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<!DOCTYPE html>
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<title>6.5 Average Value of a Function | MATH 112: Differential Calculus</title>
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<li><a href="./">M112: Differential Calculus</a></li>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>Welcome</a></li>
<li class="chapter" data-level="1" data-path="functions-and-models.html"><a href="functions-and-models.html"><i class="fa fa-check"></i><b>1</b> Functions and Models</a>
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<li class="chapter" data-level="1.1" data-path="four-ways-to-represent-a-function.html"><a href="four-ways-to-represent-a-function.html"><i class="fa fa-check"></i><b>1.1</b> Four Ways to Represent a Function</a>
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<li class="chapter" data-level="1.1.1" data-path="four-ways-to-represent-a-function.html"><a href="four-ways-to-represent-a-function.html#four-representations-of-a-function"><i class="fa fa-check"></i><b>1.1.1</b> Four Representations of a Function</a></li>
<li class="chapter" data-level="1.1.2" data-path="four-ways-to-represent-a-function.html"><a href="four-ways-to-represent-a-function.html#piecewise-defined-functions"><i class="fa fa-check"></i><b>1.1.2</b> Piecewise-Defined Functions</a></li>
<li class="chapter" data-level="1.1.3" data-path="four-ways-to-represent-a-function.html"><a href="four-ways-to-represent-a-function.html#symmetry"><i class="fa fa-check"></i><b>1.1.3</b> Symmetry</a></li>
<li class="chapter" data-level="1.1.4" data-path="four-ways-to-represent-a-function.html"><a href="four-ways-to-represent-a-function.html#increasing-and-decreasing-functions"><i class="fa fa-check"></i><b>1.1.4</b> Increasing and Decreasing Functions</a></li>
<li class="chapter" data-level="1.1.5" data-path="four-ways-to-represent-a-function.html"><a href="four-ways-to-represent-a-function.html#putting-it-all-together"><i class="fa fa-check"></i><b>1.1.5</b> Putting it All Together</a></li>
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<li class="chapter" data-level="1.2" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html"><i class="fa fa-check"></i><b>1.2</b> Mathematical Models: A Catalog of Essential Functions</a>
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<li class="chapter" data-level="1.2.1" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#linear-models"><i class="fa fa-check"></i><b>1.2.1</b> Linear Models</a></li>
<li class="chapter" data-level="1.2.2" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#polynomial-functions"><i class="fa fa-check"></i><b>1.2.2</b> Polynomial Functions</a></li>
<li class="chapter" data-level="1.2.3" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#power-functions"><i class="fa fa-check"></i><b>1.2.3</b> Power Functions</a></li>
<li class="chapter" data-level="1.2.4" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#rational-functions"><i class="fa fa-check"></i><b>1.2.4</b> Rational Functions</a></li>
<li class="chapter" data-level="1.2.5" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#algebraic-functions"><i class="fa fa-check"></i><b>1.2.5</b> Algebraic Functions</a></li>
<li class="chapter" data-level="1.2.6" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#trigonometric-functions"><i class="fa fa-check"></i><b>1.2.6</b> Trigonometric Functions</a></li>
<li class="chapter" data-level="1.2.7" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#exponential-functions"><i class="fa fa-check"></i><b>1.2.7</b> Exponential Functions</a></li>
<li class="chapter" data-level="1.2.8" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#logarithmic-functions"><i class="fa fa-check"></i><b>1.2.8</b> Logarithmic Functions</a></li>
<li class="chapter" data-level="1.2.9" data-path="mathematical-models-a-catalog-of-essential-functions.html"><a href="mathematical-models-a-catalog-of-essential-functions.html#putting-it-all-together-1"><i class="fa fa-check"></i><b>1.2.9</b> Putting it All Together</a></li>
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<li class="chapter" data-level="1.3" data-path="new-functions-from-old-functions.html"><a href="new-functions-from-old-functions.html"><i class="fa fa-check"></i><b>1.3</b> New Functions from Old Functions</a>
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<li class="chapter" data-level="1.3.1" data-path="new-functions-from-old-functions.html"><a href="new-functions-from-old-functions.html#transformations-of-functions"><i class="fa fa-check"></i><b>1.3.1</b> Transformations of Functions</a></li>
<li class="chapter" data-level="1.3.2" data-path="new-functions-from-old-functions.html"><a href="new-functions-from-old-functions.html#combinations-of-functions"><i class="fa fa-check"></i><b>1.3.2</b> Combinations of Functions</a></li>
<li class="chapter" data-level="1.3.3" data-path="new-functions-from-old-functions.html"><a href="new-functions-from-old-functions.html#putting-it-all-together-2"><i class="fa fa-check"></i><b>1.3.3</b> Putting It All Together</a></li>
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<li class="chapter" data-level="1.4" data-path="exponential-functions-1.html"><a href="exponential-functions-1.html"><i class="fa fa-check"></i><b>1.4</b> Exponential Functions</a>
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<li class="chapter" data-level="1.4.1" data-path="exponential-functions-1.html"><a href="exponential-functions-1.html#graphical-behavior-of-y-bx"><i class="fa fa-check"></i><b>1.4.1</b> Graphical Behavior of <span class="math inline">\(y = b^x\)</span></a></li>
<li class="chapter" data-level="1.4.2" data-path="exponential-functions-1.html"><a href="exponential-functions-1.html#applications-of-exponential-functions"><i class="fa fa-check"></i><b>1.4.2</b> Applications of Exponential Functions</a></li>
<li class="chapter" data-level="1.4.3" data-path="exponential-functions-1.html"><a href="exponential-functions-1.html#the-number-e-and-the-natural-exponential-function"><i class="fa fa-check"></i><b>1.4.3</b> The Number <span class="math inline">\(e\)</span> and the Natural Exponential Function</a></li>
<li class="chapter" data-level="1.4.4" data-path="exponential-functions-1.html"><a href="exponential-functions-1.html#graph-transformations-of-exponentials"><i class="fa fa-check"></i><b>1.4.4</b> Graph Transformations of Exponentials</a></li>
<li class="chapter" data-level="1.4.5" data-path="exponential-functions-1.html"><a href="exponential-functions-1.html#putting-it-all-together-3"><i class="fa fa-check"></i><b>1.4.5</b> Putting It All Together</a></li>
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<li class="chapter" data-level="1.5" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html"><i class="fa fa-check"></i><b>1.5</b> Inverse Functions and Logarithms</a>
<ul>
<li class="chapter" data-level="1.5.1" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html#graphs-of-inverse-functions"><i class="fa fa-check"></i><b>1.5.1</b> Graphs of Inverse Functions</a></li>
<li class="chapter" data-level="1.5.2" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html#logarithmic-functions-as-inverses"><i class="fa fa-check"></i><b>1.5.2</b> Logarithmic Functions as Inverses</a></li>
<li class="chapter" data-level="1.5.3" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html#solving-exponential-and-log-equations"><i class="fa fa-check"></i><b>1.5.3</b> Solving Exponential and Log Equations</a></li>
<li class="chapter" data-level="1.5.4" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html#inverse-trigonometric-functions"><i class="fa fa-check"></i><b>1.5.4</b> Inverse Trigonometric Functions</a></li>
<li class="chapter" data-level="1.5.5" data-path="inverse-functions-and-logarithms.html"><a href="inverse-functions-and-logarithms.html#pulling-it-all-together"><i class="fa fa-check"></i><b>1.5.5</b> Pulling It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="2" data-path="limits-and-derivatives.html"><a href="limits-and-derivatives.html"><i class="fa fa-check"></i><b>2</b> Limits and Derivatives</a>
<ul>
<li class="chapter" data-level="2.1" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html"><i class="fa fa-check"></i><b>2.1</b> The Tangent and Velocity Problems</a>
<ul>
<li class="chapter" data-level="2.1.1" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#the-tangent-problem"><i class="fa fa-check"></i><b>2.1.1</b> The Tangent Problem</a></li>
<li class="chapter" data-level="2.1.2" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#tangents-from-experimental-data"><i class="fa fa-check"></i><b>2.1.2</b> Tangents from Experimental Data</a></li>
<li class="chapter" data-level="2.1.3" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#the-velocity-problem"><i class="fa fa-check"></i><b>2.1.3</b> The Velocity Problem</a></li>
<li class="chapter" data-level="2.1.4" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#connection-between-tangents-and-velocity"><i class="fa fa-check"></i><b>2.1.4</b> Connection Between Tangents and Velocity</a></li>
<li class="chapter" data-level="2.1.5" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#putting-it-all-together-4"><i class="fa fa-check"></i><b>2.1.5</b> Putting It All Together</a></li>
<li class="chapter" data-level="2.1.6" data-path="the-tangent-and-velocity-problems.html"><a href="the-tangent-and-velocity-problems.html#conceptual-takeaways-5"><i class="fa fa-check"></i><b>2.1.6</b> Conceptual Takeaways</a></li>
</ul></li>
<li class="chapter" data-level="2.2" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html"><i class="fa fa-check"></i><b>2.2</b> The Limit of a Function</a>
<ul>
<li class="chapter" data-level="2.2.1" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#intuitive-idea-of-a-limit"><i class="fa fa-check"></i><b>2.2.1</b> Intuitive Idea of a Limit</a></li>
<li class="chapter" data-level="2.2.2" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#numerical-approach-to-limits"><i class="fa fa-check"></i><b>2.2.2</b> Numerical Approach to Limits</a></li>
<li class="chapter" data-level="2.2.3" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#one-sided-limits"><i class="fa fa-check"></i><b>2.2.3</b> One-Sided Limits</a></li>
<li class="chapter" data-level="2.2.4" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#infinite-limits"><i class="fa fa-check"></i><b>2.2.4</b> Infinite Limits</a></li>
<li class="chapter" data-level="2.2.5" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#vertical-asymptotes"><i class="fa fa-check"></i><b>2.2.5</b> Vertical Asymptotes</a></li>
<li class="chapter" data-level="2.2.6" data-path="the-limit-of-a-function.html"><a href="the-limit-of-a-function.html#putting-it-all-together-5"><i class="fa fa-check"></i><b>2.2.6</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="2.3" data-path="calculating-limits-using-the-limit-laws.html"><a href="calculating-limits-using-the-limit-laws.html"><i class="fa fa-check"></i><b>2.3</b> Calculating Limits Using the Limit Laws</a>
<ul>
<li class="chapter" data-level="2.3.1" data-path="calculating-limits-using-the-limit-laws.html"><a href="calculating-limits-using-the-limit-laws.html#limit-laws"><i class="fa fa-check"></i><b>2.3.1</b> Limit Laws</a></li>
<li class="chapter" data-level="2.3.2" data-path="calculating-limits-using-the-limit-laws.html"><a href="calculating-limits-using-the-limit-laws.html#the-squeeze-theorem-sandwich-theorem"><i class="fa fa-check"></i><b>2.3.2</b> The Squeeze Theorem (Sandwich Theorem)</a></li>
<li class="chapter" data-level="2.3.3" data-path="calculating-limits-using-the-limit-laws.html"><a href="calculating-limits-using-the-limit-laws.html#putting-it-all-together-6"><i class="fa fa-check"></i><b>2.3.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="2.4" data-path="the-precise-definition-of-a-limit.html"><a href="the-precise-definition-of-a-limit.html"><i class="fa fa-check"></i><b>2.4</b> The Precise Definition of a Limit</a></li>
<li class="chapter" data-level="2.5" data-path="continuity.html"><a href="continuity.html"><i class="fa fa-check"></i><b>2.5</b> Continuity</a>
<ul>
<li class="chapter" data-level="2.5.1" data-path="continuity.html"><a href="continuity.html#types-of-discontinuities"><i class="fa fa-check"></i><b>2.5.1</b> Types of Discontinuities</a></li>
<li class="chapter" data-level="2.5.2" data-path="continuity.html"><a href="continuity.html#one-sided-continuity"><i class="fa fa-check"></i><b>2.5.2</b> One-Sided Continuity</a></li>
<li class="chapter" data-level="2.5.3" data-path="continuity.html"><a href="continuity.html#composite-functions"><i class="fa fa-check"></i><b>2.5.3</b> Composite Functions</a></li>
<li class="chapter" data-level="2.5.4" data-path="continuity.html"><a href="continuity.html#intermediate-value-theorem-ivt"><i class="fa fa-check"></i><b>2.5.4</b> Intermediate Value Theorem (IVT)</a></li>
<li class="chapter" data-level="2.5.5" data-path="continuity.html"><a href="continuity.html#putting-it-all-together-7"><i class="fa fa-check"></i><b>2.5.5</b> Putting It All Together</a></li>
<li class="chapter" data-level="2.5.6" data-path="continuity.html"><a href="continuity.html#skills-you-should-be-able-to-do-7"><i class="fa fa-check"></i><b>2.5.6</b> Skills You Should Be Able To Do</a></li>
<li class="chapter" data-level="2.5.7" data-path="continuity.html"><a href="continuity.html#problems-6"><i class="fa fa-check"></i><b>2.5.7</b> Problems</a></li>
</ul></li>
<li class="chapter" data-level="2.6" data-path="limits-at-infinity-horizontal-asymptotes.html"><a href="limits-at-infinity-horizontal-asymptotes.html"><i class="fa fa-check"></i><b>2.6</b> Limits at Infinity; Horizontal Asymptotes</a>
<ul>
<li class="chapter" data-level="2.6.1" data-path="limits-at-infinity-horizontal-asymptotes.html"><a href="limits-at-infinity-horizontal-asymptotes.html#limits-at-infinity"><i class="fa fa-check"></i><b>2.6.1</b> Limits at Infinity</a></li>
<li class="chapter" data-level="2.6.2" data-path="limits-at-infinity-horizontal-asymptotes.html"><a href="limits-at-infinity-horizontal-asymptotes.html#infinite-limits-at-infinity"><i class="fa fa-check"></i><b>2.6.2</b> Infinite Limits at Infinity</a></li>
<li class="chapter" data-level="2.6.3" data-path="limits-at-infinity-horizontal-asymptotes.html"><a href="limits-at-infinity-horizontal-asymptotes.html#graph-interpretation"><i class="fa fa-check"></i><b>2.6.3</b> Graph Interpretation</a></li>
<li class="chapter" data-level="2.6.4" data-path="limits-at-infinity-horizontal-asymptotes.html"><a href="limits-at-infinity-horizontal-asymptotes.html#putting-it-all-together-8"><i class="fa fa-check"></i><b>2.6.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="2.7" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html"><i class="fa fa-check"></i><b>2.7</b> Derivatives and Rates of Change</a>
<ul>
<li class="chapter" data-level="2.7.1" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html#tangent-lines-and-limits"><i class="fa fa-check"></i><b>2.7.1</b> Tangent Lines and Limits</a></li>
<li class="chapter" data-level="2.7.2" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html#derivatives"><i class="fa fa-check"></i><b>2.7.2</b> Derivatives</a></li>
<li class="chapter" data-level="2.7.3" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html#tangent-line-using-derivatives"><i class="fa fa-check"></i><b>2.7.3</b> Tangent Line Using Derivatives</a></li>
<li class="chapter" data-level="2.7.4" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html#rates-of-change"><i class="fa fa-check"></i><b>2.7.4</b> Rates of Change</a></li>
<li class="chapter" data-level="2.7.5" data-path="derivatives-and-rates-of-change.html"><a href="derivatives-and-rates-of-change.html#putting-it-all-together-9"><i class="fa fa-check"></i><b>2.7.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="2.8" data-path="the-derivative-as-a-function.html"><a href="the-derivative-as-a-function.html"><i class="fa fa-check"></i><b>2.8</b> The Derivative as a Function</a>
<ul>
<li class="chapter" data-level="2.8.1" data-path="the-derivative-as-a-function.html"><a href="the-derivative-as-a-function.html#continuity-vs-differentiability"><i class="fa fa-check"></i><b>2.8.1</b> Continuity vs Differentiability</a></li>
<li class="chapter" data-level="2.8.2" data-path="the-derivative-as-a-function.html"><a href="the-derivative-as-a-function.html#higher-derivatives"><i class="fa fa-check"></i><b>2.8.2</b> Higher Derivatives</a></li>
<li class="chapter" data-level="2.8.3" data-path="the-derivative-as-a-function.html"><a href="the-derivative-as-a-function.html#putting-it-all-together-10"><i class="fa fa-check"></i><b>2.8.3</b> Putting It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="3" data-path="differentiation-rules.html"><a href="differentiation-rules.html"><i class="fa fa-check"></i><b>3</b> Differentiation Rules</a>
<ul>
<li class="chapter" data-level="3.1" data-path="derivatives-of-polynomials-and-exponential-functions.html"><a href="derivatives-of-polynomials-and-exponential-functions.html"><i class="fa fa-check"></i><b>3.1</b> Derivatives of Polynomials and Exponential Functions</a>
<ul>
<li class="chapter" data-level="3.1.1" data-path="derivatives-of-polynomials-and-exponential-functions.html"><a href="derivatives-of-polynomials-and-exponential-functions.html#new-derivatives-from-old"><i class="fa fa-check"></i><b>3.1.1</b> New Derivatives from Old</a></li>
<li class="chapter" data-level="3.1.2" data-path="derivatives-of-polynomials-and-exponential-functions.html"><a href="derivatives-of-polynomials-and-exponential-functions.html#exponential-functions-2"><i class="fa fa-check"></i><b>3.1.2</b> Exponential Functions</a></li>
<li class="chapter" data-level="3.1.3" data-path="derivatives-of-polynomials-and-exponential-functions.html"><a href="derivatives-of-polynomials-and-exponential-functions.html#putting-it-all-together-11"><i class="fa fa-check"></i><b>3.1.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.2" data-path="the-product-and-quotient-rules.html"><a href="the-product-and-quotient-rules.html"><i class="fa fa-check"></i><b>3.2</b> The Product and Quotient Rules</a>
<ul>
<li class="chapter" data-level="3.2.1" data-path="the-product-and-quotient-rules.html"><a href="the-product-and-quotient-rules.html#the-product-rule"><i class="fa fa-check"></i><b>3.2.1</b> The Product Rule</a></li>
<li class="chapter" data-level="3.2.2" data-path="the-product-and-quotient-rules.html"><a href="the-product-and-quotient-rules.html#the-quotient-rule"><i class="fa fa-check"></i><b>3.2.2</b> The Quotient Rule</a></li>
<li class="chapter" data-level="3.2.3" data-path="the-product-and-quotient-rules.html"><a href="the-product-and-quotient-rules.html#putting-it-all-together-12"><i class="fa fa-check"></i><b>3.2.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.3" data-path="derivatives-of-trigonometric-functions.html"><a href="derivatives-of-trigonometric-functions.html"><i class="fa fa-check"></i><b>3.3</b> Derivatives of Trigonometric Functions</a>
<ul>
<li class="chapter" data-level="3.3.1" data-path="derivatives-of-trigonometric-functions.html"><a href="derivatives-of-trigonometric-functions.html#putting-it-all-together-13"><i class="fa fa-check"></i><b>3.3.1</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.4" data-path="the-chain-rule.html"><a href="the-chain-rule.html"><i class="fa fa-check"></i><b>3.4</b> The Chain Rule</a>
<ul>
<li class="chapter" data-level="3.4.1" data-path="the-chain-rule.html"><a href="the-chain-rule.html#putting-it-all-together-14"><i class="fa fa-check"></i><b>3.4.1</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.5" data-path="implicit-differentiation.html"><a href="implicit-differentiation.html"><i class="fa fa-check"></i><b>3.5</b> Implicit Differentiation</a>
<ul>
<li class="chapter" data-level="3.5.1" data-path="implicit-differentiation.html"><a href="implicit-differentiation.html#horizontal-tangent-condition"><i class="fa fa-check"></i><b>3.5.1</b> Horizontal Tangent Condition</a></li>
<li class="chapter" data-level="3.5.2" data-path="implicit-differentiation.html"><a href="implicit-differentiation.html#inverse-trigonometric-functions-via-implicit-differentiation"><i class="fa fa-check"></i><b>3.5.2</b> Inverse Trigonometric Functions via Implicit Differentiation</a></li>
<li class="chapter" data-level="3.5.3" data-path="implicit-differentiation.html"><a href="implicit-differentiation.html#putting-it-all-together-15"><i class="fa fa-check"></i><b>3.5.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.6" data-path="derivatives-of-logarithmic-functions.html"><a href="derivatives-of-logarithmic-functions.html"><i class="fa fa-check"></i><b>3.6</b> Derivatives of Logarithmic Functions</a>
<ul>
<li class="chapter" data-level="3.6.1" data-path="derivatives-of-logarithmic-functions.html"><a href="derivatives-of-logarithmic-functions.html#logarithmic-differentiation"><i class="fa fa-check"></i><b>3.6.1</b> Logarithmic Differentiation</a></li>
<li class="chapter" data-level="3.6.2" data-path="derivatives-of-logarithmic-functions.html"><a href="derivatives-of-logarithmic-functions.html#putting-it-all-together-16"><i class="fa fa-check"></i><b>3.6.2</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.7" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html"><i class="fa fa-check"></i><b>3.7</b> Rates of Change in the Natural and Social Sciences</a>
<ul>
<li class="chapter" data-level="3.7.1" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#physics-applications"><i class="fa fa-check"></i><b>3.7.1</b> Physics Applications</a></li>
<li class="chapter" data-level="3.7.2" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#physics-density-and-current"><i class="fa fa-check"></i><b>3.7.2</b> Physics: Density and Current</a></li>
<li class="chapter" data-level="3.7.3" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#electric-current"><i class="fa fa-check"></i><b>3.7.3</b> Electric Current</a></li>
<li class="chapter" data-level="3.7.4" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#chemistry-applications"><i class="fa fa-check"></i><b>3.7.4</b> Chemistry Applications</a></li>
<li class="chapter" data-level="3.7.5" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#biology-applications"><i class="fa fa-check"></i><b>3.7.5</b> Biology Applications</a></li>
<li class="chapter" data-level="3.7.6" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#economics-applications"><i class="fa fa-check"></i><b>3.7.6</b> Economics Applications</a></li>
<li class="chapter" data-level="3.7.7" data-path="rates-of-change-in-the-natural-and-social-sciences.html"><a href="rates-of-change-in-the-natural-and-social-sciences.html#putting-it-all-together-17"><i class="fa fa-check"></i><b>3.7.7</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.8" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html"><i class="fa fa-check"></i><b>3.8</b> Exponential Growth and Decay</a>
<ul>
<li class="chapter" data-level="3.8.1" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html#the-law-of-natural-growth-and-decay"><i class="fa fa-check"></i><b>3.8.1</b> The Law of Natural Growth and Decay</a></li>
<li class="chapter" data-level="3.8.2" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html#population-growth-model"><i class="fa fa-check"></i><b>3.8.2</b> Population Growth Model</a></li>
<li class="chapter" data-level="3.8.3" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html#newtons-law-of-cooling"><i class="fa fa-check"></i><b>3.8.3</b> Newton’s Law of Cooling</a></li>
<li class="chapter" data-level="3.8.4" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html#continuously-compounded-interest"><i class="fa fa-check"></i><b>3.8.4</b> Continuously Compounded Interest</a></li>
<li class="chapter" data-level="3.8.5" data-path="exponential-growth-and-decay.html"><a href="exponential-growth-and-decay.html#putting-it-all-together-18"><i class="fa fa-check"></i><b>3.8.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.9" data-path="related-rates.html"><a href="related-rates.html"><i class="fa fa-check"></i><b>3.9</b> Related Rates</a>
<ul>
<li class="chapter" data-level="3.9.1" data-path="related-rates.html"><a href="related-rates.html#putting-it-all-together-19"><i class="fa fa-check"></i><b>3.9.1</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.10" data-path="linear-approximations-and-differentials.html"><a href="linear-approximations-and-differentials.html"><i class="fa fa-check"></i><b>3.10</b> Linear Approximations and Differentials</a>
<ul>
<li class="chapter" data-level="3.10.1" data-path="linear-approximations-and-differentials.html"><a href="linear-approximations-and-differentials.html#tangent-line-approximation"><i class="fa fa-check"></i><b>3.10.1</b> Tangent Line Approximation</a></li>
<li class="chapter" data-level="3.10.2" data-path="linear-approximations-and-differentials.html"><a href="linear-approximations-and-differentials.html#differentials"><i class="fa fa-check"></i><b>3.10.2</b> Differentials</a></li>
<li class="chapter" data-level="3.10.3" data-path="linear-approximations-and-differentials.html"><a href="linear-approximations-and-differentials.html#putting-it-all-together-20"><i class="fa fa-check"></i><b>3.10.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="3.11" data-path="hyperbolic-functions.html"><a href="hyperbolic-functions.html"><i class="fa fa-check"></i><b>3.11</b> Hyperbolic Functions</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="applications-of-differentiation.html"><a href="applications-of-differentiation.html"><i class="fa fa-check"></i><b>4</b> Applications of Differentiation</a>
<ul>
<li class="chapter" data-level="4.1" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html"><i class="fa fa-check"></i><b>4.1</b> Maximum and Minimum Values</a>
<ul>
<li class="chapter" data-level="4.1.1" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#types-of-extreme-values"><i class="fa fa-check"></i><b>4.1.1</b> Types of Extreme Values</a></li>
<li class="chapter" data-level="4.1.2" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#extreme-value-theorem-evt"><i class="fa fa-check"></i><b>4.1.2</b> Extreme Value Theorem (EVT)</a></li>
<li class="chapter" data-level="4.1.3" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#fermats-theorem"><i class="fa fa-check"></i><b>4.1.3</b> Fermat’s Theorem</a></li>
<li class="chapter" data-level="4.1.4" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#critical-numbers"><i class="fa fa-check"></i><b>4.1.4</b> Critical Numbers</a></li>
<li class="chapter" data-level="4.1.5" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#closed-interval-method"><i class="fa fa-check"></i><b>4.1.5</b> Closed Interval Method</a></li>
<li class="chapter" data-level="4.1.6" data-path="maximum-and-minimum-values.html"><a href="maximum-and-minimum-values.html#putting-it-all-together-21"><i class="fa fa-check"></i><b>4.1.6</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.2" data-path="the-mean-value-theorem.html"><a href="the-mean-value-theorem.html"><i class="fa fa-check"></i><b>4.2</b> The Mean Value Theorem</a>
<ul>
<li class="chapter" data-level="4.2.1" data-path="the-mean-value-theorem.html"><a href="the-mean-value-theorem.html#rolles-theorem"><i class="fa fa-check"></i><b>4.2.1</b> Rolle’s Theorem</a></li>
<li class="chapter" data-level="4.2.2" data-path="the-mean-value-theorem.html"><a href="the-mean-value-theorem.html#the-mean-value-theorem-1"><i class="fa fa-check"></i><b>4.2.2</b> The Mean Value Theorem</a></li>
<li class="chapter" data-level="4.2.3" data-path="the-mean-value-theorem.html"><a href="the-mean-value-theorem.html#consequences-of-the-mean-value-theorem"><i class="fa fa-check"></i><b>4.2.3</b> Consequences of the Mean Value Theorem</a></li>
<li class="chapter" data-level="4.2.4" data-path="the-mean-value-theorem.html"><a href="the-mean-value-theorem.html#putting-it-all-together-22"><i class="fa fa-check"></i><b>4.2.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.3" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html"><i class="fa fa-check"></i><b>4.3</b> How Derivatives Affect the Shape of a Graph</a>
<ul>
<li class="chapter" data-level="4.3.1" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#what-does-f-say-about-f"><i class="fa fa-check"></i><b>4.3.1</b> What Does <span class="math inline">\(f'\)</span> Say About <span class="math inline">\(f\)</span>?</a></li>
<li class="chapter" data-level="4.3.2" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#local-extreme-values"><i class="fa fa-check"></i><b>4.3.2</b> Local Extreme Values</a></li>
<li class="chapter" data-level="4.3.3" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#what-does-f-say-about-f-1"><i class="fa fa-check"></i><b>4.3.3</b> What Does <span class="math inline">\(f''\)</span> Say About <span class="math inline">\(f\)</span>?</a></li>
<li class="chapter" data-level="4.3.4" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#inflection-points"><i class="fa fa-check"></i><b>4.3.4</b> Inflection Points</a></li>
<li class="chapter" data-level="4.3.5" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#graph-shape-analysis-framework"><i class="fa fa-check"></i><b>4.3.5</b> Graph Shape Analysis Framework</a></li>
<li class="chapter" data-level="4.3.6" data-path="how-derivatives-affect-the-shape-of-a-graph.html"><a href="how-derivatives-affect-the-shape-of-a-graph.html#putting-it-all-together-23"><i class="fa fa-check"></i><b>4.3.6</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.4" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html"><i class="fa fa-check"></i><b>4.4</b> Indeterminate Forms and l’Hospital’s Rule</a>
<ul>
<li class="chapter" data-level="4.4.1" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#indeterminate-forms"><i class="fa fa-check"></i><b>4.4.1</b> Indeterminate Forms</a></li>
<li class="chapter" data-level="4.4.2" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#interpretation"><i class="fa fa-check"></i><b>4.4.2</b> Interpretation</a></li>
<li class="chapter" data-level="4.4.3" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#indeterminate-products-0-cdot-infty"><i class="fa fa-check"></i><b>4.4.3</b> Indeterminate Products <span class="math inline">\(0 \cdot \infty\)</span></a></li>
<li class="chapter" data-level="4.4.4" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#indeterminate-differences-infty---infty"><i class="fa fa-check"></i><b>4.4.4</b> Indeterminate Differences <span class="math inline">\(\infty - \infty\)</span></a></li>
<li class="chapter" data-level="4.4.5" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#indeterminate-powers"><i class="fa fa-check"></i><b>4.4.5</b> Indeterminate Powers</a></li>
<li class="chapter" data-level="4.4.6" data-path="indeterminate-forms-and-lhospitals-rule.html"><a href="indeterminate-forms-and-lhospitals-rule.html#putting-it-all-together-24"><i class="fa fa-check"></i><b>4.4.6</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.5" data-path="summary-of-curve-sketching.html"><a href="summary-of-curve-sketching.html"><i class="fa fa-check"></i><b>4.5</b> Summary of Curve Sketching</a>
<ul>
<li class="chapter" data-level="4.5.1" data-path="summary-of-curve-sketching.html"><a href="summary-of-curve-sketching.html#a-structured-framework-for-sketching-curves"><i class="fa fa-check"></i><b>4.5.1</b> A Structured Framework for Sketching Curves</a></li>
<li class="chapter" data-level="4.5.2" data-path="summary-of-curve-sketching.html"><a href="summary-of-curve-sketching.html#putting-it-all-together-25"><i class="fa fa-check"></i><b>4.5.2</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.6" data-path="graphing-with-calculus-and-calculators.html"><a href="graphing-with-calculus-and-calculators.html"><i class="fa fa-check"></i><b>4.6</b> Graphing with Calculus and Calculators</a></li>
<li class="chapter" data-level="4.7" data-path="optimization-problems.html"><a href="optimization-problems.html"><i class="fa fa-check"></i><b>4.7</b> Optimization Problems</a>
<ul>
<li class="chapter" data-level="4.7.1" data-path="optimization-problems.html"><a href="optimization-problems.html#general-strategy-for-solving-optimization-problems"><i class="fa fa-check"></i><b>4.7.1</b> General Strategy for Solving Optimization Problems</a></li>
<li class="chapter" data-level="4.7.2" data-path="optimization-problems.html"><a href="optimization-problems.html#applications-in-business-and-economics"><i class="fa fa-check"></i><b>4.7.2</b> Applications in Business and Economics</a></li>
<li class="chapter" data-level="4.7.3" data-path="optimization-problems.html"><a href="optimization-problems.html#putting-it-all-together-26"><i class="fa fa-check"></i><b>4.7.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.8" data-path="newtons-method.html"><a href="newtons-method.html"><i class="fa fa-check"></i><b>4.8</b> Newton’s Method</a>
<ul>
<li class="chapter" data-level="4.8.1" data-path="newtons-method.html"><a href="newtons-method.html#derivation-of-the-formula"><i class="fa fa-check"></i><b>4.8.1</b> Derivation of the Formula</a></li>
<li class="chapter" data-level="4.8.2" data-path="newtons-method.html"><a href="newtons-method.html#convergence-and-failure"><i class="fa fa-check"></i><b>4.8.2</b> Convergence and Failure</a></li>
<li class="chapter" data-level="4.8.3" data-path="newtons-method.html"><a href="newtons-method.html#stopping-criterion"><i class="fa fa-check"></i><b>4.8.3</b> Stopping Criterion</a></li>
<li class="chapter" data-level="4.8.4" data-path="newtons-method.html"><a href="newtons-method.html#putting-it-all-together-27"><i class="fa fa-check"></i><b>4.8.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="4.9" data-path="antiderivatives.html"><a href="antiderivatives.html"><i class="fa fa-check"></i><b>4.9</b> Antiderivatives</a>
<ul>
<li class="chapter" data-level="4.9.1" data-path="antiderivatives.html"><a href="antiderivatives.html#general-antiderivative-theorem"><i class="fa fa-check"></i><b>4.9.1</b> General Antiderivative Theorem</a></li>
<li class="chapter" data-level="4.9.2" data-path="antiderivatives.html"><a href="antiderivatives.html#basic-antidifferentiation-rules"><i class="fa fa-check"></i><b>4.9.2</b> Basic Antidifferentiation Rules</a></li>
<li class="chapter" data-level="4.9.3" data-path="antiderivatives.html"><a href="antiderivatives.html#graphical-interpretation"><i class="fa fa-check"></i><b>4.9.3</b> Graphical Interpretation</a></li>
<li class="chapter" data-level="4.9.4" data-path="antiderivatives.html"><a href="antiderivatives.html#putting-it-all-together-28"><i class="fa fa-check"></i><b>4.9.4</b> Putting It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="5" data-path="integrals.html"><a href="integrals.html"><i class="fa fa-check"></i><b>5</b> Integrals</a>
<ul>
<li class="chapter" data-level="5.1" data-path="areas-and-distances.html"><a href="areas-and-distances.html"><i class="fa fa-check"></i><b>5.1</b> Areas and Distances</a>
<ul>
<li class="chapter" data-level="5.1.1" data-path="areas-and-distances.html"><a href="areas-and-distances.html#the-area-problem"><i class="fa fa-check"></i><b>5.1.1</b> The Area Problem</a></li>
<li class="chapter" data-level="5.1.2" data-path="areas-and-distances.html"><a href="areas-and-distances.html#types-of-riemann-sums"><i class="fa fa-check"></i><b>5.1.2</b> Types of Riemann Sums</a></li>
<li class="chapter" data-level="5.1.3" data-path="areas-and-distances.html"><a href="areas-and-distances.html#the-distance-problem"><i class="fa fa-check"></i><b>5.1.3</b> The Distance Problem</a></li>
<li class="chapter" data-level="5.1.4" data-path="areas-and-distances.html"><a href="areas-and-distances.html#putting-it-all-together-29"><i class="fa fa-check"></i><b>5.1.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="5.2" data-path="the-definite-integral.html"><a href="the-definite-integral.html"><i class="fa fa-check"></i><b>5.2</b> The Definite Integral</a>
<ul>
<li class="chapter" data-level="5.2.1" data-path="the-definite-integral.html"><a href="the-definite-integral.html#riemann-sums"><i class="fa fa-check"></i><b>5.2.1</b> Riemann Sums</a></li>
<li class="chapter" data-level="5.2.2" data-path="the-definite-integral.html"><a href="the-definite-integral.html#the-midpoint-rule"><i class="fa fa-check"></i><b>5.2.2</b> The Midpoint Rule</a></li>
<li class="chapter" data-level="5.2.3" data-path="the-definite-integral.html"><a href="the-definite-integral.html#properties-of-the-definite-integral"><i class="fa fa-check"></i><b>5.2.3</b> Properties of the Definite Integral</a></li>
<li class="chapter" data-level="5.2.4" data-path="the-definite-integral.html"><a href="the-definite-integral.html#putting-it-all-together-30"><i class="fa fa-check"></i><b>5.2.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="5.3" data-path="the-fundamental-theorem-of-calculus.html"><a href="the-fundamental-theorem-of-calculus.html"><i class="fa fa-check"></i><b>5.3</b> The Fundamental Theorem of Calculus</a>
<ul>
<li class="chapter" data-level="5.3.1" data-path="the-fundamental-theorem-of-calculus.html"><a href="the-fundamental-theorem-of-calculus.html#ftc-part-1-derivative-of-an-integral-ftc1"><i class="fa fa-check"></i><b>5.3.1</b> FTC Part 1: Derivative of an Integral (FTC1)</a></li>
<li class="chapter" data-level="5.3.2" data-path="the-fundamental-theorem-of-calculus.html"><a href="the-fundamental-theorem-of-calculus.html#ftc-part-2-evaluating-definite-integrals-ftc2"><i class="fa fa-check"></i><b>5.3.2</b> FTC Part 2: Evaluating Definite Integrals (FTC2)</a></li>
<li class="chapter" data-level="5.3.3" data-path="the-fundamental-theorem-of-calculus.html"><a href="the-fundamental-theorem-of-calculus.html#putting-it-all-together-31"><i class="fa fa-check"></i><b>5.3.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="5.4" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html"><i class="fa fa-check"></i><b>5.4</b> Indefinite Integrals and the Net Change Theorem</a>
<ul>
<li class="chapter" data-level="5.4.1" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html#indefinite-integrals"><i class="fa fa-check"></i><b>5.4.1</b> Indefinite Integrals</a></li>
<li class="chapter" data-level="5.4.2" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html#fundamental-connection-to-definite-integrals"><i class="fa fa-check"></i><b>5.4.2</b> Fundamental Connection to Definite Integrals</a></li>
<li class="chapter" data-level="5.4.3" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html#the-net-change-theorem"><i class="fa fa-check"></i><b>5.4.3</b> The Net Change Theorem</a></li>
<li class="chapter" data-level="5.4.4" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html#applications-of-net-change"><i class="fa fa-check"></i><b>5.4.4</b> Applications of Net Change</a></li>
<li class="chapter" data-level="5.4.5" data-path="indefinite-integrals-and-the-net-change-theorem.html"><a href="indefinite-integrals-and-the-net-change-theorem.html#putting-it-all-together-32"><i class="fa fa-check"></i><b>5.4.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="5.5" data-path="the-substitution-rule.html"><a href="the-substitution-rule.html"><i class="fa fa-check"></i><b>5.5</b> The Substitution Rule</a>
<ul>
<li class="chapter" data-level="5.5.1" data-path="the-substitution-rule.html"><a href="the-substitution-rule.html#putting-it-all-together-33"><i class="fa fa-check"></i><b>5.5.1</b> Putting It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="6" data-path="applications-of-integration.html"><a href="applications-of-integration.html"><i class="fa fa-check"></i><b>6</b> Applications of Integration</a>
<ul>
<li class="chapter" data-level="6.1" data-path="areas-between-curves.html"><a href="areas-between-curves.html"><i class="fa fa-check"></i><b>6.1</b> Areas Between Curves</a>
<ul>
<li class="chapter" data-level="6.1.1" data-path="areas-between-curves.html"><a href="areas-between-curves.html#when-curves-cross-absolute-value-form"><i class="fa fa-check"></i><b>6.1.1</b> When Curves Cross (Absolute Value Form)</a></li>
<li class="chapter" data-level="6.1.2" data-path="areas-between-curves.html"><a href="areas-between-curves.html#horizontal-slices-integrating-with-respect-to-y"><i class="fa fa-check"></i><b>6.1.2</b> Horizontal Slices (Integrating with Respect to <span class="math inline">\(y\)</span>)</a></li>
<li class="chapter" data-level="6.1.3" data-path="areas-between-curves.html"><a href="areas-between-curves.html#putting-it-all-together-34"><i class="fa fa-check"></i><b>6.1.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="6.2" data-path="volumes.html"><a href="volumes.html"><i class="fa fa-check"></i><b>6.2</b> Volumes</a>
<ul>
<li class="chapter" data-level="6.2.1" data-path="volumes.html"><a href="volumes.html#volume-of-a-cylinder"><i class="fa fa-check"></i><b>6.2.1</b> Volume of a Cylinder</a></li>
<li class="chapter" data-level="6.2.2" data-path="volumes.html"><a href="volumes.html#cross-sections-and-slicing"><i class="fa fa-check"></i><b>6.2.2</b> Cross-Sections and Slicing</a></li>
<li class="chapter" data-level="6.2.3" data-path="volumes.html"><a href="volumes.html#solids-of-revolution"><i class="fa fa-check"></i><b>6.2.3</b> Solids of Revolution</a></li>
<li class="chapter" data-level="6.2.4" data-path="volumes.html"><a href="volumes.html#pulling-it-all-together-1"><i class="fa fa-check"></i><b>6.2.4</b> Pulling It All Together</a></li>
</ul></li>
<li class="chapter" data-level="6.3" data-path="volumes-by-cylindrical-shells.html"><a href="volumes-by-cylindrical-shells.html"><i class="fa fa-check"></i><b>6.3</b> Volumes by Cylindrical Shells</a>
<ul>
<li class="chapter" data-level="6.3.1" data-path="volumes-by-cylindrical-shells.html"><a href="volumes-by-cylindrical-shells.html#pulling-it-all-together-2"><i class="fa fa-check"></i><b>6.3.1</b> Pulling It All Together</a></li>
</ul></li>
<li class="chapter" data-level="6.4" data-path="work.html"><a href="work.html"><i class="fa fa-check"></i><b>6.4</b> Work</a></li>
<li class="chapter" data-level="6.5" data-path="average-value-of-a-function.html"><a href="average-value-of-a-function.html"><i class="fa fa-check"></i><b>6.5</b> Average Value of a Function</a>
<ul>
<li class="chapter" data-level="6.5.1" data-path="average-value-of-a-function.html"><a href="average-value-of-a-function.html#mean-value-theorem-for-integrals"><i class="fa fa-check"></i><b>6.5.1</b> Mean Value Theorem for Integrals</a></li>
</ul></li>
</ul></li>
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<h1>
<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">MATH 112: Differential Calculus</a>
</h1>
</div>
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<section class="normal" id="section-">
<div id="average-value-of-a-function" class="section level2 hasAnchor" number="6.5">
<h2><span class="header-section-number">6.5</span> Average Value of a Function<a href="average-value-of-a-function.html#average-value-of-a-function" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>Finding the average of finitely many numbers is straightforward:
<span class="math display">\[
y_{\text{avg}} = \frac{y_1 + y_2 + \cdots + y_n}{n}
\]</span></p>
<p>But many real-world quantities vary continuously:</p>
<ul>
<li>temperature over a day<br />
</li>
<li>velocity during a trip<br />
</li>
<li>power usage over time<br />
</li>
<li>density across a region</li>
</ul>
<p>In these cases, there are infinitely many values, so we use integration to define an average value for a function over an interval.</p>
<div class="note">
<p><strong>Deriving the Average Value Formula</strong></p>
<p>Let <span class="math inline">\(y = f(x)\)</span> be defined on an interval <span class="math inline">\([a,b]\)</span>.</p>
<ol style="list-style-type: decimal">
<li><p>Divide the interval into <span class="math inline">\(n\)</span> subintervals of width:
<span class="math display">\[
\Delta x = \frac{b-a}{n}
\]</span></p></li>
<li><p>Choose a sample point <span class="math inline">\(x_i^*\)</span> in each subinterval.</p></li>
<li><p>Average the function values:
<span class="math display">\[
\frac{f(x_1^*) + f(x_2^*) + \cdots + f(x_n^*)}{n}
\]</span></p></li>
</ol>
<p>Rewriting using <span class="math inline">\(\Delta x\)</span> and taking the limit as <span class="math inline">\(n \to \infty\)</span>:</p>
<p><span class="math display">\[
f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx
\]</span></p>
</div>
<p><strong>Average Value Formula</strong></p>
<p><span class="math display">\[
f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx.
\]</span></p>
<p><strong>Geometric Interpretation</strong></p>
<p>For <strong>positive functions</strong>:
<span class="math display">\[
\frac{\text{Area under curve}}{\text{Width of interval}} = \text{Average height}
\]</span>
This means the average value is the <strong>height of a rectangle</strong> over <span class="math inline">\([a,b]\)</span> that has the <strong>same area</strong> as the region under <span class="math inline">\(f(x)\)</span>.</p>
<div class="example">
<p><span id="exm:unlabeled-div-14" class="example"><strong>Example 6.10 </strong></span>Computing an Average Value</p>
<p>Find the average value of<br />
<span class="math display">\[
f(x) = 1 + x^2
\]</span>
on the interval <span class="math inline">\([-1,2]\)</span>.</p>
<p><strong>Solution:</strong></p>
<p><span class="math display">\[\begin{align*}
f_{\text{avg}} &= \frac{1}{2 - (-1)}\int_{-1}^{2} (1 + x^2)\,dx\\
& = \frac{1}{3}\int_{-1}^{2} (1 + x^2)\,dx\\
& = \frac{1}{3}\left[ x + \frac{x^3}{3} \right]_{-1}^{2}\\
& = \frac{1}{3}\Big[(2 + \tfrac{8}{3}) - (-1 - \tfrac{1}{3})\Big]\\
& = \frac{1}{3}(6)\\
& = 2.
\end{align*}\]</span></p>
</div>
<hr />
<div id="mean-value-theorem-for-integrals" class="section level3 hasAnchor" number="6.5.1">
<h3><span class="header-section-number">6.5.1</span> Mean Value Theorem for Integrals<a href="average-value-of-a-function.html#mean-value-theorem-for-integrals" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div class="definition">
<p><span id="def:unlabeled-div-15" class="definition"><strong>Definition 6.2 </strong></span><strong>MVT for Integrals</strong></p>
<p>If <span class="math inline">\(f\)</span> is <strong>continuous</strong> on <span class="math inline">\([a,b]\)</span>, then there exists at least one number<br />
<span class="math inline">\(c \in [a,b]\)</span> such that:
<span class="math display">\[
f(c) = f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx.
\]</span></p>
<p>Equivalently:
<span class="math display">\[
\int_a^b f(x)\,dx = f(c)(b-a).
\]</span></p>
</div>
<p>Geometrically, there exists a point <span class="math inline">\(c\)</span> where the function’s height equals the average height over the interval.</p>
<blockquote>
<p>This means: a rectangle of height <span class="math inline">\(f(c)\)</span> and base <span class="math inline">\([a,b]\)</span> has the same area as the area under the curve.</p>
</blockquote>
<div class="example">
<p><span id="exm:unlabeled-div-16" class="example"><strong>Example 6.11 </strong></span>Finding the Mean Value Point</p>
<p>Using the previous example, we found:
<span class="math display">\[
f_{\text{avg}} = 2
\]</span></p>
<p>Solve:
<span class="math display">\[
f(c) = 2
\]</span></p>
<p><span class="math display">\[
1 + c^2 = 2
\]</span></p>
<p><span class="math display">\[
c^2 = 1
\]</span></p>
<p><span class="math display">\[
c = \pm 1
\]</span></p>
Both <span class="math inline">\(c=-1\)</span> and <span class="math inline">\(c=1\)</span> lie in the interval <span class="math inline">\([-1,2]\)</span>, so both satisfy the theorem.
</div>
<hr />
<div id="application-average-velocity" class="section level3 hasAnchor" number="6.5.2">
<h3><span class="header-section-number">6.5.2</span> Application: Average Velocity<a href="average-value-of-a-function.html#application-average-velocity" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Let <span class="math inline">\(s(t)\)</span> be position and <span class="math inline">\(v(t) = s'(t)\)</span> be velocity.</p>
<p><strong>Average velocity (definition):</strong>
<span class="math display">\[
v_{\text{avg}} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}
\]</span></p>
<p><strong>Average value of velocity function:</strong>
<span class="math display">\[
v_{\text{avg}} = \frac{1}{t_2 - t_1}\int_{t_1}^{t_2} v(t)\,dt
\]</span></p>
<p>Using the Net Change Theorem:
<span class="math display">\[
\int_{t_1}^{t_2} v(t)\,dt = s(t_2) - s(t_1)
\]</span></p>
<p>So:
<span class="math display">\[
\text{Average velocity} = \text{Average value of velocity function}
\]</span></p>
<p>This shows that <strong>physical average velocity</strong> and <strong>integral-based average value</strong> are mathematically identical.</p>
<hr />
</div>
<div id="pulling-it-all-together-3" class="section level3 hasAnchor" number="6.5.3">
<h3><span class="header-section-number">6.5.3</span> Pulling It All Together<a href="average-value-of-a-function.html#pulling-it-all-together-3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Think:
> <strong>Average value = total accumulation ÷ interval length</strong></p>
<p>or</p>
<blockquote>
<p><strong>Area ÷ width = average height</strong></p>
</blockquote>
<p><span class="math display">\[
f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx
\]</span></p>
<p><span class="math display">\[
\int_a^b f(x)\,dx = f(c)(b-a)
\]</span></p>
<hr />
<div id="conceptual-takeaways-38" class="section level4 hasAnchor" number="6.5.3.1">
<h4><span class="header-section-number">6.5.3.1</span> Conceptual Takeaways<a href="average-value-of-a-function.html#conceptual-takeaways-38" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<ul>
<li>Average value of a function is a <strong>continuous analog</strong> of averaging numbers<br />
</li>
<li>Integration replaces summation<br />
</li>
<li>Area plays the role of total accumulation<br />
</li>
<li>Division by interval length produces an average height<br />
</li>
<li>The Mean Value Theorem for Integrals guarantees that:
<ul>
<li>the average value is actually <strong>attained</strong> by the function<br />
</li>
</ul></li>
<li>Average value has <strong>geometric meaning</strong> and <strong>physical meaning</strong></li>
</ul>
<hr />
</div>
<div id="skills-you-should-be-able-to-do-37" class="section level4 hasAnchor" number="6.5.3.2">
<h4><span class="header-section-number">6.5.3.2</span> Skills You Should Be Able To Do<a href="average-value-of-a-function.html#skills-you-should-be-able-to-do-37" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<p>After this section, you should be able to:</p>
<ul>
<li>Explain average value conceptually<br />
</li>
<li>Derive the average value formula<br />
</li>
<li>Compute average values using integrals<br />
</li>
<li>Interpret average value geometrically<br />
</li>
<li>Apply the Mean Value Theorem for Integrals<br />
</li>
<li>Find values <span class="math inline">\(c\)</span> such that <span class="math inline">\(f(c) = f_{\text{avg}}\)</span><br />
</li>
<li>Connect area to physical meaning<br />
</li>
<li>Apply average value to:
<ul>
<li>velocity<br />
</li>
<li>temperature<br />
</li>
<li>accumulation processes<br />
</li>
</ul></li>
<li>Translate between physical averages and integral averages<br />
</li>
<li>Explain the difference between:
<ul>
<li>discrete averages<br />
</li>
<li>continuous averages</li>
</ul></li>
</ul>
<hr />
</div>
<div id="problems-39" class="section level4 hasAnchor" number="6.5.3.3">
<h4><span class="header-section-number">6.5.3.3</span> Problems<a href="average-value-of-a-function.html#problems-39" class="anchor-section" aria-label="Anchor link to header"></a></h4>
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