Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the use of derivatives to solve optimization problems by finding extreme values of functions. It introduces fundamental concepts such as absolute and local extrema, critical points, and the Extreme Value Theorem, which assures the existence of extrema on closed intervals. Important theorems like Rolle’s and the Mean Value Theorem build on these concepts, providing the theoretical foundation to guarantee conditions under which functions have specific behaviors, such as horizontal tangents. These tools are essential in a wide variety of real-world and theoretical applications across mathematics, science, and engineering.

Learning Objectives

1

Apply derivatives to solve optimization problems across various fields.

2

Identify and distinguish between absolute and local extrema on closed intervals.

3

Utilize the Extreme Value Theorem and the First Derivative Test to locate extreme values.

4

Explain and apply Rolle’s Theorem and the Mean Value Theorem in problem solving.

Key Concepts

CONCEPT

DEFINITION

Optimization

The process of finding the best possible solution, such as maximum profit or minimum cost, often using derivatives to locate extreme values.

Absolute Maximum/Minimum

The highest or lowest value taken by a function on a given domain; also known as global maximum or minimum.

Local Maximum/Minimum

The highest or lowest value in a small interval around a point within the function’s domain, also called relative extrema.

Critical Point

An interior point of the domain of a function where its derivative is zero or undefined, making it a candidate for an extreme value.

Extreme Value Theorem

A theorem stating that if a function is continuous on a closed and finite interval, then it has both an absolute maximum and an absolute minimum on that interval.

Rolle’s Theorem

A special case of the Mean Value Theorem which states that if a function is continuous on a closed interval and differentiable on its interior, and its values at the endpoints are equal, then there exists at least one point in the interior where the derivative is zero.

Mean Value Theorem

A theorem that generalizes Rolle’s Theorem by guaranteeing that for a function continuous on [a, b] and differentiable on (a, b), there is at least one point c where the instantaneous rate of change equals the average rate of change over the interval.

Example Problems

Example 1

In Exercises $1-6,$ determine from the graph whether the function has any absolute extreme values on $[a, b] .$ Then explain how your answer is consistent with. Theorem 1

Example 2

In Exercises $1-6,$ determine from the graph whether the function has any absolute extreme values on $[a, b] .$ Then explain how your answer is consistent with. Theorem 1

Example 3

In Exercises $1-6,$ determine from the graph whether the function has any absolute extreme values on $[a, b] .$ Then explain how your answer is consistent with. Theorem 1

Example 4

In Exercises $1-6,$ determine from the graph whether the function has any absolute extreme values on $[a, b] .$ Then explain how your answer is consistent with. Theorem 1

Example 5

In Exercises $1-6,$ determine from the graph whether the function has any absolute extreme values on $[a, b] .$ Then explain how your answer is consistent with. Theorem 1
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Step-by-Step Explanations

QUESTION

Find the absolute maximum and minimum values of f(x) = x² on the interval [-2, 1].

STEP-BY-STEP ANSWER:

Step 1: Compute the derivative: f'(x) = 2x.
Step 2: Set the derivative equal to zero: 2x = 0, which gives x = 0 as a critical point.
Step 3: Evaluate the function at the critical point and the endpoints: f(-2) = 4, f(0) = 0, f(1) = 1.
Step 4: Compare the values: The smallest value is 0 and the largest is 4.
Final Answer: The absolute minimum is 0 at x = 0 and the absolute maximum is 4 at x = -2.

Finding Absolute Extrema

QUESTION

Verify Rolle’s Theorem for f(x) = sin(x) on the interval [0, π].

STEP-BY-STEP ANSWER:

Step 1: Confirm that the function is continuous on [0, π] and differentiable on (0, π).
Step 2: Check the endpoint values: f(0) = 0 and f(Ï€) = 0, so f(0) = f(Ï€).
Step 3: By Rolle's Theorem, there exists at least one c in (0, π) such that f'(c) = 0.
Step 4: Find the derivative: f'(x) = cos(x); setting cos(c) = 0 gives c = π/2.
Final Answer: c = π/2 is the guaranteed point where the tangent is horizontal, satisfying Rolle’s Theorem.

Verifying Rolle’s Theorem

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Common Mistakes

  • Neglecting to evaluate the function at the endpoints of a closed interval, which can lead to missing the absolute extrema.
  • Assuming that every critical point (where the derivative is zero or undefined) is an extreme point without further analysis.
  • Overlooking the requirement for the domain to be both closed and finite when applying the Extreme Value Theorem.
  • Confusing local extrema with absolute extrema, especially in cases where the function has multiple local extreme values.