Consider the following functions.G(x) = 4×2; f(x) = 8x(a) a. Verify that G is an antiderivative of f.G(x) is an antiderivative of f(x

Question

Consider the following functions.G(x) = 4×2; 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.)

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Amara 3 days 2021-10-10T19:01:25+00:00 1 Answer 0

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    2021-10-10T19:02:38+00:00

    Answer:

    (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

    Step-by-step 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

    \frac{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.

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