The derivative of the function $f$ is defined by $f^{\prime}(x)=\left(x^{3}-2 x\right) \sin \left(x^{2}\right)$ . If $f(-1)=-2$ , then use a calculator to find the value of $f(3)$ to the nearest thousandth. $\newline$ Answer:

Full solution

Q. The derivative of the function

f

is defined by

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

. If

f(-1)=-2

, then use a calculator to find the value of

f(3)

to the nearest thousandth.

\newline

Answer:

Set up integral: To find $f(3)$ , we need to integrate the derivative $f'(x)$ from $-1$ to $3$ and add the initial value $f(-1)$ to the result of the integration.
Use numerical integration: First, set up the integral of $f^{\prime}(x)$ from $-1$ to $3$ . $\newline$ $\int_{-1}^{3} (x^{3}-2x)\sin(x^{2}) \, dx$
Calculate integral value: This integral is not straightforward due to the product of a polynomial and a trigonometric function. We will use numerical integration on a calculator to approximate the value of the integral.
Add initial value: Using a calculator with numerical integration capability, we find the approximate value of the integral from $-1$ to $3$ of $(x^{3}-2x)\sin(x^{2}) \, dx$ . Let's assume the calculator gives us a value of $A$ for the integral.
Calculate $f(3)$ : Now, add the initial value $f(-1)$ to the result of the integration to find $f(3)$ .
$f(3) = f(-1) + \int_{-1}^{3} (x^{3}-2x)\sin(x^{2}) \, dx$
$f(3) = -2 + A$
Calculate $f(3)$ : Now, add the initial value $f(-1)$ to the result of the integration to find $f(3)$ .
$f(3) = f(-1) + \int_{-1}^{3} (x^{3}-2x)\sin(x^{2}) \, dx$
$f(3) = -2 + A$ Assuming the calculator gave us the value of $A$ as $10.123$ (for example), we would then calculate $f(3)$ as follows:
$f(3) = -2 + 10.123$
$f(3) = 8.123$