Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

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2,819,387 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section focuses on applying differential calculus to solve optimization problems by determining the maximum and minimum values of functions. It covers essential concepts such as absolute and local extremes, the Extreme Value Theorem, and Fermat’s Theorem, along with the use of critical numbers and the Closed Interval Method. Through worked examples and real-world applications, students learn to approach and solve a variety of problems in engineering, physics, and biology using differentiation techniques.

Learning Objectives

1

Understand and differentiate between absolute and local maximum and minimum values in optimization problems.

2

Apply the Extreme Value Theorem to continuous functions on closed intervals to guarantee the existence of extreme values.

3

Utilize Fermat’s Theorem to locate potential extreme values by identifying critical numbers where the derivative is zero or does not exist.

4

Learn and implement the Closed Interval Method to determine absolute extreme values by evaluating both critical points and endpoints.

5

Analyze real-world applications of differentiation such as optimizing manufacturing costs, maximizing acceleration, and modeling physiological phenomena.

Key Concepts

CONCEPT

DEFINITION

Absolute Maximum

The highest value of a function on its domain, f(c) is the absolute maximum on D if f(c) ≥ f(x) for all x in D.

Absolute Minimum

The lowest value of a function on its domain, f(c) is the absolute minimum on D if f(c) ≤ f(x) for all x in D.

Local Maximum and Minimum

The highest or lowest value of a function in a small open interval around a point c. f(c) is a local maximum if it is greater than nearby values, and similarly, a local minimum if it is less.

Extreme Value Theorem

A continuous function on a closed interval [a, b] attains both an absolute maximum and an absolute minimum within that interval.

Fermat’s Theorem

If a function f has a local maximum or minimum at c and f'(c) exists, then f'(c) equals 0. This only provides potential candidates for extreme values.

Critical Number

A number c in the function’s domain for which either f'(c) = 0 or f'(c) does not exist. Critical numbers are prime candidates for extreme values.

Closed Interval Method

An approach to finding absolute extremes on a closed interval by evaluating the function at its critical numbers and at the interval endpoints.

Example Problems

Example 1

Explain the difference between an absolute minimum and a local minimum.

Example 2

Suppose $ f $ is a continuous function defined on a closed interval $ [a, b] $. (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for $ f $? (b) What steps would you take to find those maximum and minimum values?

Example 3

For each of the numbers $ a $, $ b $, $ c $, $ d $, $ r $, and $ s $, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

Example 4

For each of the numbers $ a $, $ b $, $ c $, $ d $, $ r $, and $ s $, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.

Example 5

Use the graph to state the absolute and local maximum and minimum values of the function.
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Step-by-Step Explanations

QUESTION

How do you find the absolute maximum and minimum values of a continuous function on a closed interval?

STEP-BY-STEP ANSWER:

Step 1: Identify and compute the derivative of the function to find all critical numbers within the interval where f'(x) = 0 or does not exist.
Step 2: Evaluate the function at each of the critical numbers that lie within the interval.
Step 3: Evaluate the function at the endpoints of the interval.
Step 4: Compare all the values obtained; the highest value indicates the absolute maximum and the lowest value the absolute minimum.
Final Answer: The absolute extrema are found by taking the maximum and minimum of the set {f(critical numbers), f(a), f(b)}.

Finding Absolute Extreme Values

QUESTION

Why does Fermat’s Theorem suggest that local extreme points occur where the derivative is zero?

STEP-BY-STEP ANSWER:

Step 1: Understand that a local extreme value means that there is a small interval around c where f(c) is either the highest or lowest.
Step 2: Knowing that the derivative represents the slope, at a peak or valley the slope is flat or horizontal.
Step 3: Therefore, if f is differentiable at c and has a local extreme at that point, it must be that f'(c) = 0.
Final Answer: Fermat’s Theorem assures that if a differentiable function has a local extreme at c, then the tangent line at c is horizontal, hence f'(c) = 0.

Applying Fermat’s Theorem

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Common Mistakes

  • Confusing local extremes with absolute extremes without checking the endpoints of the interval.
  • Assuming that if the derivative equals zero at a point, a maximum or minimum must exist.
  • Failing to consider cases where the derivative does not exist, yet an extreme value may occur.
  • Overlooking the importance of continuity on a closed interval as stated in the Extreme Value Theorem.