Textbook Question
Checking Antiderivative Formulas
Right, or wrong? Say which for each formula and give a brief reason for each answer.
∫xsinx dx = -x cos x + C
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.3.5b
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
b. On what open intervals is f increasing or decreasing?
f′(x) = (x − 1)(x + 2)(x − 3)
Verified step by step guidance1
First, identify the critical points of the function f by setting the derivative f'(x) = (x - 1)(x + 2)(x - 3) equal to zero. Solve the equation (x - 1)(x + 2)(x - 3) = 0 to find the values of x where the derivative is zero.
The solutions to the equation are x = 1, x = -2, and x = 3. These are the critical points where the function f could change from increasing to decreasing or vice versa.
Next, determine the sign of f'(x) in the intervals defined by the critical points: (-∞, -2), (-2, 1), (1, 3), and (3, ∞). Choose test points from each interval and substitute them into f'(x) to check whether the derivative is positive or negative.
For each interval, if f'(x) is positive, the function f is increasing in that interval. If f'(x) is negative, the function f is decreasing in that interval.
Summarize the intervals where f is increasing and decreasing based on the sign of f'(x) in each interval. This will give you the open intervals where the function f is increasing or decreasing.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Points
Critical points of a function occur where its derivative is zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. In the given derivative f′(x) = (x − 1)(x + 2)(x − 3), the critical points are x = 1, x = -2, and x = 3, where the derivative equals zero.
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04:50
Critical Points
Increasing and Decreasing Intervals
A function is increasing on intervals where its derivative is positive and decreasing where its derivative is negative. By analyzing the sign of f′(x) = (x − 1)(x + 2)(x − 3) across different intervals determined by the critical points, we can determine where the function f is increasing or decreasing.
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07:32Determining Where a Function is Increasing & Decreasing
Sign Chart Analysis
A sign chart helps visualize the sign changes of a derivative across different intervals. By testing points in each interval created by the critical points, we can determine the sign of the derivative in those intervals. This analysis helps identify where the function is increasing or decreasing, based on whether the derivative is positive or negative.
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06:51Derivatives Applied To Acceleration Example 2
Related Practice
Textbook Question
Identifying Extrema
In Exercises 19–40:
b. Identify the function’s local extreme values, if any, saying where they occur.
f(θ) = 3θ² − 4θ³
Textbook Question
Identifying Extrema
In Exercises 19–40:
b. Identify the function’s local extreme values, if any, saying where they occur.
f(r) = 3r³ + 16r
Textbook Question
Theory and Examples
In Exercises 51 and 52, give reasons for your answers.
Let f(x) = |x³ − 9x|.
b. Does f'(3) exist?
Textbook Question
Analyzing Functions from Derivatives
Answer the following questions about the functions whose derivatives are given in Exercises 1–14:
b. On what open intervals is f increasing or decreasing?
f′(x) = 1− 4/x², x ≠ 0
Textbook Question
Finding Antiderivatives
In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.
x⁻³/2 + x²
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