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 4.9.15
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
p(x) = 3 sec² x
Verified step by step guidance1
Step 1: Recall the definition of an antiderivative. The antiderivative of a function is a function whose derivative is the original function. For this problem, we need to find the antiderivative of p(x) = 3 sec²(x).
Step 2: Identify the basic antiderivative rule that applies here. The derivative of tan(x) is sec²(x), so the antiderivative of sec²(x) is tan(x). Use this rule to find the antiderivative of 3 sec²(x).
Step 3: Multiply the constant 3 by the antiderivative of sec²(x). This gives us 3 tan(x) as part of the solution.
Step 4: Add the constant of integration, C, to account for all possible antiderivatives. The general antiderivative of p(x) = 3 sec²(x) is therefore 3 tan(x) + C.
Step 5: Verify your work by taking the derivative of the antiderivative you found. Differentiate 3 tan(x) + C, and confirm that the result is 3 sec²(x), which matches the original function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivatives
An antiderivative of a function is another function whose derivative is the original function. In calculus, finding antiderivatives is essential for solving problems related to integration. The process involves determining a function F(x) such that F'(x) equals the given function p(x). Antiderivatives are not unique; they can differ by a constant, represented as F(x) + C, where C is any constant.
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Integration
Integration is the process of finding the integral of a function, which is closely related to finding antiderivatives. It can be thought of as the reverse operation of differentiation. The integral of a function over an interval gives the area under the curve of that function. In this context, the integral of p(x) = 3 sec² x will yield the family of functions that are antiderivatives of p(x).
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Derivative Check
To verify that a function is indeed an antiderivative, one can take its derivative and check if it matches the original function. This process is crucial in confirming the correctness of the antiderivative found. For example, if F(x) is an antiderivative of p(x), then differentiating F(x) should yield p(x). This step ensures that the integration process was performed correctly.
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Related Practice
Textbook Question
{Use of Tech} Critical points and extreme values
a. Find the critical points of the following functions on the given interval. Use a root finder, if necessary.
b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.
c. Find the absolute maximum and minimum values on the given interval, if they exist
h(x) (5-x)/(x² + 2x - 3) on [-10,10]
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
Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.
f(x) = 8x³ + sin x; F(0) = 2
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]