Textbook Question
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
e. On what intervals (approximately) is f concave up?
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.R.1c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. F(x) = x² + 10 and G(x) = x² - 100 are antiderivatives of the same function.
Verified step by step guidance1
Recall that two functions are antiderivatives of the same function if their derivatives are equal, differing only by a constant.
Find the derivative of \( F(x) = x^2 + 10 \). Using the power rule, \( F'(x) = 2x \).
Find the derivative of \( G(x) = x^2 - 100 \). Similarly, \( G'(x) = 2x \).
Since \( F'(x) = G'(x) = 2x \), both \( F(x) \) and \( G(x) \) have the same derivative, meaning they are antiderivatives of the same function.
The difference between \( F(x) \) and \( G(x) \) is a constant (\( 10 - (-100) = 110 \)), which confirms they differ by a constant and thus are antiderivatives of the same function.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivative Definition
An antiderivative of a function f(x) is another function F(x) whose derivative is f(x). In other words, if F'(x) = f(x), then F(x) is an antiderivative of f(x). Antiderivatives differ by a constant since the derivative of a constant is zero.
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Derivative of Polynomial Functions
The derivative of a polynomial function like x² is found using the power rule: d/dx[x^n] = n*x^(n-1). For example, the derivative of x² is 2x. Constants vanish when differentiating, so terms like +10 or -100 become zero.
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Checking if Two Functions are Antiderivatives of the Same Function
Two functions are antiderivatives of the same function if their derivatives are identical. Since constants disappear upon differentiation, functions differing only by a constant are antiderivatives of the same function. If their derivatives differ, they are not.
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Related Practice
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Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
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