The derivative of the function f is defined by f^(')(x)=(x^(3)+1)cos(3x). If f(5)=-8, then use a calculator to find the value of f(0) to the nearest thousandth. Answer:

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The derivative of the function  f is defined by  f^(')(x)=(x^(3)+1)cos(3x). If  f(5)=-8, then use a calculator to find the value of  f(0) to the nearest thousandth. Answer:

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Integrate f'(x) for f(0): To find f(0), we need to integrate the derivative f'(x) to get the original function f(x). The integral of f'(x) will include an arbitrary constant C, which we can solve for using the given condition f(5) = -8.Perform integration by parts: Let's integrate f'(x) = (x^3 + 1)cos(3x). 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.Calculate integrated function: Using a calculator to integrate (x^3 + 1)cos(3x) with respect to x, we get f(x) = (1/3)x^3 sin(3x) - x^2 cos(3x)/3 + x sin(3x) + C, where C is the constant of integration.Solve for constant C: Now we use the given condition f(5) = -8 to solve for C. We substitute x = 5 into the integrated function and set it equal to -8: (-8) = (1/3)(5)^3 sin(15) - (5)^2 cos(15)/3 + (5) sin(15) + C.Compute trigonometric values: Using a calculator to compute the trigonometric values and solve for C, we find that C is approximately equal to a certain value. We need to be careful with rounding here to ensure accuracy.Find exact value of C: After calculating the trigonometric values and solving for C, we find that C ≈ -8 - [(1/3)(125) sin(15) - (25) cos(15)/3 + 5 sin(15)]. We use a calculator to find the exact value of C.Substitute x=0 into function: Now that we have the value of C, we can find f(0) by substituting x = 0 into the integrated function f(x) = (1/3)x^3 sin(3x) - x^2 cos(3x)/3 + x sin(3x) + C.Calculate f(0): Substituting x = 0 into the function, we get f(0) = (1/3)(0)^3 sin(0) - (0)^2 cos(0)/3 + (0) sin(0) + C = C.Final answer: We use the previously calculated value of C to find f(0). Since f(0) = C, we can now state the final answer to the nearest thousandth.

The derivative of the function $f$ is defined by $f^{\prime}(x)=\left(x^{3}+1\right) \cos (3 x)$ . If $f(5)=-8$ , then use a calculator to find the value of $f(0)$ to the nearest thousandth. $\newline$ Answer:

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The derivative of the function f f f is defined by f′(x)=(x3+1)cos⁡(3x) f^{\prime}(x)=\left(x^{3}+1\right) \cos (3 x) f′(x)=(x3+1)cos(3x). If f(5)=−8 f(5)=-8 f(5)=−8, then use a calculator to find the value of f(0) f(0) f(0) to the nearest thousandth.\newlineAnswer:

The derivative of the function $f$ is defined by $f^{\prime}(x)=\left(x^{3}+1\right) \cos (3 x)$ . If $f(5)=-8$ , then use a calculator to find the value of $f(0)$ to the nearest thousandth. $\newline$ Answer: