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

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2} \cos (3 x)$ . If $f(-2)=-2$ , then use a calculator to find the value of $f(3)$ 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 (3 x)

. If

f(-2)=-2

, then use a calculator to find the value of

f(3)

to the nearest thousandth.

\newline

Answer:

Integrate $f'(x)$ : To find the value of $f(3)$ , we need to integrate the derivative $f'(x) = x^2\cos(3x)$ to get the original function $f(x)$ . We will then use the initial condition $f(-2) = -2$ to solve for the constant of integration.
Find Constant $C$ : We integrate $f'(x) = x^2\cos(3x)$ with respect to $x$ . This requires integration by parts or a special technique since it is a product of a polynomial and a trigonometric function. We will use a calculator to perform this integration.
Substitute $x = -2$ : After integrating, we get $f(x) = \frac{x^2\sin(3x)}{3} - \frac{2}{9}x\sin(3x) - \frac{2}{27}\cos(3x) + C$ , where $C$ is the constant of integration.
Calculate Constant C: We use the initial condition $f(-2) = -2$ to find the constant C. We substitute $x = -2$ into the integrated function and set it equal to $-2$ . $\newline$ $-2 = (-2)^2\sin(3(-2))/3 - (2/9)(-2)\sin(3(-2)) - (2/27)\cos(3(-2)) + C$
Substitute $x = 3$ : We simplify and solve for $C$ . $-2 = \frac{4\sin(-6)}{3} + \frac{4}{9}\sin(-6) - \frac{2}{27}\cos(-6) + C$
Calculate $f(3)$ : Using a calculator to find the values of $\sin(-6)$ and $\cos(-6)$ , we get: $\newline$ $\sin(-6) \approx -0.2794$ $\newline$ $\cos(-6) \approx 0.9602$
Calculate $f(3)$ : Using a calculator to find the values of $\sin(-6)$ and $\cos(-6)$ , we get: $\newline$ $\sin(-6) \approx -0.2794$ $\newline$ $\cos(-6) \approx 0.9602$ Substitute these values into the equation to solve for $C$ . $\newline$ $-2 = 4(-0.2794)/3 + (4/9)(-0.2794) - (2/27)(0.9602) + C$
Calculate $f(3)$ : Using a calculator to find the values of $\sin(-6)$ and $\cos(-6)$ , we get: $\newline$ $\sin(-6) \approx -0.2794$ $\newline$ $\cos(-6) \approx 0.9602$ Substitute these values into the equation to solve for $C$ . $\newline$ $-2 = 4(-0.2794)/3 + (4/9)(-0.2794) - (2/27)(0.9602) + C$ Perform the calculations to find $C$ . $\newline$ $-2 \approx -0.3725 - 0.1243 - 0.0711 + C$ $\newline$ $C \approx -2 + 0.3725 + 0.1243 + 0.0711$ $\newline$ $\sin(-6)$ $0$
Calculate $f(3)$ : Using a calculator to find the values of $\sin(-6)$ and $\cos(-6)$ , we get: $\newline$ $\sin(-6) \approx -0.2794$ $\newline$ $\cos(-6) \approx 0.9602$ Substitute these values into the equation to solve for $C$ . $\newline$ $-2 = 4(-0.2794)/3 + (4/9)(-0.2794) - (2/27)(0.9602) + C$ Perform the calculations to find $C$ . $\newline$ $-2 \approx -0.3725 - 0.1243 - 0.0711 + C$ $\newline$ $C \approx -2 + 0.3725 + 0.1243 + 0.0711$ $\newline$ $\sin(-6)$ $0$ Now that we have the constant $C$ , we can find $f(3)$ by substituting $\sin(-6)$ $3$ into the integrated function $\sin(-6)$ $4$ . $\newline$ $\sin(-6)$ $5$
Calculate $f(3)$ : Using a calculator to find the values of $\sin(-6)$ and $\cos(-6)$ , we get: $\newline$ $\sin(-6) \approx -0.2794$ $\newline$ $\cos(-6) \approx 0.9602$ Substitute these values into the equation to solve for $C$ . $\newline$ $-2 = 4(-0.2794)/3 + (4/9)(-0.2794) - (2/27)(0.9602) + C$ Perform the calculations to find $C$ . $\newline$ $-2 \approx -0.3725 - 0.1243 - 0.0711 + C$ $\newline$ $C \approx -2 + 0.3725 + 0.1243 + 0.0711$ $\newline$ $\sin(-6)$ $0$ Now that we have the constant $C$ , we can find $f(3)$ by substituting $\sin(-6)$ $3$ into the integrated function $\sin(-6)$ $4$ . $\newline$ $\sin(-6)$ $5$ Using a calculator to find the values of $\sin(-6)$ $6$ and $\sin(-6)$ $7$ , we get: $\newline$ $\sin(-6)$ $8$ $\newline$ $\sin(-6)$ $9$
Calculate $f(3)$ : Using a calculator to find the values of $\sin(-6)$ and $\cos(-6)$ , we get: $\newline$ $\sin(-6) \approx -0.2794$ $\newline$ $\cos(-6) \approx 0.9602$ Substitute these values into the equation to solve for $C$ . $\newline$ $-2 = 4(-0.2794)/3 + (4/9)(-0.2794) - (2/27)(0.9602) + C$ Perform the calculations to find $C$ . $\newline$ $-2 \approx -0.3725 - 0.1243 - 0.0711 + C$ $\newline$ $C \approx -2 + 0.3725 + 0.1243 + 0.0711$ $\newline$ $\sin(-6)$ $0$ Now that we have the constant $C$ , we can find $f(3)$ by substituting $\sin(-6)$ $3$ into the integrated function $\sin(-6)$ $4$ . $\newline$ $\sin(-6)$ $5$ Using a calculator to find the values of $\sin(-6)$ $6$ and $\sin(-6)$ $7$ , we get: $\newline$ $\sin(-6)$ $8$ $\newline$ $\sin(-6)$ $9$ Substitute these values into the equation to find $f(3)$ . $\newline$ $\cos(-6)$ $1$
Calculate $f(3)$ : Using a calculator to find the values of $\sin(-6)$ and $\cos(-6)$ , we get: $\newline$ $\sin(-6) \approx -0.2794$ $\newline$ $\cos(-6) \approx 0.9602$ Substitute these values into the equation to solve for $C$ . $\newline$ $-2 = 4(-0.2794)/3 + (4/9)(-0.2794) - (2/27)(0.9602) + C$ Perform the calculations to find $C$ . $\newline$ $-2 \approx -0.3725 - 0.1243 - 0.0711 + C$ $\newline$ $C \approx -2 + 0.3725 + 0.1243 + 0.0711$ $\newline$ $\sin(-6)$ $0$ Now that we have the constant $C$ , we can find $f(3)$ by substituting $\sin(-6)$ $3$ into the integrated function $\sin(-6)$ $4$ . $\newline$ $\sin(-6)$ $5$ Using a calculator to find the values of $\sin(-6)$ $6$ and $\sin(-6)$ $7$ , we get: $\newline$ $\sin(-6)$ $8$ $\newline$ $\sin(-6)$ $9$ Substitute these values into the equation to find $f(3)$ . $\newline$ $\cos(-6)$ $1$ Perform the calculations to find $f(3)$ . $\newline$ $\cos(-6)$ $3$ $\newline$ $\cos(-6)$ $4$ $\newline$ $\cos(-6)$ $5$
Calculate $f(3)$ : Using a calculator to find the values of $\sin(-6)$ and $\cos(-6)$ , we get: $\newline$ $\sin(-6) \approx -0.2794$ $\newline$ $\cos(-6) \approx 0.9602$ Substitute these values into the equation to solve for $C$ . $\newline$ $-2 = 4(-0.2794)/3 + (4/9)(-0.2794) - (2/27)(0.9602) + C$ Perform the calculations to find $C$ . $\newline$ $-2 \approx -0.3725 - 0.1243 - 0.0711 + C$ $\newline$ $C \approx -2 + 0.3725 + 0.1243 + 0.0711$ $\newline$ $\sin(-6)$ $0$ Now that we have the constant $C$ , we can find $f(3)$ by substituting $\sin(-6)$ $3$ into the integrated function $\sin(-6)$ $4$ . $\newline$ $\sin(-6)$ $5$ Using a calculator to find the values of $\sin(-6)$ $6$ and $\sin(-6)$ $7$ , we get: $\newline$ $\sin(-6)$ $8$ $\newline$ $\sin(-6)$ $9$ Substitute these values into the equation to find $f(3)$ . $\newline$ $\cos(-6)$ $1$ Perform the calculations to find $f(3)$ . $\newline$ $\cos(-6)$ $3$ $\newline$ $\cos(-6)$ $4$ $\newline$ $\cos(-6)$ $5$ Round the result to the nearest thousandth. $\newline$ $\cos(-6)$ $6$

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