Textbook Question
Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.
ƒ(x) = 3x² - 4x + 2
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.9.71
Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.
f(x) = 8x³ + sin x; F(0) = 2
Verified step by step guidance1
Identify the given function to integrate: \(f(x) = 8x^{3} + \sin x\).
Find the general antiderivative \(F(x)\) by integrating each term separately: \(\int 8x^{3} \, dx\) and \(\int \sin x \, dx\).
Recall the integral formulas: \(\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C\) for \(n \neq -1\), and \(\int \sin x \, dx = -\cos x + C\).
Apply these formulas to get \(F(x) = 2x^{4} - \cos x + C\), where \(C\) is the constant of integration.
Use the initial condition \(F(0) = 2\) to solve for \(C\) by substituting \(x=0\) into \(F(x)\) and setting the expression equal to 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivative (Indefinite Integral)
An antiderivative of a function f(x) is a function F(x) whose derivative is f(x). It represents the reverse process of differentiation and is expressed as an indefinite integral, including an arbitrary constant C since differentiation eliminates constants.
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Introduction to Indefinite Integrals
Initial Condition for Particular Solution
An initial condition like F(0) = 2 specifies the value of the antiderivative at a particular point, allowing us to determine the constant of integration C. This transforms the general antiderivative into a unique particular solution.
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Integration of Basic Functions
To find the antiderivative, one must integrate each term separately using known formulas: the integral of x^n is (x^(n+1))/(n+1), and the integral of sin x is -cos x. Combining these results forms the general antiderivative.
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Related Practice
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_v→3 (v-1-√(v²-5)) / (v-3)
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 (x + cos x)¹/ˣ
Textbook Question
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
p(x) = 3 sec² x
Textbook Question
{Use of Tech} Absolute maxima and minima
a. Find the critical points of f on the given interval.
b. Determine the absolute extreme values of f on the given interval.
c. Use a graphing utility to confirm your conclusions.
f(x) = 2ᶻ sin x on [-2,6]
Textbook Question
{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.
f(x) = cos⁻¹ x - x; x₀ = 0.75
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