Derivatives - First derivative test

</section> <section style="font-size: 50px";> <h2 id="slide--1">Slide 1</h2> <ul> <li>Introduction to the first derivative test</li> <li>Explain the concept of the first derivative</li> <li>Discuss the importance of finding critical points in a function</li> <li>Define a critical point as a point where the first derivative is either 0 or undefined</li> <li>State the goal of the first derivative test

</section> <section style="font-size: 50px";> <h2 id="slide--2">Slide 2</h2> <ul> <li>The first derivative test helps determine whether a critical point is a maximum, minimum, or neither</li> <li>Explain the behavior of a function near a critical point using the first derivative</li> <li>Discuss the sign changes of the first derivative on either side of a critical point</li> <li>Example: f(x) = x^3 - 3x^2 - 4x + 5

</section> <section style="font-size: 50px";> <h2 id="slide--3">Slide 3</h2> <ul> <li>Step 1: Find the critical points by setting the first derivative equal to zero and solve for x</li> <li>Provide an example equation to demonstrate this step</li> <li>Show the process of finding critical points by taking the first derivative and finding where it equals zero</li> <li>Example: f’(x) = 3x^2 - 6x - 4

</section> <section style="font-size: 50px";> <h2 id="slide--4">Slide 4</h2> <ul> <li>Step 2: Determine the sign of the first derivative to the left and right of each critical point</li> <li>Explain how to use test points to evaluate the sign of the first derivative</li> <li>Discuss the intervals where the first derivative is negative or positive</li> <li>Example: Interval analysis for f’(x)

</section> <section style="font-size: 50px";> <h2 id="slide--5">Slide 5</h2> <ul> <li>Step 3: Use the sign changes of the first derivative to classify each critical point</li> <li>Explain the rules for classifying maxima, minima, and inflection points</li> <li>Define a relative maximum and minimum</li> <li>Discuss how to identify inflection points</li> <li>Example: Analyzing the critical points of f(x)

</section> <section style="font-size: 50px";> <h2 id="slide--6">Slide 6</h2> <ul> <li>Recap the steps of the first derivative test</li> <li>Emphasize the need to check the behavior of the function near each critical point</li> <li>Discuss the limitations of the first derivative test</li> <li>Remind students to incorporate second derivative analysis for a comprehensive understanding</li> <li>Example: Applying the first derivative test to real-world scenarios

</section> <section style="font-size: 50px";> <h2 id="slide--7">Slide 7</h2> <ul> <li>Application: Optimization problems using the first derivative test</li> <li>Explain how to use the first derivative to find maximum or minimum values</li> <li>Discuss real-life examples where optimization is required</li> <li>Show the step-by-step process of solving an optimization problem</li> <li>Example: Maximizing the area of a rectangle with a fixed perimeter

</section> <section style="font-size: 50px";> <h2 id="slide--8">Slide 8</h2> <ul> <li>Application: Concavity and the second derivative test</li> <li>Introduce the concept of concavity and the second derivative</li> <li>Define concave up and concave down regions</li> <li>Discuss how the second derivative indicates concavity</li> <li>Example: Analyzing the concavity of f(x) = x^2 - 3x + 2

</section> <section style="font-size: 50px";> <h2 id="slide--9">Slide 9</h2> <ul> <li>Step 1: Find the critical points using the first derivative</li> <li>Recap the process of finding critical points discussed in earlier slides</li> <li>Example: Finding the critical points of f(x) = x^3 - 9x^2 + 24x - 8

</section> <section style="font-size: 50px";> <h2 id="slide--10">Slide 10</h2> <ul> <li>Step 2: Find the second derivative and analyze its sign</li> <li>Explain how to find the second derivative by differentiating the first derivative</li> <li>Discuss the significance of the sign of the second derivative</li> <li>Example: Analyzing the concavity of f(x) using the second derivative, f’’(x)

Slide 11

• Step 3: Determine the concavity of the function based on the sign of the second derivative
• Explain the relationship between concavity and the second derivative
• Discuss concave up and concave down regions
• Identify the intervals where the second derivative is positive or negative
• Example: Analyzing the concavity of f(x) = x^3 - 3x^2 - 4x + 5

Slide 12

• Step 4: Combine the information from steps 2 and 3 to classify each critical point
• Discuss how to use the information from both the first and second derivatives
• Define points of inflection and their significance
• Example: Classifying the critical points of f(x) = x^3 - 3x^2 - 4x + 5

Slide 13

• Recap the steps of the second derivative test
• Emphasize the importance of analyzing both the first and second derivatives
• Discuss the comprehensive understanding achieved through both tests
• Provide examples where the first derivative test may fail but the second derivative test helps identify characteristics of the function
• Example: Applying the second derivative test to determine maxima and minima

Slide 14

• Application: Optimization problems using the second derivative test
• Explain how to use the second derivative to find the maximum or minimum values of a function
• Discuss real-life examples where optimization is required and the second derivative test can be applied
• Show the step-by-step process of solving optimization problems with the second derivative test
• Example: Maximizing the volume of a box with a fixed surface area

Slide 15

• Application: Sketching the graph of a function using the first and second derivative tests
• Explain how to use the information from the first and second derivatives to sketch the graph of a function
• Discuss the characteristics of the graph at critical points, points of inflection, and regions of concavity
• Provide step-by-step instructions on sketching the graph using the derivative tests
• Example: Sketching the graph of f(x) = x^3 - 3x^2 - 4x + 5

Slide 16

• Related rates problems
• Introduce the concept of related rates and their application in real-world scenarios
• Discuss how to approach related rates problems and the use of derivatives
• Provide examples of related rates problems and the step-by-step process of solving them
• Example: Finding the rate at which a shadow is moving along a wall

Slide 17

• Newton’s method
• Explain the basics of Newton’s method and its application in solving equations
• Discuss the process of finding roots or solutions using Newton’s method
• Show step-by-step instructions on applying Newton’s method to find solutions
• Example: Using Newton’s method to find the root of f(x) = x^3 - 3x^2 - 4x + 5

Slide 18

• Limits and continuity
• Define limits and explain their importance in calculus
• Discuss the concept of continuity and its relation to limits
• Explain how to determine if a function is continuous at a point
• Example: Finding the limit and evaluating the continuity of a function

Slide 19

• Intermediate value theorem
• Introduce the intermediate value theorem and its applications
• Explain how the intermediate value theorem is used to show the existence of zeros or solutions
• Discuss the conditions required for the intermediate value theorem to hold true
• Example: Proving the existence of a root using the intermediate value theorem

Slide 20

• Summary and conclusion
• Recap the main topics covered in the lecture
• Emphasize the importance of derivative tests in analyzing the behavior of functions
• Encourage further practice and exploration of calculus topics
• Provide additional resources for reference and practice

</section> <section style="font-size: 50px";> <h2 id="slide--21">Slide 21</h2> <p>Applications of Derivatives - Rates of Change</p> <ul> <li>Introduction to rates of change and their applications</li> <li>Explain how derivatives can be used to find rates of change</li> <li>Discuss real-life examples where rates of change are important</li> <li>Example: Finding the rate at which the volume of a sphere is changing with respect to its radius

</section> <section style="font-size: 50px";> <h2 id="slide--22">Slide 22</h2> <p>Applications of Derivatives - Related Rates</p> <ul> <li>Discuss related rates problems and their application in real-world scenarios</li> <li>Explain how to approach related rates problems and the use of derivatives</li> <li>Show step-by-step examples of solving related rates problems</li> <li>Example: Finding the rate at which the distance between two cars is changing

</section> <section style="font-size: 50px";> <h2 id="slide--23">Slide 23</h2> <p>Applications of Derivatives - Optimization Problems</p> <ul> <li>Discuss optimization problems and their significance in real-life situations</li> <li>Explain how to use derivatives to solve optimization problems</li> <li>Provide step-by-step instructions on solving optimization problems</li> <li>Example: Finding the dimensions of a rectangle with maximum area given a fixed perimeter

</section> <section style="font-size: 50px";> <h2 id="slide--24">Slide 24</h2> <p>Applications of Derivatives - Curve Sketching</p> <ul> <li>Introduce curve sketching and its importance in understanding the behavior of a function</li> <li>Discuss the steps involved in curve sketching, including finding critical points, inflection points, and regions of concavity</li> <li>Provide examples of curve sketching using derivatives</li> <li>Example: Sketching the graph of y = sin(x)

</section> <section style="font-size: 50px";> <h2 id="slide--25">Slide 25</h2> <p>Applications of Derivatives - Improper Integrals</p> <ul> <li>Explain improper integrals and their significance in calculus</li> <li>Discuss the conditions under which improper integrals exist</li> <li>Provide examples of evaluating improper integrals using derivatives</li> <li>Example: Evaluating the integral of 1/x from 1 to infinity

</section> <section style="font-size: 50px";> <h2 id="slide--26">Slide 26</h2> <p>Applications of Derivatives - Differential Equations</p> <ul> <li>Introduction to differential equations and their applications</li> <li>Discuss how derivatives are used to express relationships between variables</li> <li>Explain the process of solving differential equations using integration</li> <li>Example: Solving a simple first-order linear differential equation

</section> <section style="font-size: 50px";> <h2 id="slide--27">Slide 27</h2> <p>Applications of Derivatives - Taylor Series</p> <ul> <li>Introduce Taylor series and their applications in approximation</li> <li>Discuss how Taylor series can be used to represent functions as series expansions</li> <li>Provide examples of finding Taylor series expansions</li> <li>Example: Finding the Taylor series expansion of e^x

</section> <section style="font-size: 50px";> <h2 id="slide--28">Slide 28</h2> <p>Applications of Derivatives - Mean Value Theorem</p> <ul> <li>Explain the mean value theorem and its applications in calculus</li> <li>Discuss the conditions required for the mean value theorem to hold true</li> <li>Provide examples of applying the mean value theorem to solve problems</li> <li>Example: Proving the existence of a point where the derivative is equal to the average rate of change

</section> <section style="font-size: 50px";> <h2 id="slide--29">Slide 29</h2> <p>Applications of Derivatives - L’Hopital’s Rule</p> <ul> <li>Introduce L’Hopital’s rule and its use in evaluating limits</li> <li>Discuss the conditions required for applying L’Hopital’s rule</li> <li>Provide examples of using L’Hopital’s rule to evaluate limits</li> <li>Example: Evaluating the limit of (sin(x))/x as x approaches 0

</section> <section style="font-size: 50px";> <h2 id="slide--30">Slide 30</h2> <p>Applications of Derivatives - Series Convergence</p> <ul> <li>Explain series convergence and its importance in calculus</li> <li>Discuss various tests for series convergence, such as the ratio test and the comparison test</li> <li>Provide examples of applying convergence tests to determine the convergence of a series</li> <li>Example: Analyzing the convergence of the series sum(1/n^2)