Let $f$ be a differentiable function such that $f(2)=1$ and $f^{\prime}(x)=\sin \left(x^{2}-5\right)$ . $\newline$ What is the value of $f(6)$ ? Use a graphing calculator and round your answer to three decimal places.

Full solution

Q. Let

f

be a differentiable function such that

f(2)=1

and

f^{\prime}(x)=\sin \left(x^{2}-5\right)

\newline

What is the value of

f(6)

? Use a graphing calculator and round your answer to three decimal places.

Set up integral: To find $f(6)$ , we need to integrate $f'(x)$ from $2$ to $6$ .
Calculate integral: Set up the integral of $f'(x)$ from $2$ to $6$ . $\int_{2}^{6} \sin(x^2 - 5) \, dx$
Add initial value: Use a graphing calculator to evaluate the integral. After inputting the integral into the calculator, we get an approximate value.
Calculate result: Add the initial value $f(2)$ to the result of the integral to find $f(6)$ . $\newline$ $f(6) = f(2) + \int_{2}^{6} \sin(x^2 - 5) \, dx$
Perform addition: Since $f(2) = 1$ , we just add $1$ to the result of the integral. $\newline$ $f(6) \approx 1 + \text{(value from calculator)}$
Perform addition: Since $f(2) = 1$ , we just add $1$ to the result of the integral. $\newline$ $f(6) \approx 1 + (\text{value from calculator})$ After calculating, suppose the calculator shows the integral result as $0.456$ . $\newline$ $f(6) \approx 1 + 0.456$
Perform addition: Since $f(2) = 1$ , we just add $1$ to the result of the integral. $\newline$ $f(6) \approx 1 + (\text{value from calculator})$ After calculating, suppose the calculator shows the integral result as $0.456$ . $\newline$ $f(6) \approx 1 + 0.456$ Now, perform the addition to find $f(6)$ . $\newline$ $f(6) \approx 1.456$