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.
1 / 2x³
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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 4.3.23b
Identifying Extrema
In Exercises 19–40:
b. Identify the function’s local extreme values, if any, saying where they occur.
f(θ) = 3θ² − 4θ³
Verified step by step guidance1
To identify the local extrema of the function \( f(\theta) = 3\theta^2 - 4\theta^3 \), first find the derivative \( f'(\theta) \). This will help us locate critical points where extrema might occur.
Calculate the derivative: \( f'(\theta) = \frac{d}{d\theta}(3\theta^2 - 4\theta^3) = 6\theta - 12\theta^2 \).
Set the derivative equal to zero to find critical points: \( 6\theta - 12\theta^2 = 0 \). Factor the equation: \( 6\theta(1 - 2\theta) = 0 \). This gives us \( \theta = 0 \) and \( \theta = \frac{1}{2} \) as critical points.
Determine the nature of these critical points by using the second derivative test. Find the second derivative: \( f''(\theta) = \frac{d}{d\theta}(6\theta - 12\theta^2) = 6 - 24\theta \).
Evaluate \( f''(\theta) \) at the critical points: For \( \theta = 0 \), \( f''(0) = 6 \) (positive, indicating a local minimum). For \( \theta = \frac{1}{2} \), \( f''\left(\frac{1}{2}\right) = 6 - 24\left(\frac{1}{2}\right) = -6 \) (negative, indicating a local maximum). Thus, the function has a local minimum at \( \theta = 0 \) and a local maximum at \( \theta = \frac{1}{2} \).
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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 potential locations for local extrema, as they indicate where the function's slope changes direction. To find critical points, compute the derivative of the function and solve for values of the variable where the derivative equals zero or is undefined.
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04:50
Critical Points
First Derivative Test
The First Derivative Test helps determine whether a critical point is a local maximum or minimum. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function. If the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it is a local minimum.
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07:09The First Derivative Test: Finding Local Extrema
Second Derivative Test
The Second Derivative Test provides another method to classify critical points. If the second derivative at a critical point is positive, the function is concave up, indicating a local minimum. Conversely, if the second derivative is negative, the function is concave down, indicating a local maximum. If the second derivative is zero, the test is inconclusive.
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06:02The Second Derivative Test: Finding Local Extrema
Related Practice
Textbook Question
[Technology Exercise] 17. Designing a suitcase A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid.
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b. Find the domain of V for the problem situation and graph V over this domain.
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
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)
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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