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The derivative of the function
f is defined by
f^(')(x)=(x^(2)+3)sin(3x). If
f(2)=7, 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 $f^{\prime}(x)=\left(x^{2}+3\right) \sin (3 x)$ . If $f(2)=7$ , then use a calculator to find the value of $f(6)$ to the nearest thousandth. $\newline$ Answer:

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Find trigonometric ratios using multiple identities

Full solution

Q. The derivative of the function

f

is defined by

f^{\prime}(x)=\left(x^{2}+3\right) \sin (3 x)

. If

f(2)=7

, then use a calculator to find the value of

f(6)

to the nearest thousandth.

\newline

Answer:

Integrate $f'(x)$ from $2$ to $6$ : To find $f(6)$ , we need to integrate $f'(x)$ from $2$ to $6$ and add the result to $f(2)$ since $f(2) = 7$ .
Set up integral: Set up the integral of $f'(x)$ from $2$ to $6$ . $\int_{2}^{6} (x^2 + 3)\sin(3x) \, dx$
Evaluate integral numerically: This integral does not have an elementary antiderivative, so we will use a calculator to evaluate it numerically. $\newline$ Use a calculator to evaluate the integral.
Calculate numerical value: After calculating the integral on a calculator, we get a numerical value. Let's denote this value as $I$ . $I \approx \text{[Calculator Output]}$
Find $f(6)$ : Now, add the value of $f(2)$ to the integral result to find $f(6)$ .
$f(6) = f(2) + I$
$f(6) = 7 + I$
Add $f(2)$ to integral result: Replace $I$ with the numerical value obtained from the calculator to find $f(6)$ . $\newline$ $f(6) \approx 7 + [\text{Calculator Output}]$
Replace $I$ with value: Round the result to the nearest thousandth as required by the question prompt. $f(6) \approx \text{[Rounded Value]}$

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