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<!DOCTYPE html>
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<title>9.2 Direction Fields and Euler’s Method | 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>
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<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>
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<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>
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<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></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></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></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></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></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></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></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></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></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></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: Differential Calculus</a>
</h1>
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<section class="normal" id="section-">
<div id="direction-fields-and-eulers-method" class="section level2 hasAnchor" number="9.2">
<h2><span class="header-section-number">9.2</span> Direction Fields and Euler’s Method<a href="direction-fields-and-eulers-method.html#direction-fields-and-eulers-method" class="anchor-section" aria-label="Anchor link to header"></a></h2>
<p>In most real-world applications, differential equations cannot be solved explicitly in closed form. This means we often cannot write solutions as formulas. However, we can still understand solution behavior using:</p>
<ul>
<li><strong>Graphical methods</strong> → direction (slope) fields<br />
</li>
<li><strong>Numerical methods</strong> → Euler’s method</li>
</ul>
<p>These approaches allow us to:</p>
<ul>
<li>Visualize solution curves<br />
</li>
<li>Predict long-term behavior<br />
</li>
<li>Approximate solutions at specific points<br />
</li>
<li>Interpret physical and biological models</li>
</ul>
<hr />
<div id="direction-fields-slope-fields" class="section level3 hasAnchor" number="9.2.1">
<h3><span class="header-section-number">9.2.1</span> Direction Fields (Slope Fields)<a href="direction-fields-and-eulers-method.html#direction-fields-slope-fields" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>A first-order differential equation has the form
<span class="math display">\[
y' = F(x,y).
\]</span></p>
<p>At each point <span class="math inline">\((x,y)\)</span>, the value <span class="math inline">\(F(x,y)\)</span> gives the <strong>slope</strong> of the solution curve at that point. A <strong>direction field</strong> is a collection of short line segments drawn at many points <span class="math inline">\((x,y)\)</span>, each with slope <span class="math inline">\(F(x,y)\)</span>. These line segments show the <em>direction</em> in which solution curves must flow. This allows us to sketch solution curves without solving the equation.</p>
<div class="example">
<p><span id="exm:unlabeled-div-5" class="example"><strong>Example 9.4 </strong></span>Basic Direction Field</p>
<p><span class="math display">\[
y' = x + y, \qquad y(0) = 1
\]</span></p>
<p>At any point <span class="math inline">\((x,y)\)</span>, the slope is:
<span class="math display">\[
\text{slope} = x + y.
\]</span></p>
<p>So:</p>
<ul>
<li>At <span class="math inline">\((0,1)\)</span>: slope = <span class="math inline">\(1\)</span><br />
</li>
<li>At <span class="math inline">\((1,2)\)</span>: slope = <span class="math inline">\(3\)</span><br />
</li>
<li>At <span class="math inline">\((-1,1)\)</span>: slope = <span class="math inline">\(0\)</span></li>
</ul>
<p><strong>Solution Behavior</strong></p>
<ol style="list-style-type: decimal">
<li>Draw short line segments with slope <span class="math inline">\(x+y\)</span> at many points.<br />
</li>
<li>Start at the initial point <span class="math inline">\((0,1)\)</span>.<br />
</li>
<li>Trace a curve that stays <strong>parallel to the local line segments</strong>.</li>
</ol>
<p>This produces a smooth solution curve following the direction field. See the textbook.</p>
</div>
<hr />
</div>
<div id="autonomous-differential-equations" class="section level3 hasAnchor" number="9.2.2">
<h3><span class="header-section-number">9.2.2</span> Autonomous Differential Equations<a href="direction-fields-and-eulers-method.html#autonomous-differential-equations" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>An equation of the form
<span class="math display">\[
y' = f(y)
\]</span>
is called <strong>autonomous</strong> (no explicit <span class="math inline">\(x\)</span> dependence).</p>
<p><strong>Key Properties:</strong></p>
<ul>
<li>Slopes depend only on <span class="math inline">\(y\)</span><br />
</li>
<li>Direction field has <strong>horizontal translation symmetry</strong><br />
</li>
<li>Solutions can be shifted left/right in <span class="math inline">\(x\)</span><br />
</li>
<li>Constant solutions <span class="math inline">\(y = C\)</span> where <span class="math inline">\(f(C)=0\)</span> are called equilibrium solutions</li>
</ul>
<div class="example">
<p><span id="exm:unlabeled-div-6" class="example"><strong>Example 9.5 </strong></span>Equilibrium Solutions</p>
<p><span class="math display">\[
\frac{dI}{dt} = 15 - 3I
\]</span></p>
<p><strong>Equilibrium Condition</strong>
<span class="math display">\[
0 = 15 - 3I \Rightarrow I = 5
\]</span></p>
<p>So:
<span class="math display">\[
I(t) = 5
\]</span>
is an equilibrium solution.</p>
<p><strong>Interpretation</strong></p>
<p>All nearby solutions move toward <span class="math inline">\(I=5\)</span>. This represents stable equilibrium behavior.</p>
</div>
<hr />
</div>
<div id="eulers-method" class="section level3 hasAnchor" number="9.2.3">
<h3><span class="header-section-number">9.2.3</span> Euler’s Method<a href="direction-fields-and-eulers-method.html#eulers-method" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<p>Euler’s method is a numerical algorithm for approximating solutions of:
<span class="math display">\[
y' = F(x,y), \qquad y(x_0)=y_0
\]</span>
Instead of solving exactly, we approximate step-by-step using tangent lines.</p>
<div class="note">
<p><strong>Euler’s Method Formula</strong></p>
<p>Let step size be <span class="math inline">\(h\)</span>. Define points:
<span class="math display">\[
x_n = x_0 + nh
\]</span>
Then the recursion formula is:
<span class="math display">\[
y_{n} = y_{n-1} + h\,F(x_{n-1}, y_{n-1})
\]</span></p>
</div>
<p>This means:</p>
<blockquote>
<p>New value = old value + (step size × slope)</p>
</blockquote>
<div class="example">
<p><span id="exm:unlabeled-div-7" class="example"><strong>Example 9.6 </strong></span>Euler Approximation</p>
<p>Approximate <span class="math inline">\(y\)</span> near <span class="math inline">\(x = 0.5\)</span> given
<span class="math display">\[
y' = x + y, \qquad y(0)=1, \qquad h=0.1
\]</span></p>
<p>So:
<span class="math display">\[
F(x,y) = x+y.
\]</span></p>
<p><strong>Step 1:</strong>
<span class="math display">\[
y_1 = 1 + 0.1(0+1) = 1.1
\]</span></p>
<p><strong>Step 2:</strong>
<span class="math display">\[
y_2 = 1.1 + 0.1(0.1+1.1) = 1.22
\]</span></p>
<p><strong>Step 3:</strong>
<span class="math display">\[
y_3 = 1.22 + 0.1(0.2+1.22) = 1.362
\]</span></p>
<p><strong>Table</strong></p>
<table>
<thead>
<tr class="header">
<th><span class="math inline">\(n\)</span></th>
<th><span class="math inline">\(x_n\)</span></th>
<th><span class="math inline">\(y_n\)</span></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>0</td>
<td>0.0</td>
<td>1.0000</td>
</tr>
<tr class="even">
<td>1</td>
<td>0.1</td>
<td>1.1000</td>
</tr>
<tr class="odd">
<td>2</td>
<td>0.2</td>
<td>1.2200</td>
</tr>
<tr class="even">
<td>3</td>
<td>0.3</td>
<td>1.3620</td>
</tr>
<tr class="odd">
<td>4</td>
<td>0.4</td>
<td>1.5282</td>
</tr>
<tr class="even">
<td>5</td>
<td>0.5</td>
<td>1.7210</td>
</tr>
</tbody>
</table>
<p>So,
<span class="math display">\[
y(0.5) \approx 1.721
\]</span></p>
</div>
<p><strong>Accuracy and Step Size</strong></p>
<ul>
<li>Smaller <span class="math inline">\(h\)</span> → better accuracy<br />
</li>
<li>Larger <span class="math inline">\(h\)</span> → faster computation, lower accuracy<br />
</li>
<li>As <span class="math inline">\(h \to 0\)</span>, Euler’s method approaches the true solution</li>
</ul>
<p>Euler’s method is the foundation of modern numerical solvers used in scientific computing.</p>
<hr />
</div>
<div id="putting-it-all-together-43" class="section level3 hasAnchor" number="9.2.4">
<h3><span class="header-section-number">9.2.4</span> Putting It All Together<a href="direction-fields-and-eulers-method.html#putting-it-all-together-43" class="anchor-section" aria-label="Anchor link to header"></a></h3>
<blockquote>
<p>Differential equations describe change. Direction fields show <strong>how change behaves</strong>. Euler’s method shows <strong>how change accumulates</strong>.</p>
</blockquote>
<p>Together, they provide a complete framework for understanding systems when exact solutions are impossible.</p>
<div id="conceptual-takeaways-47" class="section level4 hasAnchor" number="9.2.4.1">
<h4><span class="header-section-number">9.2.4.1</span> Conceptual Takeaways<a href="direction-fields-and-eulers-method.html#conceptual-takeaways-47" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<ul>
<li>Most differential equations <strong>cannot be solved explicitly</strong><br />
</li>
<li>Direction fields show <strong>global behavior</strong> of solutions<br />
</li>
<li>Euler’s method gives <strong>local numerical approximations</strong><br />
</li>
<li>Direction fields → qualitative understanding<br />
</li>
<li>Euler’s method → quantitative approximation<br />
</li>
<li>Equilibrium solutions model <strong>steady states</strong><br />
</li>
<li>Autonomous equations exhibit <strong>horizontal shift symmetry</strong></li>
</ul>
<hr />
</div>
<div id="skills-you-should-be-able-to-do-45" class="section level4 hasAnchor" number="9.2.4.2">
<h4><span class="header-section-number">9.2.4.2</span> Skills You Should Be Able to Do<a href="direction-fields-and-eulers-method.html#skills-you-should-be-able-to-do-45" class="anchor-section" aria-label="Anchor link to header"></a></h4>
<p>After this section, you should be able to:</p>
<p><strong>Direction Fields</strong></p>
<ul>
<li>Interpret <span class="math inline">\(y' = F(x,y)\)</span> as a slope rule<br />
</li>
<li>Construct and read direction fields<br />
</li>
<li>Sketch solution curves using slope fields<br />
</li>
<li>Identify equilibrium solutions<br />
</li>
<li>Predict long-term behavior from direction fields</li>
</ul>
<p><strong>Euler’s Method</strong></p>
<ul>
<li>Apply the Euler update formula<br />
</li>
<li>Build numerical solution tables<br />
</li>
<li>Approximate <span class="math inline">\(y(x)\)</span> values<br />
</li>
<li>Understand the effect of step size <span class="math inline">\(h\)</span><br />
</li>
<li>Interpret numerical solutions physically</li>
</ul>
<hr />
</div>
<div id="problems-49" class="section level4 hasAnchor" number="9.2.4.3">
<h4><span class="header-section-number">9.2.4.3</span> Problems<a href="direction-fields-and-eulers-method.html#problems-49" class="anchor-section" aria-label="Anchor link to header"></a></h4>
</div>
</div>
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