Book cover for Calculus Early Transcendental Functions

Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

ISBN #9781285774770

6th Edition

8,973 Questions

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64,375 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section focuses on finding extrema of functions using differentiation. It emphasizes the importance of evaluating functions at critical numbers and endpoints on closed intervals using the Extreme Value Theorem. Additionally, the section introduces Rolleā€™s Theorem by providing conditions under which a functionā€™s derivative must be zero, illustrating the use of derivative sign changes to identify relative extrema. Understanding these theoretical concepts helps in solving real-world optimization problems.

Learning Objectives

1

Explain the definitions of absolute (global) and relative (local) extrema and understand their significance on different types of intervals.

2

Identify and compute critical numbers of a function by analyzing where the derivative is zero or does not exist.

3

Apply differentiation techniques to determine extrema on closed intervals, including evaluating endpoints.

4

Utilize Rolleā€™s Theorem to conclude the existence of horizontal tangents in functions that satisfy its conditions.

5

Interpret real-life applications of extrema and the Mean Value Theorem using technology and problem-solving strategies.

Key Concepts

CONCEPT

DEFINITION

Extrema

The maximum or minimum value of a function on an interval. When a function attains its absolute highest or lowest point, these are called absolute (global) extrema; when they occur in a neighborhood or open interval, they are relative (local) extrema.

Absolute Extrema

The absolute maximum or minimum values that a function reaches on a closed interval.

Relative Extrema

Points where the function attains a local maximum or minimum value, typically occurring at interior points, which may not be the overall highest or lowest values on the interval.

Critical Number

A number c in the domain of a function where the derivative is zero or does not exist. These points are potential candidates for relative extrema.

Extreme Value Theorem

A theorem stating that if a function is continuous on a closed interval, then it must have both a minimum and a maximum on that interval.

Rolleā€™s Theorem

A theorem which states that if a function is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) where f'(c) = 0.

Example Problems

Example 1

Find the value of the derivative (if it exists) at each indicated extremum. $$f(x)=\frac{x^{2}}{x^{2}+4}$$

Example 2

Find the value of the derivative (if it exists) at each indicated extremum. $$f(x)=\cos \frac{\pi x}{2}$$

Example 3

Find the value of the derivative (if it exists) at each indicated extremum. $$g(x)=x+\frac{4}{x^{2}}$$

Example 4

Find the value of the derivative (if it exists) at each indicated extremum. $$f(x)=-3 x \sqrt{x+1}$$

Example 5

Find the value of the derivative (if it exists) at each indicated extremum. $$f(x)=(x+2)^{2 / 3}$$

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Step-by-Step Explanations

QUESTION

Find the absolute maximum and minimum of f(x) = 3xā“ - 4xĀ³ on the closed interval [-1, 2].

STEP-BY-STEP ANSWER:

Step 1: Differentiate f(x) to obtain f'(x) = 12xĀ³ - 12xĀ².
Step 2: Factor the derivative: f'(x) = 12xĀ²(x - 1) and set it equal to zero to find critical numbers. This yields x = 0 and x = 1.
Step 3: Evaluate f(x) at the critical numbers and at the endpoints of the interval: compute f(-1), f(0), f(1), and f(2).
Step 4: Compare these function values to determine which is the absolute maximum and which is the absolute minimum.
Final Answer: The highest and lowest values among these evaluations give the absolute maximum and minimum of f on [-1, 2].

Finding Absolute Extrema on a Closed Interval

QUESTION

Verify Rolleā€™s Theorem for f(x) = sin(x) on the interval [0, Ļ€].

STEP-BY-STEP ANSWER:

Step 1: Confirm continuity and differentiability. f(x) = sin(x) is continuous on [0, Ļ€] and differentiable on (0, Ļ€).
Step 2: Check the endpoint values: f(0) = 0 and f(Ļ€) = 0, satisfying the condition f(0) = f(Ļ€).
Step 3: Differentiate f(x) to find f'(x) = cos(x) and solve the equation cos(x) = 0 on the interval (0, Ļ€).
Step 4: One solution is x = Ļ€/2, where the derivative is zero.
Final Answer: x = Ļ€/2 is a value in (0, Ļ€) where f'(x) = 0, thereby verifying Rolleā€™s Theorem.

Applying Rolleā€™s Theorem

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Common Mistakes

  • Failing to evaluate endpoints when searching for absolute extrema on a closed interval.
  • Assuming that every critical number (where f'(x) = 0 or undefined) results in a relative extremum.
  • Overlooking the necessity of continuity and differentiability conditions required by the Extreme Value Theorem and Rolleā€™s Theorem.
  • Mixing up the definitions of absolute and relative extrema, leading to incorrect conclusions during analysis.