Question

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.)

Answer 1

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

(b) Every function of the form
`4x^2+C` is an antiderivative of 8x

Explanation:

A function F is an antiderivative of the function f if

`F'(x)=f(x)`

for all x in the domain of f.

(a) If
`f(x) = 8x`, then
`G(x)=4x^2` is an antiderivative of f because

`G'(x)=8x=f(x)`

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

`(d)/(dx)(4x^2)=8x`

then
`G(x)=4x^2` is an antiderivative of
`f(x) = 8x`. Therefore, every antiderivative of 8x is of the form
`4x^2+C` for some constant C, and every function of the form
`4x^2+C` is an antiderivative of 8x.