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The derivative of the function f is defined by f^(')(x)=(x^(3)-1)sin(x). If f(4)=3, then use a calculator to find the value of f(-1) to the nearest thousandth. Answer:

The derivative of the function f f is defined by f′(x)=(x3−1)sin⁡(x) f^{\prime}(x)=\left(x^{3}-1\right) \sin (x) . If f(4)=3 f(4)=3 , then use a calculator to find the value of f(−1) f(-1) to the nearest thousandth.\newlineAnswer:

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Full solution

Q. The derivative of the function f f is defined by f′(x)=(x3−1)sin⁡(x) f^{\prime}(x)=\left(x^{3}-1\right) \sin (x) . If f(4)=3 f(4)=3 , then use a calculator to find the value of f(−1) f(-1) to the nearest thousandth.\newlineAnswer:
  1. Identify Initial Condition: To find the value of f(−1)f(-1), we need to integrate the derivative f′(x)f'(x) to get the original function f(x)f(x). However, we are given a specific value f(4)=3f(4) = 3, which will help us determine the constant of integration after finding the indefinite integral of f′(x)f'(x).
  2. Find Indefinite Integral: First, we find the indefinite integral of f′(x)=(x3−1)sin⁡(x)f'(x) = (x^3 - 1)\sin(x). This requires integration by parts or a special technique since it is a product of a polynomial and a trigonometric function. However, this is a non-standard integral that does not have an elementary antiderivative, so we cannot express the integral of f′(x)f'(x) in terms of elementary functions.
  3. Non-Standard Integral: Since we cannot find an elementary antiderivative for f′(x)f'(x), we cannot directly integrate to find f(x)f(x). Therefore, we cannot proceed with the usual method of finding f(−1)f(-1) by integrating f′(x)f'(x) and then applying the initial condition f(4)=3f(4) = 3 to solve for the constant of integration.
  4. Unable to Solve: Given that we cannot find f(−1)f(-1) through integration, we must acknowledge that the problem as stated cannot be solved with the information provided. We need either a different method or additional information to find f(−1)f(-1).

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