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The derivative of the function
f is defined by
f^(')(x)=x^(2)cos(2x+3). If
f(0)=4, then use a calculator to find the value of
f(6) to the nearest thousandth.
Answer:

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2} \cos (2 x+3)$ . If $f(0)=4$ , then use a calculator to find the value of $f(6)$ to the nearest thousandth. $\newline$ Answer:

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Evaluate variable expressions for sequences

Full solution

Q. The derivative of the function

f

is defined by

f^{\prime}(x)=x^{2} \cos (2 x+3)

. If

f(0)=4

, then use a calculator to find the value of

f(6)

to the nearest thousandth.

\newline

Answer:

Set up integral: To find $f(6)$ , we need to integrate the derivative $f'(x)$ from $0$ to $6$ and then add the initial value $f(0)$ to the result of the integration.
Evaluate integral: First, we set up the integral of $f'(x)$ from $0$ to $6$ : $\newline$ $\int_{0}^{6} x^2\cos(2x+3) \, dx$
Add initial value: We use a calculator to evaluate the definite integral. This step involves numerical integration, which is typically not done by hand for functions like this one.
Calculate $f(6)$ : After calculating the integral on a calculator, we add the result to the initial value $f(0) = 4$ to find $f(6)$ .
Report final value: Assuming the calculator gave us the correct value of the integral, we would then report the value of $f(6)$ to the nearest thousandth.

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