Book cover for Calculus Applications and Technology

Calculus Applications and Technology

Edmond C. Tomastik

ISBN #9780534464967

3rd Edition

2,380 Questions

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8,785 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

The section focuses on using the first derivative to determine the behavior of functions. It introduces the concepts of critical values, the constant sign theorem, and the first derivative test, which allow us to identify intervals of increase or decrease and to classify relative extrema. Through both graphical and analytic methods, one learns how to apply these concepts in optimization problems across various fields such as economics and biology.

Learning Objectives

1

Explain how the first derivative determines where a function is increasing or decreasing.

2

Identify and compute critical values where f?(x) is zero or undefined.

3

Apply the constant sign theorem and the first derivative test to determine relative extrema.

4

Analyze real-world applications of derivative tests in optimization and graph sketching.

Key Concepts

CONCEPT

DEFINITION

First Derivative Test

A method for determining where a function is increasing or decreasing based on the sign of its derivative. It also identifies relative maxima and minima when the derivative changes sign around a critical value.

Critical Value

A value x = c in the domain of the function such that either f′(c) = 0 or f′(c) does not exist.

Constant Sign Theorem

A theorem that states if a continuous function f(x) never equals zero on an interval, then f(x) keeps a constant sign (always positive or always negative) on that interval.

Relative Maximum and Minimum

Points where the function attains a peak (maximum) or a valley (minimum) locally, determined by changes in the sign of the first derivative around the point.

Example Problems

Example 1

Various graphs are shown. In each case, find all values where the function attains a relative maximum. A relative minimum.

Example 2

Various graphs are shown. In each case, find all values where the function attains a relative maximum. A relative minimum.

Example 3

Various graphs are shown. In each case, find all values where the function attains a relative maximum. A relative minimum.

Example 4

Various graphs are shown. In each case, find all values where the function attains a relative maximum. A relative minimum.

Example 5

Various graphs are shown. In each case, find all values where the function attains a relative maximum. A relative minimum.
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Step-by-Step Explanations

QUESTION

For the function f(x) = x³ − 3x² + 1, determine the intervals on which the function is increasing or decreasing.

STEP-BY-STEP ANSWER:

Step 1: Find the derivative. f′(x) = 3x² − 6x.
Step 2: Set the derivative equal to zero: 3x² − 6x = 0, which factors as 3x(x − 2) = 0. The solutions are x = 0 and x = 2, these are the critical values.
Step 3: Divide the number line into intervals: (−∞, 0), (0, 2) and (2, ∞).
Step 4: Choose test points in each interval (for example, x = −1, x = 1, and x = 3) and evaluate f′(x).
Step 5: f′(−1) is positive (since 3(1 + 6) > 0); f′(1) is negative; f′(3) is positive.
Step 6: Conclude that f(x) is increasing on (−∞, 0) and (2, ∞), and decreasing on (0, 2).
Final Answer: f(x) increases on (−∞, 0) and (2, ∞), and decreases on (0, 2).

Determining increasing/decreasing behavior

QUESTION

Given a graph of a function where the derivative equals zero at points a, b and c, and does not exist at point d, identify the critical values.

STEP-BY-STEP ANSWER:

Step 1: Recognize that any point where the derivative is zero is a critical value. Thus, a, b, and c are critical values.
Step 2: Identify any point where the derivative does not exist but the function is defined, which makes d a critical value.
Step 3: List all identified critical values: a, b, c, and d.
Final Answer: The critical values are a, b, c, and d.

Identifying Critical Values Graphically

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Common Mistakes

  • Canceling a common factor (like x) without considering that it may eliminate a valid critical point.
  • Forgetting to check points where the derivative is undefined even if the function is defined, which can overlook important critical values.
  • Assuming that the derivative’s sign is non-changing without testing at representative points in each interval.
  • Confusing relative extrema with absolute extrema, as the first derivative test only provides local information.