Book cover for Calculus with Applications

Calculus with Applications

Margaret L. Lial • Raymond N. Greenwell • Nathan P. Ritchey

ISBN #9781292108971

11th Edition

3,612 Questions

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224,424 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers the analysis of increasing and decreasing functions using graph interpretation and derivative tests. It shows that by examining the sign of the derivative, one can determine the intervals over which a function rises or falls. Critical numbers, identified when the derivative equals zero or is undefined, are essential for subdividing the domain and understanding the function's behavior. The applications range from profit maximization to optimizing viewer attention, making these techniques valuable for both academic and practical problem-solving.

Learning Objectives

1

Explain how to determine intervals where functions are increasing or decreasing using both the graph and derivative tests.

2

Identify and compute critical numbers by finding where the derivative is zero or does not exist.

3

Analyze the behavior of functions through the sign of their derivative and relate this to features like horizontal tangents and increased/decreased intervals.

4

Apply these techniques in real-world contexts such as profit analysis and optimizing production.

5

Graph functions by combining information from critical points, test intervals, and derivative sign analysis.

Key Concepts

CONCEPT

DEFINITION

Increasing Function

A function f(x) is increasing on an interval if for any two numbers x1 < x2 in that interval, f(x1) < f(x2); its graph rises from left to right.

Decreasing Function

A function f(x) is decreasing on an interval if for any two numbers x1 < x2 in that interval, f(x1) > f(x2); its graph falls from left to right.

Derivative

The derivative f'(x) of a function gives the slope of the tangent line at x; it indicates how the function is changing at that point.

Critical Number

A number c in the domain of f where f'(c) = 0 or f'(c) does not exist. Critical points (c, f(c)) are potential sites for local maximums or minimums.

Tangent Line

A line that touches a curve at one point and has the same slope as the curve at that point; used to approximate the behavior of the function.

Example Problems

Example 1

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing. Graph

Example 2

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing. Graph

Example 3

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing. Graph

Example 4

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing. Graph

Example 5

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing. Graph
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Step-by-Step Explanations

QUESTION

Determine where the function f(x) = x³ + 3x² - 9x + 4 is increasing and decreasing, and locate where the tangent line is horizontal.

STEP-BY-STEP ANSWER:

Step 1: Compute the derivative of the function: f'(x) = 3x² + 6x - 9.
Step 2: Find the critical numbers by setting f'(x) = 0. Solve: 3x² + 6x - 9 = 0, which simplifies to x² + 2x - 3 = 0. Factor to get (x + 3)(x - 1) = 0 so that x = -3 and x = 1.
Step 3: Divide the real number line into intervals using the critical numbers: (-∞, -3), (-3, 1), and (1, ∞).
Step 4: Select test points from each interval (e.g., x = -4, 0, and 2) and evaluate f'(x) at these points.
Step 5: Determine the sign of f'(x) in each interval: f'(-4) > 0 (increasing), f'(0) < 0 (decreasing), and f'(2) > 0 (increasing).
Final Answer: f(x) is increasing on (-∞, -3) and (1, ∞), and decreasing on (-3, 1). The tangent line is horizontal at x = -3 and x = 1.

Test for Increasing/Decreasing Functions

QUESTION

For a profit function P(x) = 0.0008x³ - 2.2x² + 1400x, determine the intervals on which profit is increasing, given the domain 0 ≤ x ≤ 1000.

STEP-BY-STEP ANSWER:

Step 1: Compute the derivative of the profit function: P'(x) = 0.0024x² - 4.4x + 1400.
Step 2: Set P'(x) = 0 and solve for x. The quadratic formula gives approximate solutions x ≈ 409.8 and x ≈ 1423.6. Since x must be in the domain [0, 1000], only x ≈ 409.8 is valid.
Step 3: Divide the domain into intervals using x = 409.8: [0, 409.8) and (409.8, 1000].
Step 4: Choose test points in each interval (e.g., x = 0 and x = 1000) and evaluate P'(x). The computations show that P'(0) > 0 and P'(1000) < 0.
Final Answer: The profit function is increasing on the interval (0, 409.8) and decreasing on (409.8, 1000).

Profit Function Analysis

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Common Mistakes

  • Assuming that a function is constant at a point where the derivative is zero; a zero derivative indicates a horizontal tangent but not necessarily constancy.
  • Overlooking critical numbers where the derivative does not exist, which can lead to incomplete analysis of the function’s behavior.
  • Failing to check that a critical number lies within the domain of the function.
  • Incorrectly assuming that the sign of the derivative always alternates on either side of every critical number; factors raised to an even power may prevent sign changes.