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<!DOCTYPE html>
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<title>1.4 Exponential Functions | MATH 112/ 122: Differential and Integral 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>
</ul></li>
<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>
<ul>
<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>
</ul></li>
<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>
<li class="chapter" data-level="6.5.2" data-path="average-value-of-a-function.html"><a href="average-value-of-a-function.html#application-average-velocity"><i class="fa fa-check"></i><b>6.5.2</b> Application: Average Velocity</a></li>
<li class="chapter" data-level="6.5.3" data-path="average-value-of-a-function.html"><a href="average-value-of-a-function.html#pulling-it-all-together-3"><i class="fa fa-check"></i><b>6.5.3</b> Pulling It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="7" data-path="techniques-of-integration.html"><a href="techniques-of-integration.html"><i class="fa fa-check"></i><b>7</b> Techniques of Integration</a>
<ul>
<li class="chapter" data-level="7.1" data-path="integration-by-parts.html"><a href="integration-by-parts.html"><i class="fa fa-check"></i><b>7.1</b> Integration by Parts</a>
<ul>
<li class="chapter" data-level="7.1.1" data-path="integration-by-parts.html"><a href="integration-by-parts.html#from-the-product-rule-to-integration-by-parts"><i class="fa fa-check"></i><b>7.1.1</b> From the Product Rule to Integration by Parts</a></li>
<li class="chapter" data-level="7.1.2" data-path="integration-by-parts.html"><a href="integration-by-parts.html#integration-by-parts-formula"><i class="fa fa-check"></i><b>7.1.2</b> Integration by Parts Formula</a></li>
<li class="chapter" data-level="7.1.3" data-path="integration-by-parts.html"><a href="integration-by-parts.html#choosing-u-and-dv"><i class="fa fa-check"></i><b>7.1.3</b> Choosing <span class="math inline">\(u\)</span> and <span class="math inline">\(dv\)</span></a></li>
<li class="chapter" data-level="7.1.4" data-path="integration-by-parts.html"><a href="integration-by-parts.html#putting-it-all-together-35"><i class="fa fa-check"></i><b>7.1.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.2" data-path="trigonometric-integrals.html"><a href="trigonometric-integrals.html"><i class="fa fa-check"></i><b>7.2</b> Trigonometric Integrals</a>
<ul>
<li class="chapter" data-level="7.2.1" data-path="trigonometric-integrals.html"><a href="trigonometric-integrals.html#powers-of-sine-and-cosine"><i class="fa fa-check"></i><b>7.2.1</b> Powers of Sine and Cosine</a></li>
<li class="chapter" data-level="7.2.2" data-path="trigonometric-integrals.html"><a href="trigonometric-integrals.html#tangent-and-secant-integrals"><i class="fa fa-check"></i><b>7.2.2</b> Tangent and Secant Integrals</a></li>
<li class="chapter" data-level="7.2.3" data-path="trigonometric-integrals.html"><a href="trigonometric-integrals.html#special-integrals"><i class="fa fa-check"></i><b>7.2.3</b> Special Integrals</a></li>
<li class="chapter" data-level="7.2.4" data-path="trigonometric-integrals.html"><a href="trigonometric-integrals.html#putting-it-all-together-36"><i class="fa fa-check"></i><b>7.2.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.3" data-path="trigonometric-substitution.html"><a href="trigonometric-substitution.html"><i class="fa fa-check"></i><b>7.3</b> Trigonometric Substitution</a>
<ul>
<li class="chapter" data-level="7.3.1" data-path="trigonometric-substitution.html"><a href="trigonometric-substitution.html#inverse-substitution-idea"><i class="fa fa-check"></i><b>7.3.1</b> Inverse Substitution Idea</a></li>
<li class="chapter" data-level="7.3.2" data-path="trigonometric-substitution.html"><a href="trigonometric-substitution.html#putting-it-all-together-37"><i class="fa fa-check"></i><b>7.3.2</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.4" data-path="integration-of-rational-functions-by-partial-fractions.html"><a href="integration-of-rational-functions-by-partial-fractions.html"><i class="fa fa-check"></i><b>7.4</b> Integration of Rational Functions by Partial Fractions</a>
<ul>
<li class="chapter" data-level="7.4.1" data-path="integration-of-rational-functions-by-partial-fractions.html"><a href="integration-of-rational-functions-by-partial-fractions.html#general-strategy"><i class="fa fa-check"></i><b>7.4.1</b> General Strategy</a></li>
<li class="chapter" data-level="7.4.2" data-path="integration-of-rational-functions-by-partial-fractions.html"><a href="integration-of-rational-functions-by-partial-fractions.html#special-technique-rationalizing-substitutions"><i class="fa fa-check"></i><b>7.4.2</b> Special Technique: Rationalizing Substitutions</a></li>
<li class="chapter" data-level="7.4.3" data-path="integration-of-rational-functions-by-partial-fractions.html"><a href="integration-of-rational-functions-by-partial-fractions.html#putting-it-all-together-38"><i class="fa fa-check"></i><b>7.4.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.5" data-path="strategy-for-integration.html"><a href="strategy-for-integration.html"><i class="fa fa-check"></i><b>7.5</b> Strategy for Integration</a>
<ul>
<li class="chapter" data-level="7.5.1" data-path="strategy-for-integration.html"><a href="strategy-for-integration.html#a-practical-strategy-for-integration"><i class="fa fa-check"></i><b>7.5.1</b> A Practical Strategy for Integration</a></li>
<li class="chapter" data-level="7.5.2" data-path="strategy-for-integration.html"><a href="strategy-for-integration.html#important-theoretical-insight"><i class="fa fa-check"></i><b>7.5.2</b> Important Theoretical Insight</a></li>
<li class="chapter" data-level="7.5.3" data-path="strategy-for-integration.html"><a href="strategy-for-integration.html#putting-it-all-together-39"><i class="fa fa-check"></i><b>7.5.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.6" data-path="integration-using-tables-and-computer-algebra-systems.html"><a href="integration-using-tables-and-computer-algebra-systems.html"><i class="fa fa-check"></i><b>7.6</b> Integration Using Tables and Computer Algebra Systems</a></li>
<li class="chapter" data-level="7.7" data-path="approximate-integration.html"><a href="approximate-integration.html"><i class="fa fa-check"></i><b>7.7</b> Approximate Integration</a>
<ul>
<li class="chapter" data-level="7.7.1" data-path="approximate-integration.html"><a href="approximate-integration.html#riemann-sum-approximations"><i class="fa fa-check"></i><b>7.7.1</b> Riemann Sum Approximations</a></li>
<li class="chapter" data-level="7.7.2" data-path="approximate-integration.html"><a href="approximate-integration.html#trapezoidal-rule"><i class="fa fa-check"></i><b>7.7.2</b> Trapezoidal Rule</a></li>
<li class="chapter" data-level="7.7.3" data-path="approximate-integration.html"><a href="approximate-integration.html#simpsons-rule"><i class="fa fa-check"></i><b>7.7.3</b> Simpson’s Rule</a></li>
<li class="chapter" data-level="7.7.4" data-path="approximate-integration.html"><a href="approximate-integration.html#error-behavior-conceptual"><i class="fa fa-check"></i><b>7.7.4</b> Error Behavior (Conceptual)</a></li>
<li class="chapter" data-level="7.7.5" data-path="approximate-integration.html"><a href="approximate-integration.html#putting-it-all-together-40"><i class="fa fa-check"></i><b>7.7.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="7.8" data-path="improper-integrals.html"><a href="improper-integrals.html"><i class="fa fa-check"></i><b>7.8</b> Improper Integrals</a>
<ul>
<li class="chapter" data-level="7.8.1" data-path="improper-integrals.html"><a href="improper-integrals.html#type-1-improper-integrals-infinite-intervals"><i class="fa fa-check"></i><b>7.8.1</b> Type 1 Improper Integrals — Infinite Intervals</a></li>
<li class="chapter" data-level="7.8.2" data-path="improper-integrals.html"><a href="improper-integrals.html#the-p-test-for-infinite-intervals"><i class="fa fa-check"></i><b>7.8.2</b> The <span class="math inline">\(p\)</span>-Test for Infinite Intervals</a></li>
<li class="chapter" data-level="7.8.3" data-path="improper-integrals.html"><a href="improper-integrals.html#type-2-improper-integrals-infinite-discontinuities"><i class="fa fa-check"></i><b>7.8.3</b> Type 2 Improper Integrals — Infinite Discontinuities</a></li>
<li class="chapter" data-level="7.8.4" data-path="improper-integrals.html"><a href="improper-integrals.html#comparison-test-for-improper-integrals"><i class="fa fa-check"></i><b>7.8.4</b> Comparison Test for Improper Integrals</a></li>
<li class="chapter" data-level="7.8.5" data-path="improper-integrals.html"><a href="improper-integrals.html#putting-it-all-together-41"><i class="fa fa-check"></i><b>7.8.5</b> Putting It All Together</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="8" data-path="further-applications-of-integration.html"><a href="further-applications-of-integration.html"><i class="fa fa-check"></i><b>8</b> Further Applications of Integration</a></li>
<li class="chapter" data-level="9" data-path="differential-equations.html"><a href="differential-equations.html"><i class="fa fa-check"></i><b>9</b> Differential Equations</a>
<ul>
<li class="chapter" data-level="9.1" data-path="modeling-with-differential-equations.html"><a href="modeling-with-differential-equations.html"><i class="fa fa-check"></i><b>9.1</b> Modeling with Differential Equations</a>
<ul>
<li class="chapter" data-level="9.1.1" data-path="modeling-with-differential-equations.html"><a href="modeling-with-differential-equations.html#population-growth-models"><i class="fa fa-check"></i><b>9.1.1</b> Population Growth Models</a></li>
<li class="chapter" data-level="9.1.2" data-path="modeling-with-differential-equations.html"><a href="modeling-with-differential-equations.html#physical-model-motion-of-a-spring"><i class="fa fa-check"></i><b>9.1.2</b> Physical Model: Motion of a Spring</a></li>
<li class="chapter" data-level="9.1.3" data-path="modeling-with-differential-equations.html"><a href="modeling-with-differential-equations.html#general-differential-equations"><i class="fa fa-check"></i><b>9.1.3</b> General Differential Equations</a></li>
<li class="chapter" data-level="9.1.4" data-path="modeling-with-differential-equations.html"><a href="modeling-with-differential-equations.html#putting-it-all-together-42"><i class="fa fa-check"></i><b>9.1.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="9.2" data-path="direction-fields-and-eulers-method.html"><a href="direction-fields-and-eulers-method.html"><i class="fa fa-check"></i><b>9.2</b> Direction Fields and Euler’s Method</a>
<ul>
<li class="chapter" data-level="9.2.1" data-path="direction-fields-and-eulers-method.html"><a href="direction-fields-and-eulers-method.html#direction-fields-slope-fields"><i class="fa fa-check"></i><b>9.2.1</b> Direction Fields (Slope Fields)</a></li>
<li class="chapter" data-level="9.2.2" data-path="direction-fields-and-eulers-method.html"><a href="direction-fields-and-eulers-method.html#autonomous-differential-equations"><i class="fa fa-check"></i><b>9.2.2</b> Autonomous Differential Equations</a></li>
<li class="chapter" data-level="9.2.3" data-path="direction-fields-and-eulers-method.html"><a href="direction-fields-and-eulers-method.html#eulers-method"><i class="fa fa-check"></i><b>9.2.3</b> Euler’s Method</a></li>
<li class="chapter" data-level="9.2.4" data-path="direction-fields-and-eulers-method.html"><a href="direction-fields-and-eulers-method.html#putting-it-all-together-43"><i class="fa fa-check"></i><b>9.2.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="9.3" data-path="separable-equations.html"><a href="separable-equations.html"><i class="fa fa-check"></i><b>9.3</b> Separable Equations</a>
<ul>
<li class="chapter" data-level="9.3.1" data-path="separable-equations.html"><a href="separable-equations.html#physical-application-example"><i class="fa fa-check"></i><b>9.3.1</b> Physical Application Example</a></li>
<li class="chapter" data-level="9.3.2" data-path="separable-equations.html"><a href="separable-equations.html#putting-it-all-together-44"><i class="fa fa-check"></i><b>9.3.2</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="9.4" data-path="models-for-population-growth.html"><a href="models-for-population-growth.html"><i class="fa fa-check"></i><b>9.4</b> Models for Population Growth</a>
<ul>
<li class="chapter" data-level="9.4.1" data-path="models-for-population-growth.html"><a href="models-for-population-growth.html#the-law-of-natural-growth-exponential-model"><i class="fa fa-check"></i><b>9.4.1</b> The Law of Natural Growth (Exponential Model)</a></li>
<li class="chapter" data-level="9.4.2" data-path="models-for-population-growth.html"><a href="models-for-population-growth.html#the-logistic-growth-model"><i class="fa fa-check"></i><b>9.4.2</b> The Logistic Growth Model</a></li>
<li class="chapter" data-level="9.4.3" data-path="models-for-population-growth.html"><a href="models-for-population-growth.html#comparing-the-models"><i class="fa fa-check"></i><b>9.4.3</b> Comparing the Models</a></li>
<li class="chapter" data-level="9.4.4" data-path="models-for-population-growth.html"><a href="models-for-population-growth.html#putting-it-all-together-45"><i class="fa fa-check"></i><b>9.4.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="9.5" data-path="linear-equations.html"><a href="linear-equations.html"><i class="fa fa-check"></i><b>9.5</b> Linear Equations</a>
<ul>
<li class="chapter" data-level="9.5.1" data-path="linear-equations.html"><a href="linear-equations.html#the-integrating-factor-method"><i class="fa fa-check"></i><b>9.5.1</b> The Integrating Factor Method</a></li>
<li class="chapter" data-level="9.5.2" data-path="linear-equations.html"><a href="linear-equations.html#application-example-electric-circuits-rl-circuit"><i class="fa fa-check"></i><b>9.5.2</b> Application Example: Electric Circuits (RL Circuit)</a></li>
<li class="chapter" data-level="9.5.3" data-path="linear-equations.html"><a href="linear-equations.html#putting-it-all-together-46"><i class="fa fa-check"></i><b>9.5.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="9.6" data-path="predator-prey-systems.html"><a href="predator-prey-systems.html"><i class="fa fa-check"></i><b>9.6</b> Predator-Prey Systems</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="parametric-equations-and-polar-coordinates.html"><a href="parametric-equations-and-polar-coordinates.html"><i class="fa fa-check"></i><b>10</b> Parametric Equations and Polar Coordinates</a></li>
<li class="chapter" data-level="11" data-path="infinite-sequences-and-series.html"><a href="infinite-sequences-and-series.html"><i class="fa fa-check"></i><b>11</b> Infinite Sequences and Series</a>
<ul>
<li class="chapter" data-level="11.1" data-path="sequences.html"><a href="sequences.html"><i class="fa fa-check"></i><b>11.1</b> Sequences</a>
<ul>
<li class="chapter" data-level="11.1.1" data-path="sequences.html"><a href="sequences.html#limits-of-sequences"><i class="fa fa-check"></i><b>11.1.1</b> Limits of Sequences</a></li>
<li class="chapter" data-level="11.1.2" data-path="sequences.html"><a href="sequences.html#limit-laws-for-sequences"><i class="fa fa-check"></i><b>11.1.2</b> Limit Laws for Sequences</a></li>
<li class="chapter" data-level="11.1.3" data-path="sequences.html"><a href="sequences.html#some-additional-important-theorems"><i class="fa fa-check"></i><b>11.1.3</b> Some Additional Important Theorems</a></li>
<li class="chapter" data-level="11.1.4" data-path="sequences.html"><a href="sequences.html#special-sequences"><i class="fa fa-check"></i><b>11.1.4</b> Special Sequences</a></li>
<li class="chapter" data-level="11.1.5" data-path="sequences.html"><a href="sequences.html#putting-it-all-together-47"><i class="fa fa-check"></i><b>11.1.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.2" data-path="series.html"><a href="series.html"><i class="fa fa-check"></i><b>11.2</b> Series</a>
<ul>
<li class="chapter" data-level="11.2.1" data-path="series.html"><a href="series.html#sequences-vs.-series"><i class="fa fa-check"></i><b>11.2.1</b> Sequences vs. Series</a></li>
<li class="chapter" data-level="11.2.2" data-path="series.html"><a href="series.html#partial-sums-and-the-meaning-of-sum"><i class="fa fa-check"></i><b>11.2.2</b> Partial Sums and the Meaning of “Sum”</a></li>
<li class="chapter" data-level="11.2.3" data-path="series.html"><a href="series.html#geometric-series"><i class="fa fa-check"></i><b>11.2.3</b> Geometric Series</a></li>
<li class="chapter" data-level="11.2.4" data-path="series.html"><a href="series.html#telescoping-series"><i class="fa fa-check"></i><b>11.2.4</b> Telescoping Series</a></li>
<li class="chapter" data-level="11.2.5" data-path="series.html"><a href="series.html#fundamental-tests-and-theorems"><i class="fa fa-check"></i><b>11.2.5</b> Fundamental Tests and Theorems</a></li>
<li class="chapter" data-level="11.2.6" data-path="series.html"><a href="series.html#putting-it-all-together-48"><i class="fa fa-check"></i><b>11.2.6</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.3" data-path="the-integral-test-and-estimates-of-sums.html"><a href="the-integral-test-and-estimates-of-sums.html"><i class="fa fa-check"></i><b>11.3</b> The Integral Test and Estimates of Sums</a>
<ul>
<li class="chapter" data-level="11.3.1" data-path="the-integral-test-and-estimates-of-sums.html"><a href="the-integral-test-and-estimates-of-sums.html#the-p-series"><i class="fa fa-check"></i><b>11.3.1</b> The p-Series</a></li>
<li class="chapter" data-level="11.3.2" data-path="the-integral-test-and-estimates-of-sums.html"><a href="the-integral-test-and-estimates-of-sums.html#error-estimates"><i class="fa fa-check"></i><b>11.3.2</b> Error Estimates</a></li>
<li class="chapter" data-level="11.3.3" data-path="the-integral-test-and-estimates-of-sums.html"><a href="the-integral-test-and-estimates-of-sums.html#putting-it-all-together-49"><i class="fa fa-check"></i><b>11.3.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.4" data-path="the-comparison-tests.html"><a href="the-comparison-tests.html"><i class="fa fa-check"></i><b>11.4</b> The Comparison Tests</a>
<ul>
<li class="chapter" data-level="11.4.1" data-path="the-comparison-tests.html"><a href="the-comparison-tests.html#the-comparison-test"><i class="fa fa-check"></i><b>11.4.1</b> The Comparison Test</a></li>
<li class="chapter" data-level="11.4.2" data-path="the-comparison-tests.html"><a href="the-comparison-tests.html#the-limit-comparison-test"><i class="fa fa-check"></i><b>11.4.2</b> The Limit Comparison Test</a></li>
<li class="chapter" data-level="11.4.3" data-path="the-comparison-tests.html"><a href="the-comparison-tests.html#estimating-sums-using-comparison"><i class="fa fa-check"></i><b>11.4.3</b> Estimating Sums Using Comparison</a></li>
<li class="chapter" data-level="11.4.4" data-path="the-comparison-tests.html"><a href="the-comparison-tests.html#putting-it-all-together-50"><i class="fa fa-check"></i><b>11.4.4</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.5" data-path="alternating-series.html"><a href="alternating-series.html"><i class="fa fa-check"></i><b>11.5</b> Alternating Series</a>
<ul>
<li class="chapter" data-level="11.5.1" data-path="alternating-series.html"><a href="alternating-series.html#alternating-series-test-ast"><i class="fa fa-check"></i><b>11.5.1</b> Alternating Series Test (AST)</a></li>
<li class="chapter" data-level="11.5.2" data-path="alternating-series.html"><a href="alternating-series.html#estimating-sums-of-alternating-series"><i class="fa fa-check"></i><b>11.5.2</b> Estimating Sums of Alternating Series</a></li>
<li class="chapter" data-level="11.5.3" data-path="alternating-series.html"><a href="alternating-series.html#putting-it-all-together-51"><i class="fa fa-check"></i><b>11.5.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.6" data-path="absolute-convergence-and-the-ratio-and-root-tests.html"><a href="absolute-convergence-and-the-ratio-and-root-tests.html"><i class="fa fa-check"></i><b>11.6</b> Absolute Convergence and the Ratio and Root Tests</a>
<ul>
<li class="chapter" data-level="11.6.1" data-path="absolute-convergence-and-the-ratio-and-root-tests.html"><a href="absolute-convergence-and-the-ratio-and-root-tests.html#absolute-convergence"><i class="fa fa-check"></i><b>11.6.1</b> Absolute Convergence</a></li>
<li class="chapter" data-level="11.6.2" data-path="absolute-convergence-and-the-ratio-and-root-tests.html"><a href="absolute-convergence-and-the-ratio-and-root-tests.html#the-ratio-test"><i class="fa fa-check"></i><b>11.6.2</b> The Ratio Test</a></li>
<li class="chapter" data-level="11.6.3" data-path="absolute-convergence-and-the-ratio-and-root-tests.html"><a href="absolute-convergence-and-the-ratio-and-root-tests.html#the-root-test"><i class="fa fa-check"></i><b>11.6.3</b> The Root Test</a></li>
<li class="chapter" data-level="11.6.4" data-path="absolute-convergence-and-the-ratio-and-root-tests.html"><a href="absolute-convergence-and-the-ratio-and-root-tests.html#rearrangements-of-series"><i class="fa fa-check"></i><b>11.6.4</b> Rearrangements of Series</a></li>
<li class="chapter" data-level="11.6.5" data-path="absolute-convergence-and-the-ratio-and-root-tests.html"><a href="absolute-convergence-and-the-ratio-and-root-tests.html#putting-it-all-together-52"><i class="fa fa-check"></i><b>11.6.5</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.7" data-path="strategy-for-testing-series.html"><a href="strategy-for-testing-series.html"><i class="fa fa-check"></i><b>11.7</b> Strategy for Testing Series</a>
<ul>
<li class="chapter" data-level="11.7.1" data-path="strategy-for-testing-series.html"><a href="strategy-for-testing-series.html#worked-strategy-examples"><i class="fa fa-check"></i><b>11.7.1</b> Worked Strategy Examples</a></li>
<li class="chapter" data-level="11.7.2" data-path="strategy-for-testing-series.html"><a href="strategy-for-testing-series.html#putting-it-all-together-53"><i class="fa fa-check"></i><b>11.7.2</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.8" data-path="power-series.html"><a href="power-series.html"><i class="fa fa-check"></i><b>11.8</b> Power Series</a>
<ul>
<li class="chapter" data-level="11.8.1" data-path="power-series.html"><a href="power-series.html#power-series-as-functions"><i class="fa fa-check"></i><b>11.8.1</b> Power Series as Functions</a></li>
<li class="chapter" data-level="11.8.2" data-path="power-series.html"><a href="power-series.html#radius-and-interval-of-convergence"><i class="fa fa-check"></i><b>11.8.2</b> Radius and Interval of Convergence</a></li>
<li class="chapter" data-level="11.8.3" data-path="power-series.html"><a href="power-series.html#putting-it-all-together-54"><i class="fa fa-check"></i><b>11.8.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.9" data-path="representations-of-functions-as-power-series.html"><a href="representations-of-functions-as-power-series.html"><i class="fa fa-check"></i><b>11.9</b> Representations of Functions as Power Series</a>
<ul>
<li class="chapter" data-level="11.9.1" data-path="representations-of-functions-as-power-series.html"><a href="representations-of-functions-as-power-series.html#the-fundamental-geometric-series-model"><i class="fa fa-check"></i><b>11.9.1</b> The Fundamental Geometric Series Model</a></li>
<li class="chapter" data-level="11.9.2" data-path="representations-of-functions-as-power-series.html"><a href="representations-of-functions-as-power-series.html#differentiation-and-integration-of-power-series"><i class="fa fa-check"></i><b>11.9.2</b> Differentiation and Integration of Power Series</a></li>
<li class="chapter" data-level="11.9.3" data-path="representations-of-functions-as-power-series.html"><a href="representations-of-functions-as-power-series.html#putting-it-all-together-55"><i class="fa fa-check"></i><b>11.9.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.10" data-path="taylor-and-maclaurin-series.html"><a href="taylor-and-maclaurin-series.html"><i class="fa fa-check"></i><b>11.10</b> Taylor and Maclaurin Series</a>
<ul>
<li class="chapter" data-level="11.10.1" data-path="taylor-and-maclaurin-series.html"><a href="taylor-and-maclaurin-series.html#taylor-polynomials-finite-approximations"><i class="fa fa-check"></i><b>11.10.1</b> Taylor Polynomials (Finite Approximations)</a></li>
<li class="chapter" data-level="11.10.2" data-path="taylor-and-maclaurin-series.html"><a href="taylor-and-maclaurin-series.html#remainder-and-convergence-idea"><i class="fa fa-check"></i><b>11.10.2</b> Remainder and Convergence Idea</a></li>
<li class="chapter" data-level="11.10.3" data-path="taylor-and-maclaurin-series.html"><a href="taylor-and-maclaurin-series.html#putting-it-all-together-56"><i class="fa fa-check"></i><b>11.10.3</b> Putting It All Together</a></li>
</ul></li>
<li class="chapter" data-level="11.11" data-path="applications-of-taylor-polynomials.html"><a href="applications-of-taylor-polynomials.html"><i class="fa fa-check"></i><b>11.11</b> Applications of Taylor Polynomials</a></li>
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<h1>
<i class="fa fa-circle-o-notch fa-spin"></i><a href="./">MATH 112/ 122: Differential and Integral Calculus</a>
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<div id="exponential-functions-1" class="section level2 hasAnchor" number="1.4">
<h2><span class="header-section-number">1.4</span> Exponential Functions<a href="exponential-functions-1.html#exponential-functions-1" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<div class="definition">
<p><span id="def:unlabeled-div-38" class="definition"><strong>Definition 1.17 </strong></span>An <strong>exponential function</strong> is a function where the variable appears in the exponent:
<span class="math display">\[
f(x) = b^x, \quad b > 0,\; b \neq 1.
\]</span></p>
</div>
<p>This should not be confused with a <strong>power function</strong>:
<span class="math display">\[
g(x) = x^n,
\]</span>
where the variable is the base, not the exponent.</p>
<table>
<thead>
<tr class="header">
<th>Type</th>
<th>Form</th>
<th>Variable Location</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Exponential</td>
<td><span class="math inline">\(b^x\)</span></td>
<td>Variable in exponent</td>
</tr>
<tr class="even">
<td>Power</td>
<td><span class="math inline">\(x^n\)</span></td>
<td>Variable in base</td>
</tr>
</tbody>
</table>
<div class="theorem">
<p><span id="thm:unlabeled-div-39" class="theorem"><strong>Theorem 1.1 </strong></span>For a base <span class="math inline">\(b>0\)</span>:</p>
<ul>
<li>Integer exponents:
<span class="math display">\[
b^n = \underbrace{b \cdot b \cdot \dots \cdot b}_{n \text{ times}}
\]</span><br />
</li>
<li>Zero exponent:
<span class="math display">\[
b^0 = 1
\]</span><br />
</li>
<li>Negative exponent:
<span class="math display">\[
b^{-n} = \frac{1}{b^n}
\]</span><br />
</li>
<li>Rational exponents:
<span class="math display">\[
b^{p/q} = \sqrt[q]{b^p}
\]</span><br />
</li>
<li><strong>Irrational exponents</strong> are defined using limits and approximations, ensuring:
<span class="math display">\[
b^x \text{ is defined for all real } x
\]</span></li>
</ul>
<p>This allows exponential functions to have domain <span class="math inline">\(\mathbb{R}\)</span> (all real numbers).</p>
</div>
<hr />
<div id="graphical-behavior-of-y-bx" class="section level3 hasAnchor" number="1.4.1">
<h3><span class="header-section-number">1.4.1</span> Graphical Behavior of <span class="math inline">\(y = b^x\)</span><a href="exponential-functions-1.html#graphical-behavior-of-y-bx" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>There are three Fundamental Cases</p>
<ol style="list-style-type: decimal">
<li><strong>Growth</strong>: <span class="math inline">\(b > 1\)</span> → Increasing function<br />
</li>
<li><strong>Decay</strong>: <span class="math inline">\(0 < b < 1\)</span> → Decreasing function<br />
</li>
<li><strong>Constant</strong>: <span class="math inline">\(b = 1\)</span> → <span class="math inline">\(y = 1\)</span></li>
</ol>
<div class="note">
<p><strong>Properties</strong></p>
<ul>
<li>Domain: <span class="math inline">\((-\infty, \infty)\)</span><br />
</li>
<li>Range: <span class="math inline">\((0, \infty)\)</span><br />
</li>
<li>Always passes through: <span class="math inline">\((0,1)\)</span><br />
</li>
<li>Horizontal asymptote: <span class="math inline">\(y = 0\)</span><br />
</li>
</ul>
</div>
<div class="theorem">
<p><span id="thm:unlabeled-div-40" class="theorem"><strong>Theorem 1.2 </strong></span><strong>Laws of Exponents</strong></p>
<p>For <span class="math inline">\(a,b>0\)</span> and real <span class="math inline">\(x,y\)</span>:</p>
<p><span class="math display">\[
\begin{aligned}
b^{x+y} &= b^x b^y \\
b^{x-y} &= \frac{b^x}{b^y} \\
(b^x)^y &= b^{xy} \\
(ab)^x &= a^x b^x
\end{aligned}
\]</span></p>
</div>
<div class="example">
<p><span id="exm:unlabeled-div-41" class="example"><strong>Example 1.22 </strong></span><strong>Example 1: Product Rule</strong></p>
<p>Simplify:
<span class="math display">\[
2^x \cdot 2^3 \cdot 2^{x-1}
\]</span></p>
<p><strong>Solution</strong></p>
<p>All terms have the same base <span class="math inline">\(2\)</span>, so we add the exponents:</p>
<p><span class="math display">\[
2^x \cdot 2^3 \cdot 2^{x-1}
= 2^{x + 3 + (x-1)}
= 2^{2x + 2}.
\]</span></p>
<p><strong>Example 2: Quotient Rule</strong></p>
<p>Simplify:
<span class="math display">\[
\frac{5^{2x+1}}{5^{x-4}}
\]</span></p>
<p><strong>Solution</strong></p>
<p>Same base <span class="math inline">\(5\)</span>, so subtract the exponents:</p>
<p><span class="math display">\[
\frac{5^{2x+1}}{5^{x-4}}
= 5^{(2x+1) - (x-4)}
= 5^{2x+1-x+4}
= 5^{x+5}.
\]</span></p>
<p><strong>Example 3: Power of a Power</strong></p>
<p>Simplify:
<span class="math display">\[
(3^{x-2})^4
\]</span></p>
<p><strong>Solution</strong></p>
<p>Multiply the exponents:
<span class="math display">\[
(3^{x-2})^4 = 3^{4(x-2)} = 3^{4x-8}.
\]</span></p>
<p><strong>Example 4: Product to a Power</strong></p>
<p>Simplify:
<span class="math display">\[
(2x^3)^4
\]</span></p>
<p><strong>Solution</strong></p>
<p>Apply the power to each factor:
<span class="math display">\[
(2x^3)^4 = 2^4 \cdot (x^3)^4 = 16x^{12}.
\]</span></p>
<p><strong>Example 5: Rational Exponents</strong></p>
<p>Simplify:
<span class="math display">\[
16^{3/4}
\]</span></p>
<p><strong>Solution</strong></p>
<p>Rewrite using roots:
<span class="math display">\[
16^{3/4} = (\sqrt[4]{16})^3.
\]</span></p>
<p>Since <span class="math inline">\(\sqrt[4]{16} = 2\)</span>:
<span class="math display">\[
16^{3/4} = 2^3 = 8.
\]</span></p>
</div>
<p>Exponential functions <strong>eventually grow faster than any power function</strong>, no matter the degree of the power. This is a fundamental concept in calculus and modeling.</p>
<div class="example">
<p><span id="exm:unlabeled-div-42" class="example"><strong>Example 1.23 </strong></span><strong>Mixed Laws</strong></p>
<p>Simplify:
<span class="math display">\[
\frac{(4^x \cdot 2^{2x})^2}{8^x}
\]</span></p>
<p><strong>Solution</strong></p>
<p>Step 1: Rewrite everything in base 2:
<span class="math display">\[
4 = 2^2, \quad 8 = 2^3
\]</span>
<span class="math display">\[
(4^x \cdot 2^{2x})^2 = ((2^2)^x \cdot 2^{2x})^2 = (2^{2x} \cdot 2^{2x})^2
\]</span>
<span class="math display">\[
= (2^{4x})^2 = 2^{8x}
\]</span>
Denominator:
<span class="math display">\[
8^x = (2^3)^x = 2^{3x}
\]</span></p>
<p>Step 2: Apply quotient rule:
<span class="math display">\[
\frac{2^{8x}}{2^{3x}} = 2^{5x}
\]</span></p>
</div>
</div>
<div id="applications-of-exponential-functions" class="section level3 hasAnchor" number="1.4.2">
<h3><span class="header-section-number">1.4.2</span> Applications of Exponential Functions<a href="exponential-functions-1.html#applications-of-exponential-functions" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p><strong>1. Population Growth</strong></p>
<p>If a population doubles every unit of time:
<span class="math display">\[
P(t) = P_0 \cdot 2^t
\]</span></p>
<div class="example">
<p><span id="exm:unlabeled-div-43" class="example"><strong>Example 1.24 </strong></span>If the initial population is 1000 and doubles every year, then
<span class="math display">\[
P(0)=1000 \Rightarrow P(t)=1000 \cdot 2^t.
\]</span></p>
</div>
<p><strong>2. Exponential Decay (Half-Life Model)</strong></p>
<p>If a quantity halves every <span class="math inline">\(h\)</span> units of time:
<span class="math display">\[
M(t) = M_0 \left(\frac{1}{2}\right)^{t/h}
\]</span></p>
<div class="example">
<p><span id="exm:unlabeled-div-44" class="example"><strong>Example 1.25 </strong></span>If the initial mass of a redioactive element is 24 and it has a half life of 25 years, then
<span class="math display">\[
M(t)=24\left(\frac12\right)^{t/25}
\]</span></p>
</div>
<p><strong>3. General Exponential Model</strong></p>
<p>The general exponential model is
<span class="math display">\[
y = a b^t
\]</span>
where<br />
- <span class="math inline">\(a\)</span> = initial value<br />
- <span class="math inline">\(b\)</span> = growth/decay factor<br />
- <span class="math inline">\(t\)</span> = time</p>
<hr />
</div>
<div id="the-number-e-and-the-natural-exponential-function" class="section level3 hasAnchor" number="1.4.3">
<h3><span class="header-section-number">1.4.3</span> The Number <span class="math inline">\(e\)</span> and the Natural Exponential Function<a href="exponential-functions-1.html#the-number-e-and-the-natural-exponential-function" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<div class="definition">
<p><span id="def:unlabeled-div-45" class="definition"><strong>Definition 1.18 </strong></span>There exists a special base <span class="math inline">\(e \approx 2.71828\)</span> such that:
<span class="math display">\[
f(x) = e^x
\]</span>
has a <strong>slope of 1 at the point (0,1)</strong>.</p>
</div>
<p>This property makes it the most natural base for calculus.</p>
<div class="note">
<p><strong>Properties of the Natural Exponential Function</strong>
<span class="math display">\[
f(x) = e^x
\]</span></p>
<ul>
<li>Domain: <span class="math inline">\(\mathbb{R}\)</span></li>
<li>Range: <span class="math inline">\((0,\infty)\)</span></li>
<li>Passes through <span class="math inline">\((0,1)\)</span></li>
<li>Growth is extremely rapid</li>
</ul>
</div>
<hr />
</div>
<div id="graph-transformations-of-exponentials" class="section level3 hasAnchor" number="1.4.4">
<h3><span class="header-section-number">1.4.4</span> Graph Transformations of Exponentials<a href="exponential-functions-1.html#graph-transformations-of-exponentials" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Starting from <span class="math inline">\(y = b^x\)</span> or <span class="math inline">\(y = e^x\)</span>:</p>
<table>
<thead>
<tr class="header">
<th>Transformation</th>
<th>Effect</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><span class="math inline">\(b^{-x}\)</span></td>
<td>Reflect across y-axis</td>
</tr>
<tr class="even">
<td><span class="math inline">\(-b^x\)</span></td>
<td>Reflect across x-axis</td>
</tr>
<tr class="odd">
<td><span class="math inline">\(c b^x\)</span></td>
<td>Vertical stretch/compression</td>
</tr>
<tr class="even">
<td><span class="math inline">\(b^x + k\)</span></td>
<td>Vertical shift</td>
</tr>
</tbody>
</table>
<div class="example">
<p><span id="exm:unlabeled-div-46" class="example"><strong>Example 1.26 </strong></span>Describe the graph of:
<span class="math display">\[
y = \frac{1}{2} e^{-x} - 1.
\]</span></p>
<p><strong>Solution:</strong></p>
<p>Steps:</p>
<ol style="list-style-type: decimal">
<li>Reflect <span class="math inline">\(e^x\)</span> → <span class="math inline">\(e^{-x}\)</span><br />
</li>
<li>Vertical compression by factor <span class="math inline">\(\frac{1}{2}\)</span><br />
</li>
<li>Shift down 1 unit<br />
</li>
</ol>
</div>
<hr />
</div>
<div id="putting-it-all-together-3" class="section level3 hasAnchor" number="1.4.5">
<h3><span class="header-section-number">1.4.5</span> Putting It All Together<a href="exponential-functions-1.html#putting-it-all-together-3" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p><strong>Exponential functions describe systems where the rate of change is proportional to the current size of the system.</strong> This single idea explains why they appear everywhere in science, engineering, economics, biology, computing, and data science.</p>
<div id="conceptual-takeaways-3" class="section level4 hasAnchor" number="1.4.5.1">
<h4><span class="header-section-number">1.4.5.1</span> Conceptual Takeaways<a href="exponential-functions-1.html#conceptual-takeaways-3" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<ul>
<li>Exponential functions model <strong>multiplicative change</strong>, not additive change<br />
</li>
<li>Small growth rates → massive long-term effects<br />
</li>
<li>Exponential growth eventually dominates polynomial growth<br />
</li>
<li>The base controls behavior:
<ul>
<li><span class="math inline">\(b>1\)</span>: growth<br />
</li>
<li><span class="math inline">\(0<b<1\)</span>: decay<br />
</li>
</ul></li>
<li><span class="math inline">\(e^x\)</span> is special because it behaves optimally for calculus<br />
</li>
<li>Real-world systems often follow exponential laws:
<ul>
<li>populations<br />
</li>
<li>finance<br />
</li>
<li>radioactive decay<br />
</li>
<li>virus spread<br />
</li>
<li>charging/discharging systems<br />
</li>
<li>learning curves<br />
</li>
<li>neural activity<br />
</li>
<li>data growth<br />
</li>
<li>computing performance scaling</li>
</ul></li>
</ul>
<hr />
</div>
<div id="skills-you-should-be-able-to-do-3" class="section level4 hasAnchor" number="1.4.5.2">
<h4><span class="header-section-number">1.4.5.2</span> Skills You Should Be Able To Do<a href="exponential-functions-1.html#skills-you-should-be-able-to-do-3" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<ul>
<li>Identify exponential functions<br />
</li>
<li>Distinguish exponential vs power functions<br />
</li>
<li>Interpret growth vs decay from equations<br />
</li>
<li>Find domain and range<br />
</li>
<li>Explain why exponential growth is “fast”<br />
</li>
<li>Compare growth rates of functions<br />
</li>
<li>Understand the meaning of base <span class="math inline">\(b\)</span><br />
</li>
<li>Understand why <span class="math inline">\(e\)</span> is special<br />
</li>
<li>Apply laws of exponents<br />
</li>
<li>Graph exponential functions<br />
</li>
<li>Perform graph transformations<br />
</li>
<li>Solve basic exponential equations</li>
</ul>
<hr />
</div>
<div id="problems-2" class="section level4 hasAnchor" number="1.4.5.3">
<h4><span class="header-section-number">1.4.5.3</span> Problems<a href="exponential-functions-1.html#problems-2" class="anchor-section" aria-label="Anchor link to header"></a></h4>
</div>
</div>
</div>
</section>
</div>
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