Consider the following functions. g(x) = 4x2; f(x) = 8x(a)
a. verify that g is an antiderivative of f. g(x) is an antiderivative of f(x) because f '(x) = g(x) for all x.

a. g(x) is an antiderivative of f(x) because g(x) = f(x) for all x.
b. g(x) is an antiderivative of f(x) because g'(x) = f(x) for all x.
c. g(x) is an antiderivative of f(x) because g(x) = f(x) + c for all x.
d. g(x) is an antiderivative of f(x) because f(x) = g(x) + c for all x.

b. find all antiderivatives of f. (use c for the constant of integration.)

(a) B. G(x) is an antiderivative of f(x) because G'(x) = f(x) for all x.

(b) Every function of the form is an antiderivative of 8x

Step-by-step explanation:

A function F is an antiderivative of the function f if

for all x in the domain of f.

(a) If , then is an antiderivative of f because

Therefore, G(x) is an antiderivative of f(x) because G'(x) = f(x) for all x.

Let F be an antiderivative of f. Then, for each constant C, the function F(x) + C is also an antiderivative of f.

(b) Because

then is an antiderivative of . Therefore, every antiderivative of 8x is of the form for some constant C, and every function of the form is an antiderivative of 8x.

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