교과서 질문
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
a. Find the critical points of f and determine where f is increasing and where it is decreasing.
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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
4장, 문제 4.9.111a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.
검증된 단계별 안내1
Recall that two functions are antiderivatives of the same function if their derivatives are equal, which means they differ by only a constant.
Find the derivative of \( F(x) = x^3 - 4x + 100 \). Using the power rule and constant rule, \( F'(x) = 3x^2 - 4 \).
Find the derivative of \( G(x) = x^3 - 4x - 100 \). Similarly, \( G'(x) = 3x^2 - 4 \).
Since \( F'(x) = G'(x) = 3x^2 - 4 \), both \( F(x) \) and \( G(x) \) are antiderivatives of the same function \( f(x) = 3x^2 - 4 \).
The difference between \( F(x) \) and \( G(x) \) is a constant (\( 200 \)), which confirms they are antiderivatives of the same function.
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주요 개념
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Antiderivative (Indefinite Integral)
An antiderivative of a function f(x) is another 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.
추천 영상:
가이드 코스
05:04
Introduction to Indefinite Integrals
Constant of Integration
When finding antiderivatives, any constant term disappears upon differentiation. Therefore, all antiderivatives of a function differ by a constant, called the constant of integration, which accounts for all possible vertical shifts of the antiderivative graph.
추천 영상:
가이드 코스
06:18Integration by Parts for Definite Integrals
Verifying Antiderivatives by Differentiation
To determine if two functions are antiderivatives of the same function, differentiate both and compare their derivatives. If the derivatives are identical, the functions differ only by a constant and are antiderivatives of the same function.
추천 영상:
가이드 코스
05:50Antiderivatives
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Optimal popcorn box A small popcorn box is created from a 12" x 12" sheet of paperboard by first cutting out four shaded rectangles, each of length x and width x/2 (see figure). The remaining paperboard is folded along the solid lines to form a box. What dimensions of the box maximize the volume of the box? <IMAGE>
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교과서 질문
Rectangles beneath a line
a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?
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교과서 질문
Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.
a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?
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교과서 질문
Sketch a graph of a function f with the following properties.
f' < 0 and f" < 0, for x < -1
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교과서 질문
Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.
a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.
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