{"id":3593,"date":"2024-06-28T17:12:00","date_gmt":"2024-06-28T17:12:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3593"},"modified":"2024-08-05T01:41:39","modified_gmt":"2024-08-05T01:41:39","slug":"introduction-to-derivatives-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/introduction-to-derivatives-cheat-sheet\/","title":{"raw":"Introduction to Derivatives: Cheat Sheet","rendered":"Introduction to 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_+Introduction+to+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>Defining the Derivative<\/strong><\/p>\r\n<ul id=\"fs-id1169736611561\">\r\n\t<li>The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment [latex]h[\/latex].<\/li>\r\n\t<li>The derivative of a function [latex]f(x)[\/latex] at a value [latex]a[\/latex] is found using either of the definitions for the slope of the tangent line.<\/li>\r\n\t<li>Velocity is the rate of change of position. As such, the velocity [latex]v(t)[\/latex] at time [latex]t[\/latex] is the derivative of the position [latex]s(t)[\/latex] at time [latex]t[\/latex].\r\n\r\n<ul>\r\n\t<li>Average velocity is given by<br \/>\r\n<div id=\"fs-id1169739286403\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]v_{\\text{avg}}=\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/div>\r\n<\/li>\r\n\t<li>\r\n<div id=\"fs-id1169739286403\" class=\"equation unnumbered\" style=\"text-align: left;\">Instantaneous velocity is given by<\/div>\r\n<div id=\"fs-id1169739191149\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]v(a)=s^{\\prime}(a)=\\underset{t\\to a}{\\lim}\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/div>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>We may estimate a derivative by using a table of values.<\/li>\r\n<\/ul>\r\n<p><strong>The Derivative as a Function<\/strong><\/p>\r\n<ul id=\"fs-id1169737935224\">\r\n\t<li>The derivative of a function [latex]f(x)[\/latex] is the function whose value at [latex]x[\/latex] is [latex]f^{\\prime}(x)[\/latex].<\/li>\r\n\t<li>The graph of a derivative of a function [latex]f(x)[\/latex] is related to the graph of [latex]f(x)[\/latex]. Where [latex]f(x)[\/latex] has a tangent line with positive slope, [latex]f^{\\prime}(x)&gt;0[\/latex]. Where [latex]f(x)[\/latex] has a tangent line with negative slope, [latex]f^{\\prime}(x)&lt;0[\/latex]. Where [latex]f(x)[\/latex] has a horizontal tangent line, [latex]f^{\\prime}(x)=0[\/latex].<\/li>\r\n\t<li>If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.<\/li>\r\n\t<li>Higher-order derivatives are derivatives of derivatives, from the second derivative to the [latex]n\\text{th}[\/latex] derivative.<\/li>\r\n<\/ul>\r\n<p><strong>Differentiation Rules<\/strong><\/p>\r\n<ul id=\"fs-id1169736659069\">\r\n\t<li>The derivative of a constant function is zero.<\/li>\r\n\t<li>The derivative of a power function is a function in which the power on [latex]x[\/latex] becomes the coefficient of the term and the power on [latex]x[\/latex] in the derivative decreases by 1.<\/li>\r\n\t<li>The derivative of a constant [latex]c[\/latex] multiplied by a function [latex]f[\/latex] is the same as the constant multiplied by the derivative.<\/li>\r\n\t<li>The derivative of the sum of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the sum of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex]<em>.<\/em><\/li>\r\n\t<li>The derivative of the difference of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the difference of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex]<em>.<\/em><\/li>\r\n\t<li>The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.<\/li>\r\n\t<li>The derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function.<\/li>\r\n\t<li>We used the limit definition of the derivative to develop formulas that allow us to find derivatives without resorting to the definition of the derivative. These formulas can be used singly or in combination with each other.<\/li>\r\n<\/ul>\r\n<p><br \/>\r\n<strong>Derivatives as Rates of Change<\/strong><\/p>\r\n<ul id=\"fs-id1169739269667\">\r\n\t<li>Using [latex]f(a+h)\\approx f(a)+f^{\\prime}(a)h[\/latex], it is possible to estimate [latex]f(a+h)[\/latex] given [latex]f^{\\prime}(a)[\/latex] and [latex]f(a)[\/latex].<\/li>\r\n\t<li>The rate of change of position is velocity, and the rate of change of velocity is acceleration. Speed is the absolute value, or magnitude, of velocity.<\/li>\r\n\t<li>The population growth rate and the present population can be used to predict the size of a future population.<\/li>\r\n\t<li>Marginal cost, marginal revenue, and marginal profit functions can be used to predict, respectively, the cost of producing one more item, the revenue obtained by selling one more item, and the profit obtained by producing and selling one more item.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169739274323\">\r\n\t<li><strong>Difference quotient<\/strong><br \/>\r\n[latex]Q=\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/li>\r\n\t<li><strong>Difference quotient with increment\u00a0[latex]h[\/latex]<\/strong><br \/>\r\n[latex]Q=\\dfrac{f(a+h)-f(a)}{a+h-a}=\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/li>\r\n\t<li><strong>Slope of tangent line<\/strong><br \/>\r\n[latex]m_{\\tan}=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<br \/>\r\n[latex]m_{\\tan}=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/li>\r\n\t<li><strong>Derivative of [latex]f(x)[\/latex] at [latex]a[\/latex]<\/strong><br \/>\r\n[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex] <br \/>\r\n[latex]f^{\\prime}(a)=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/li>\r\n\t<li><strong>Average velocity<br \/>\r\n<\/strong>[latex]v_{\\text{avg}}=\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/li>\r\n\t<li><strong>Instantaneous velocity<\/strong><br \/>\r\n[latex]v(a)=s^{\\prime}(a)=\\underset{t\\to a}{\\lim}\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/li>\r\n\t<li><strong>The derivative function<\/strong><br \/>\r\n[latex]f^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1169739252840\" class=\"definition\">\r\n<dt>acceleration<\/dt>\r\n<dd id=\"fs-id1169739252845\">is the rate of change of the velocity, that is, the derivative of velocity<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739252849\" class=\"definition\">\r\n<dt>amount of change<\/dt>\r\n<dd id=\"fs-id1169739252854\">the amount of a function [latex]f(x)[\/latex] over an interval [latex][x,x+h][\/latex] is [latex]f(x+h)-f(x)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739190003\" class=\"definition\">\r\n<dt>average rate of change<\/dt>\r\n<dd id=\"fs-id1169739190009\">is a function [latex]f(x)[\/latex] over an interval [latex][x,x+h][\/latex] is [latex]\\dfrac{f(x+h)-f(a)}{b-a}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739289276\" class=\"definition\">\r\n<dt>constant multiple rule<\/dt>\r\n<dd id=\"fs-id1169739289282\">the derivative of a constant [latex]c[\/latex] multiplied by a function [latex]f[\/latex] is the same as the constant multiplied by the derivative: [latex]\\frac{d}{dx}(cf(x))=cf^{\\prime}(x)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739289342\" class=\"definition\">\r\n<dt>constant rule<\/dt>\r\n<dd id=\"fs-id1169739289348\">the derivative of a constant function is zero: [latex]\\frac{d}{dx}(c)=0[\/latex], where [latex]c[\/latex] is a constant<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739111084\" class=\"definition\">\r\n<dt>derivative<\/dt>\r\n<dd id=\"fs-id1169739111089\">the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738071113\" class=\"definition\">\r\n<dt>derivative function<\/dt>\r\n<dd id=\"fs-id1169738071118\">gives the derivative of a function at each point in the domain of the original function for which the derivative is defined<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>\r\ndifference quotient<\/dt>\r\n<dd id=\"fs-id1169739111100\">\r\n<p>of a function [latex]f(x)[\/latex] at [latex]a[\/latex] is given by<\/p>\r\n<p>[latex]\\dfrac{f(a+h)-f(a)}{h}[\/latex] or [latex]\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/p>\r\n<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739289383\" class=\"definition\">\r\n<dt>difference rule<\/dt>\r\n<dd id=\"fs-id1169739289388\">the derivative of the difference of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the difference of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex]: [latex]\\frac{d}{dx}(f(x)-g(x))=f^{\\prime}(x)-g^{\\prime}(x)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>\r\ndifferentiable at [latex]a[\/latex]<\/dt>\r\n<dd id=\"fs-id1169738071129\">a function for which [latex]f^{\\prime}(a)[\/latex] exists is differentiable at [latex]a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738071160\" class=\"definition\">\r\n<dt>differentiable on [latex]S[\/latex]<\/dt>\r\n<dd id=\"fs-id1169738071165\">a function for which [latex]f^{\\prime}(x)[\/latex] exists for each [latex]x[\/latex] in the open set [latex]S[\/latex] is differentiable on [latex]S[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738071204\" class=\"definition\">\r\n<dt>differentiable function<\/dt>\r\n<dd id=\"fs-id1169738071210\">a function for which [latex]f^{\\prime}(x)[\/latex] exists is a differentiable function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>\r\ndifferentiation<\/dt>\r\n<dd id=\"fs-id1169739111211\">the process of taking a derivative<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169738071234\" class=\"definition\">\r\n<dt>higher-order derivative<\/dt>\r\n<dd id=\"fs-id1169738071240\">a derivative of a derivative, from the second derivative to the [latex]n[\/latex]th derivative, is called a higher-order derivative<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739111216\" class=\"definition\">\r\n<dt>instantaneous rate of change<\/dt>\r\n<dd id=\"fs-id1169736619746\">the rate of change of a function at any point along the function [latex]a[\/latex], also called [latex]f^{\\prime}(a)[\/latex], or the derivative of the function at [latex]a[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>\r\nmarginal cost<\/dt>\r\n<dd id=\"fs-id1169739190090\">is the derivative of the cost function, or the approximate cost of producing one more item<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739190095\" class=\"definition\">\r\n<dt>marginal revenue<\/dt>\r\n<dd id=\"fs-id1169739190100\">is the derivative of the revenue function, or the approximate revenue obtained by selling one more item<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739190106\" class=\"definition\">\r\n<dt>marginal profit<\/dt>\r\n<dd id=\"fs-id1169739190111\">is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169739190116\" class=\"definition\">\r\n<dt>population growth rate<\/dt>\r\n<dd id=\"fs-id1169739190122\">is the derivative of the population with respect to time<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736592414\" class=\"definition\">\r\n<dt>power rule<\/dt>\r\n<dd id=\"fs-id1169736592419\">the derivative of a power function is a function in which the power on [latex]x[\/latex] becomes the coefficient of the term and the power on [latex]x[\/latex] in the derivative decreases by 1: If [latex]n[\/latex] is an integer, then [latex]\\frac{d}{dx}(x^n)=nx^{n-1}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736592467\" class=\"definition\">\r\n<dt>product rule<\/dt>\r\n<dd id=\"fs-id1169736592472\">the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function: [latex]\\frac{d}{dx}(f(x)g(x))=f^{\\prime}(x)g(x)+g^{\\prime}(x)f(x)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736592566\" class=\"definition\">\r\n<dt>quotient rule<\/dt>\r\n<dd id=\"fs-id1169736592572\">the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function: [latex]\\frac{d}{dx}\\left(\\dfrac{f(x)}{g(x)}\\right)=\\dfrac{f^{\\prime}(x)g(x)-g^{\\prime}(x)f(x)}{(g(x))^2}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>\r\nspeed<\/dt>\r\n<dd>is the absolute value of velocity, that is, [latex]|v(t)|[\/latex] is the speed of an object at time [latex]t[\/latex] whose velocity is given by [latex]v(t)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1169736661322\" class=\"definition\">\r\n<dt>sum rule<\/dt>\r\n<dd id=\"fs-id1169736661328\">the derivative of the sum of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the sum of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex]: [latex]\\frac{d}{dx}(f(x)+g(x))=f^{\\prime}(x)+g^{\\prime}(x)[\/latex]<\/dd>\r\n<\/dl>\r\n<h2>Study Tips<\/h2>\r\n<p><strong>Tangent Lines<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice calculating difference quotients for various functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize the transition from secant lines to the tangent line.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that the tangent line represents the instantaneous rate of change.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When finding equations of tangent lines, always identify a point and calculate the slope.<\/li>\r\n<\/ul>\r\n<p><strong>The Derivative of a Function at a Point<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice computing derivatives using both definitions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Create tables of difference quotients to estimate derivatives.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Relate the concept of derivative to the slope of the tangent line.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that the derivative represents an instantaneous rate of change.<\/li>\r\n<\/ul>\r\n<p><strong>Velocities and Rates of Change<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Use tables of values to estimate instantaneous velocities.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Compare and contrast average velocity and instantaneous velocity.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that instantaneous velocity is just one example of an instantaneous rate of change.<\/li>\r\n<\/ul>\r\n<p><strong>Derivative Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Familiarize yourself with different notations for derivatives.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember common algebraic techniques like factoring and using conjugates.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize the connection between the derivative at a point and the derivative function.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice interpreting the meaning of differentiability in terms of the function's graph.<\/li>\r\n<\/ul>\r\n<p><strong>Graphing a Derivative<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice sketching [latex]f'(x)[\/latex] given the graph of [latex]f(x)[\/latex] and vice versa.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pay attention to where [latex]f(x)[\/latex] is increasing, decreasing, or has horizontal tangents.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Look for points where the concavity of [latex]f(x)[\/latex] changes.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that the [latex]y[\/latex]-value of [latex]f'(x)[\/latex] represents the slope of [latex]f(x)[\/latex] at that [latex]x[\/latex]-value.<\/li>\r\n<\/ul>\r\n<p><strong>Derivatives and Continuity<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">A function can be continuous at a point but fail to be differentiable there.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Graphically, non-differentiable points often appear as \"corners,\" \"cusps,\" or \"jumps\" in the function.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When joining functions, ensure both continuity and matching derivatives at the transition point for smoothness.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The limit definition of the derivative can be used to check differentiability when other methods are unclear.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">\u00a0<\/li>\r\n<\/ul>\r\n<p><strong>Higher-Order Derivatives<\/strong><\/p>\r\n<ul>\r\n\t<li>Familiarize yourself with different notations for higher-order derivatives.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that not all functions have derivatives of all orders.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Look for patterns in successive derivatives of common functions.<\/li>\r\n<\/ul>\r\n<p><strong>The Basic Rules<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice applying these rules to various functions, starting with simple ones and progressing to more complex combinations.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that the Constant Rule means horizontal lines have a slope of zero.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When using the Power Rule, the exponent becomes the coefficient, and the new exponent decreases by [latex]1[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For the Sum and Difference Rules, differentiate each term separately and then combine.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">The Constant Multiple Rule allows you to factor out constants before differentiating.<\/li>\r\n<\/ul>\r\n<p><strong>The Advanced Rules<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying when to use each rule based on the function's structure.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember the order of terms in the quotient rule: \"derivative of top times bottom minus derivative of bottom times top, all over bottom squared.\"<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For functions with negative exponents, consider rewriting them before applying the extended power rule.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Look for opportunities to simplify before and after applying any of the rules.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When using the product rule, the order of terms doesn't matter due to the commutative property of addition.<\/li>\r\n<\/ul>\r\n<p><strong>Combining Differentiation Rules<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying which rules to apply based on the function's structure<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When dealing with rational functions, apply the quotient rule first, then handle the numerator and denominator separately<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For problems involving horizontal tangent lines, set the derivative equal to zero and solve<\/li>\r\n<\/ul>\r\n<p><strong>Amount of Change Formula<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice estimating function values for small [latex]h[\/latex] values.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Compare estimates to actual function values to understand accuracy<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that larger [latex]h[\/latex] values generally lead to less accurate estimates<\/li>\r\n<\/ul>\r\n<p><strong>Rate of Change Applications<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice interpreting derivatives in context:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For motion problems, remember that velocity is the derivative of position, and acceleration is the derivative of velocity.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">In economics, think of marginal functions as the rate of change of the corresponding total function.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Sketch graphs to visualize problems:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">For motion problems, plot position vs. time and velocity vs. time graphs.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">For economic problems, draw cost, revenue, and profit curves.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Develop a systematic approach to word problems:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Identify the variable(s) and what they represent.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Determine which function you're working with (position, population, cost, etc.).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Decide what information the problem is asking for (rate of change, maximum\/minimum, etc.).<\/li>\r\n<\/ul>\r\n<\/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_+Introduction+to+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>Defining the Derivative<\/strong><\/p>\n<ul id=\"fs-id1169736611561\">\n<li>The slope of the tangent line to a curve measures the instantaneous rate of change of a curve. We can calculate it by finding the limit of the difference quotient or the difference quotient with increment [latex]h[\/latex].<\/li>\n<li>The derivative of a function [latex]f(x)[\/latex] at a value [latex]a[\/latex] is found using either of the definitions for the slope of the tangent line.<\/li>\n<li>Velocity is the rate of change of position. As such, the velocity [latex]v(t)[\/latex] at time [latex]t[\/latex] is the derivative of the position [latex]s(t)[\/latex] at time [latex]t[\/latex].\n<ul>\n<li>Average velocity is given by\n<div id=\"fs-id1169739286403\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]v_{\\text{avg}}=\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/div>\n<\/li>\n<li>\n<div class=\"equation unnumbered\" style=\"text-align: left;\">Instantaneous velocity is given by<\/div>\n<div id=\"fs-id1169739191149\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]v(a)=s^{\\prime}(a)=\\underset{t\\to a}{\\lim}\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<li>We may estimate a derivative by using a table of values.<\/li>\n<\/ul>\n<p><strong>The Derivative as a Function<\/strong><\/p>\n<ul id=\"fs-id1169737935224\">\n<li>The derivative of a function [latex]f(x)[\/latex] is the function whose value at [latex]x[\/latex] is [latex]f^{\\prime}(x)[\/latex].<\/li>\n<li>The graph of a derivative of a function [latex]f(x)[\/latex] is related to the graph of [latex]f(x)[\/latex]. Where [latex]f(x)[\/latex] has a tangent line with positive slope, [latex]f^{\\prime}(x)>0[\/latex]. Where [latex]f(x)[\/latex] has a tangent line with negative slope, [latex]f^{\\prime}(x)<0[\/latex]. Where [latex]f(x)[\/latex] has a horizontal tangent line, [latex]f^{\\prime}(x)=0[\/latex].<\/li>\n<li>If a function is differentiable at a point, then it is continuous at that point. A function is not differentiable at a point if it is not continuous at the point, if it has a vertical tangent line at the point, or if the graph has a sharp corner or cusp.<\/li>\n<li>Higher-order derivatives are derivatives of derivatives, from the second derivative to the [latex]n\\text{th}[\/latex] derivative.<\/li>\n<\/ul>\n<p><strong>Differentiation Rules<\/strong><\/p>\n<ul id=\"fs-id1169736659069\">\n<li>The derivative of a constant function is zero.<\/li>\n<li>The derivative of a power function is a function in which the power on [latex]x[\/latex] becomes the coefficient of the term and the power on [latex]x[\/latex] in the derivative decreases by 1.<\/li>\n<li>The derivative of a constant [latex]c[\/latex] multiplied by a function [latex]f[\/latex] is the same as the constant multiplied by the derivative.<\/li>\n<li>The derivative of the sum of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the sum of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex]<em>.<\/em><\/li>\n<li>The derivative of the difference of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the difference of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex]<em>.<\/em><\/li>\n<li>The derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function.<\/li>\n<li>The derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function.<\/li>\n<li>We used the limit definition of the derivative to develop formulas that allow us to find derivatives without resorting to the definition of the derivative. These formulas can be used singly or in combination with each other.<\/li>\n<\/ul>\n<p>\n<strong>Derivatives as Rates of Change<\/strong><\/p>\n<ul id=\"fs-id1169739269667\">\n<li>Using [latex]f(a+h)\\approx f(a)+f^{\\prime}(a)h[\/latex], it is possible to estimate [latex]f(a+h)[\/latex] given [latex]f^{\\prime}(a)[\/latex] and [latex]f(a)[\/latex].<\/li>\n<li>The rate of change of position is velocity, and the rate of change of velocity is acceleration. Speed is the absolute value, or magnitude, of velocity.<\/li>\n<li>The population growth rate and the present population can be used to predict the size of a future population.<\/li>\n<li>Marginal cost, marginal revenue, and marginal profit functions can be used to predict, respectively, the cost of producing one more item, the revenue obtained by selling one more item, and the profit obtained by producing and selling one more item.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169739274323\">\n<li><strong>Difference quotient<\/strong><br \/>\n[latex]Q=\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/li>\n<li><strong>Difference quotient with increment\u00a0[latex]h[\/latex]<\/strong><br \/>\n[latex]Q=\\dfrac{f(a+h)-f(a)}{a+h-a}=\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/li>\n<li><strong>Slope of tangent line<\/strong><br \/>\n[latex]m_{\\tan}=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex]<br \/>\n[latex]m_{\\tan}=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/li>\n<li><strong>Derivative of [latex]f(x)[\/latex] at [latex]a[\/latex]<\/strong><br \/>\n[latex]f^{\\prime}(a)=\\underset{x\\to a}{\\lim}\\dfrac{f(x)-f(a)}{x-a}[\/latex] <br \/>\n[latex]f^{\\prime}(a)=\\underset{h\\to 0}{\\lim}\\dfrac{f(a+h)-f(a)}{h}[\/latex]<\/li>\n<li><strong>Average velocity<br \/>\n<\/strong>[latex]v_{\\text{avg}}=\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/li>\n<li><strong>Instantaneous velocity<\/strong><br \/>\n[latex]v(a)=s^{\\prime}(a)=\\underset{t\\to a}{\\lim}\\dfrac{s(t)-s(a)}{t-a}[\/latex]<\/li>\n<li><strong>The derivative function<\/strong><br \/>\n[latex]f^{\\prime}(x)=\\underset{h\\to 0}{\\lim}\\dfrac{f(x+h)-f(x)}{h}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1169739252840\" class=\"definition\">\n<dt>acceleration<\/dt>\n<dd id=\"fs-id1169739252845\">is the rate of change of the velocity, that is, the derivative of velocity<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739252849\" class=\"definition\">\n<dt>amount of change<\/dt>\n<dd id=\"fs-id1169739252854\">the amount of a function [latex]f(x)[\/latex] over an interval [latex][x,x+h][\/latex] is [latex]f(x+h)-f(x)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739190003\" class=\"definition\">\n<dt>average rate of change<\/dt>\n<dd id=\"fs-id1169739190009\">is a function [latex]f(x)[\/latex] over an interval [latex][x,x+h][\/latex] is [latex]\\dfrac{f(x+h)-f(a)}{b-a}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739289276\" class=\"definition\">\n<dt>constant multiple rule<\/dt>\n<dd id=\"fs-id1169739289282\">the derivative of a constant [latex]c[\/latex] multiplied by a function [latex]f[\/latex] is the same as the constant multiplied by the derivative: [latex]\\frac{d}{dx}(cf(x))=cf^{\\prime}(x)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739289342\" class=\"definition\">\n<dt>constant rule<\/dt>\n<dd id=\"fs-id1169739289348\">the derivative of a constant function is zero: [latex]\\frac{d}{dx}(c)=0[\/latex], where [latex]c[\/latex] is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739111084\" class=\"definition\">\n<dt>derivative<\/dt>\n<dd id=\"fs-id1169739111089\">the slope of the tangent line to a function at a point, calculated by taking the limit of the difference quotient, is the derivative<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738071113\" class=\"definition\">\n<dt>derivative function<\/dt>\n<dd id=\"fs-id1169738071118\">gives the derivative of a function at each point in the domain of the original function for which the derivative is defined<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>\ndifference quotient<\/dt>\n<dd id=\"fs-id1169739111100\">\n<p>of a function [latex]f(x)[\/latex] at [latex]a[\/latex] is given by<\/p>\n<p>[latex]\\dfrac{f(a+h)-f(a)}{h}[\/latex] or [latex]\\dfrac{f(x)-f(a)}{x-a}[\/latex]<\/p>\n<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739289383\" class=\"definition\">\n<dt>difference rule<\/dt>\n<dd id=\"fs-id1169739289388\">the derivative of the difference of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the difference of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex]: [latex]\\frac{d}{dx}(f(x)-g(x))=f^{\\prime}(x)-g^{\\prime}(x)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>\ndifferentiable at [latex]a[\/latex]<\/dt>\n<dd id=\"fs-id1169738071129\">a function for which [latex]f^{\\prime}(a)[\/latex] exists is differentiable at [latex]a[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738071160\" class=\"definition\">\n<dt>differentiable on [latex]S[\/latex]<\/dt>\n<dd id=\"fs-id1169738071165\">a function for which [latex]f^{\\prime}(x)[\/latex] exists for each [latex]x[\/latex] in the open set [latex]S[\/latex] is differentiable on [latex]S[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738071204\" class=\"definition\">\n<dt>differentiable function<\/dt>\n<dd id=\"fs-id1169738071210\">a function for which [latex]f^{\\prime}(x)[\/latex] exists is a differentiable function<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>\ndifferentiation<\/dt>\n<dd id=\"fs-id1169739111211\">the process of taking a derivative<\/dd>\n<\/dl>\n<dl id=\"fs-id1169738071234\" class=\"definition\">\n<dt>higher-order derivative<\/dt>\n<dd id=\"fs-id1169738071240\">a derivative of a derivative, from the second derivative to the [latex]n[\/latex]th derivative, is called a higher-order derivative<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739111216\" class=\"definition\">\n<dt>instantaneous rate of change<\/dt>\n<dd id=\"fs-id1169736619746\">the rate of change of a function at any point along the function [latex]a[\/latex], also called [latex]f^{\\prime}(a)[\/latex], or the derivative of the function at [latex]a[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>\nmarginal cost<\/dt>\n<dd id=\"fs-id1169739190090\">is the derivative of the cost function, or the approximate cost of producing one more item<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739190095\" class=\"definition\">\n<dt>marginal revenue<\/dt>\n<dd id=\"fs-id1169739190100\">is the derivative of the revenue function, or the approximate revenue obtained by selling one more item<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739190106\" class=\"definition\">\n<dt>marginal profit<\/dt>\n<dd id=\"fs-id1169739190111\">is the derivative of the profit function, or the approximate profit obtained by producing and selling one more item<\/dd>\n<\/dl>\n<dl id=\"fs-id1169739190116\" class=\"definition\">\n<dt>population growth rate<\/dt>\n<dd id=\"fs-id1169739190122\">is the derivative of the population with respect to time<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736592414\" class=\"definition\">\n<dt>power rule<\/dt>\n<dd id=\"fs-id1169736592419\">the derivative of a power function is a function in which the power on [latex]x[\/latex] becomes the coefficient of the term and the power on [latex]x[\/latex] in the derivative decreases by 1: If [latex]n[\/latex] is an integer, then [latex]\\frac{d}{dx}(x^n)=nx^{n-1}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736592467\" class=\"definition\">\n<dt>product rule<\/dt>\n<dd id=\"fs-id1169736592472\">the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function: [latex]\\frac{d}{dx}(f(x)g(x))=f^{\\prime}(x)g(x)+g^{\\prime}(x)f(x)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736592566\" class=\"definition\">\n<dt>quotient rule<\/dt>\n<dd id=\"fs-id1169736592572\">the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function: [latex]\\frac{d}{dx}\\left(\\dfrac{f(x)}{g(x)}\\right)=\\dfrac{f^{\\prime}(x)g(x)-g^{\\prime}(x)f(x)}{(g(x))^2}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>\nspeed<\/dt>\n<dd>is the absolute value of velocity, that is, [latex]|v(t)|[\/latex] is the speed of an object at time [latex]t[\/latex] whose velocity is given by [latex]v(t)[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1169736661322\" class=\"definition\">\n<dt>sum rule<\/dt>\n<dd id=\"fs-id1169736661328\">the derivative of the sum of a function [latex]f[\/latex] and a function [latex]g[\/latex] is the same as the sum of the derivative of [latex]f[\/latex] and the derivative of [latex]g[\/latex]: [latex]\\frac{d}{dx}(f(x)+g(x))=f^{\\prime}(x)+g^{\\prime}(x)[\/latex]<\/dd>\n<\/dl>\n<h2>Study Tips<\/h2>\n<p><strong>Tangent Lines<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice calculating difference quotients for various functions.<\/li>\n<li class=\"whitespace-normal break-words\">Visualize the transition from secant lines to the tangent line.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the tangent line represents the instantaneous rate of change.<\/li>\n<li class=\"whitespace-normal break-words\">When finding equations of tangent lines, always identify a point and calculate the slope.<\/li>\n<\/ul>\n<p><strong>The Derivative of a Function at a Point<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice computing derivatives using both definitions.<\/li>\n<li class=\"whitespace-normal break-words\">Create tables of difference quotients to estimate derivatives.<\/li>\n<li class=\"whitespace-normal break-words\">Relate the concept of derivative to the slope of the tangent line.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the derivative represents an instantaneous rate of change.<\/li>\n<\/ul>\n<p><strong>Velocities and Rates of Change<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Use tables of values to estimate instantaneous velocities.<\/li>\n<li class=\"whitespace-normal break-words\">Compare and contrast average velocity and instantaneous velocity.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that instantaneous velocity is just one example of an instantaneous rate of change.<\/li>\n<\/ul>\n<p><strong>Derivative Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Familiarize yourself with different notations for derivatives.<\/li>\n<li class=\"whitespace-normal break-words\">Remember common algebraic techniques like factoring and using conjugates.<\/li>\n<li class=\"whitespace-normal break-words\">Visualize the connection between the derivative at a point and the derivative function.<\/li>\n<li class=\"whitespace-normal break-words\">Practice interpreting the meaning of differentiability in terms of the function&#8217;s graph.<\/li>\n<\/ul>\n<p><strong>Graphing a Derivative<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice sketching [latex]f'(x)[\/latex] given the graph of [latex]f(x)[\/latex] and vice versa.<\/li>\n<li class=\"whitespace-normal break-words\">Pay attention to where [latex]f(x)[\/latex] is increasing, decreasing, or has horizontal tangents.<\/li>\n<li class=\"whitespace-normal break-words\">Look for points where the concavity of [latex]f(x)[\/latex] changes.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the [latex]y[\/latex]-value of [latex]f'(x)[\/latex] represents the slope of [latex]f(x)[\/latex] at that [latex]x[\/latex]-value.<\/li>\n<\/ul>\n<p><strong>Derivatives and Continuity<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">A function can be continuous at a point but fail to be differentiable there.<\/li>\n<li class=\"whitespace-normal break-words\">Graphically, non-differentiable points often appear as &#8220;corners,&#8221; &#8220;cusps,&#8221; or &#8220;jumps&#8221; in the function.<\/li>\n<li class=\"whitespace-normal break-words\">When joining functions, ensure both continuity and matching derivatives at the transition point for smoothness.<\/li>\n<li class=\"whitespace-normal break-words\">The limit definition of the derivative can be used to check differentiability when other methods are unclear.<\/li>\n<li class=\"whitespace-normal break-words\">\u00a0<\/li>\n<\/ul>\n<p><strong>Higher-Order Derivatives<\/strong><\/p>\n<ul>\n<li>Familiarize yourself with different notations for higher-order derivatives.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that not all functions have derivatives of all orders.<\/li>\n<li class=\"whitespace-normal break-words\">Look for patterns in successive derivatives of common functions.<\/li>\n<\/ul>\n<p><strong>The Basic Rules<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice applying these rules to various functions, starting with simple ones and progressing to more complex combinations.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that the Constant Rule means horizontal lines have a slope of zero.<\/li>\n<li class=\"whitespace-normal break-words\">When using the Power Rule, the exponent becomes the coefficient, and the new exponent decreases by [latex]1[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">For the Sum and Difference Rules, differentiate each term separately and then combine.<\/li>\n<li class=\"whitespace-normal break-words\">The Constant Multiple Rule allows you to factor out constants before differentiating.<\/li>\n<\/ul>\n<p><strong>The Advanced Rules<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying when to use each rule based on the function&#8217;s structure.<\/li>\n<li class=\"whitespace-normal break-words\">Remember the order of terms in the quotient rule: &#8220;derivative of top times bottom minus derivative of bottom times top, all over bottom squared.&#8221;<\/li>\n<li class=\"whitespace-normal break-words\">For functions with negative exponents, consider rewriting them before applying the extended power rule.<\/li>\n<li class=\"whitespace-normal break-words\">Look for opportunities to simplify before and after applying any of the rules.<\/li>\n<li class=\"whitespace-normal break-words\">When using the product rule, the order of terms doesn&#8217;t matter due to the commutative property of addition.<\/li>\n<\/ul>\n<p><strong>Combining Differentiation Rules<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying which rules to apply based on the function&#8217;s structure<\/li>\n<li class=\"whitespace-normal break-words\">When dealing with rational functions, apply the quotient rule first, then handle the numerator and denominator separately<\/li>\n<li class=\"whitespace-normal break-words\">For problems involving horizontal tangent lines, set the derivative equal to zero and solve<\/li>\n<\/ul>\n<p><strong>Amount of Change Formula<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice estimating function values for small [latex]h[\/latex] values.<\/li>\n<li class=\"whitespace-normal break-words\">Compare estimates to actual function values to understand accuracy<\/li>\n<li class=\"whitespace-normal break-words\">Remember that larger [latex]h[\/latex] values generally lead to less accurate estimates<\/li>\n<\/ul>\n<p><strong>Rate of Change Applications<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice interpreting derivatives in context:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For motion problems, remember that velocity is the derivative of position, and acceleration is the derivative of velocity.<\/li>\n<li class=\"whitespace-normal break-words\">In economics, think of marginal functions as the rate of change of the corresponding total function.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Sketch graphs to visualize problems:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For motion problems, plot position vs. time and velocity vs. time graphs.<\/li>\n<li class=\"whitespace-normal break-words\">For economic problems, draw cost, revenue, and profit curves.<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Develop a systematic approach to word problems:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the variable(s) and what they represent.<\/li>\n<li class=\"whitespace-normal break-words\">Determine which function you&#8217;re working with (position, population, cost, etc.).<\/li>\n<li class=\"whitespace-normal break-words\">Decide what information the problem is asking for (rate of change, maximum\/minimum, etc.).<\/li>\n<\/ul>\n<\/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":503,"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\/3593"}],"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\/3593\/revisions"}],"predecessor-version":[{"id":4074,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3593\/revisions\/4074"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/503"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3593\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3593"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3593"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3593"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3593"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}