguish between local and absolute extrema.<\/li>\n<li class=\"whitespace-normal break-words\">When finding critical points, check both for [latex]f'(x) = 0[\/latex] and where [latex]f'(x)[\/latex] is undefined.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that critical points are only candidates for local extrema, not guarantees.<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to endpoints of closed intervals, as they may contain extrema but are not considered local extrema.<\/li>\n<\/ul>\n<p><strong>Locating Absolute Extrema<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Always check both endpoints of the interval, even if they&#8217;re not critical points.<\/li>\n<li class=\"whitespace-normal break-words\">When finding critical points, check for both [latex]f'(x) = 0[\/latex] and where [latex]f'(x)[\/latex] is undefined.<\/li>\n<li class=\"whitespace-normal break-words\">Don&#8217;t forget to evaluate the function at each critical point within the interval.<\/li>\n<li class=\"whitespace-normal break-words\">Organize your work: list all candidate points (endpoints and critical points) with their function values.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that a function can have multiple points with the same extreme value.<\/li>\n<\/ul>\n<p><strong>Rolle\u2019s Theorem<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Verify all conditions of Rolle&#8217;s Theorem before applying it.<\/li>\n<li class=\"whitespace-normal break-words\">Practice identifying functions that satisfy or violate the conditions.<\/li>\n<li class=\"whitespace-normal break-words\">Use Rolle&#8217;s Theorem to prove the existence of zeros of a derivative, not to find them explicitly.<\/li>\n<\/ul>\n<p><strong>The Mean Value Theorem and Its Meaning<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Visualize the theorem geometrically: think of a secant line and parallel tangent line(s).<\/li>\n<li class=\"whitespace-normal break-words\">Connect the Mean Value Theorem with the concept of average rate of change.<\/li>\n<li class=\"whitespace-normal break-words\">Use the theorem to solve problems involving motion, particularly relating average and instantaneous velocities.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the theorem doesn&#8217;t tell you how to find [latex]c[\/latex], just that it exists.<\/li>\n<\/ul>\n<p><strong>Corollaries of the Mean Value Theorem<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Understand the relationship between these corollaries and the Mean Value Theorem.<\/li>\n<li class=\"whitespace-normal break-words\">Practice applying each corollary to different types of functions and scenarios.<\/li>\n<li class=\"whitespace-normal break-words\">For Corollary 1, remember that it applies to an entire interval, not just a single point.<\/li>\n<li class=\"whitespace-normal break-words\">For Corollary 2, think about how this relates to the concept of antiderivatives.<\/li>\n<li class=\"whitespace-normal break-words\">For Corollary 3, visualize how the sign of the derivative affects the graph of the function.<\/li>\n<\/ul>\n<p><strong>The First Derivative Test<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice finding critical points for various function types<\/li>\n<li class=\"whitespace-normal break-words\">Visualize the relationship between [latex]f'(x)[\/latex] and the graph of [latex]f(x)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Remember that not all critical points are local extrema<\/li>\n<li class=\"whitespace-normal break-words\">Use a number line to organize information about [latex]f'(x)[\/latex] signs<\/li>\n<\/ul>\n<p><strong>Concavity and Points of Inflection<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice finding [latex]f''(x)[\/latex] for various function types<\/li>\n<li class=\"whitespace-normal break-words\">Visualize how [latex]f''(x)[\/latex] relates to the &#8220;bending&#8221; of the graph<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]f''(x) = 0[\/latex] doesn&#8217;t guarantee an inflection point<\/li>\n<li class=\"whitespace-normal break-words\">Use a number line to organize information about [latex]f''(x)[\/latex] signs<\/li>\n<li class=\"whitespace-normal break-words\">Compare concavity analysis with first derivative analysis for a complete picture<\/li>\n<li class=\"whitespace-normal break-words\">Sketch graphs to confirm your analytical results<\/li>\n<\/ul>\n<p><strong>The Second Derivative Test<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Use this test in conjunction with other analytical tools (e.g., concavity analysis)<\/li>\n<li class=\"whitespace-normal break-words\">Be prepared to use the First Derivative Test when this test is inconclusive<\/li>\n<\/ul>\n","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":652,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3597"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3597\/revisions"}],"predecessor-version":[{"id":4075,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3597\/revisions\/4075"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/652"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3597\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3597"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3597"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3597"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3597"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}