Skip to main content
Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.9.111a
Chapter 4, Problem 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.

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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 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.
Recommended video:
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.
Recommended video:
06:18
Integration 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.
Recommended video:
05:50
Antiderivatives
Related Practice
Textbook Question

Folded boxes


a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.

319
views
Textbook Question

Optimal soda can


a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm³ that minimize the surface area.

292
views
Textbook Question

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>

355
views
Textbook Question

{Use of Tech } Minimizing sound intensity Two sound speakers are 100 m apart and one speaker is three times as loud as the other speaker. At what point on a line segment between the speakers is the sound intensity the weakest? (Hint: Sound intensity is directly proportional to the sound level and inversely proportional to the square of the distance from the sound source.)

281
views
Textbook Question

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?

333
views
Textbook Question

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.

276
views