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Anit [1.1K]
3 years ago
10

Consider the following functions.G(x) = 4x2; f(x) = 8x(a)

Mathematics
1 answer:
noname [10]3 years ago
8 0

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 <em>F </em>is an antiderivative of the function <em>f</em> if

F'(x)=f(x)

for all x in the domain of <em>f.</em>

(a) If f(x) = 8x, then G(x)=4x^2 is an antiderivative of <em>f </em>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 <em>f</em>.

(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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