The derivative of the function f is defined by f^(')(x)=x^(2)-4x+2cos(2x-4). If f(0)=1, then use a calculator to find the value of f(6) to the nearest thousandth. Answer:

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The derivative of the function  f is defined by  f^(')(x)=x^(2)-4x+2cos(2x-4). If  f(0)=1, then use a calculator to find the value of  f(6) to the nearest thousandth. Answer:

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Set up integral: To find the value of f(6), we need to integrate the derivative f'(x) from 0 to 6 and then add the initial value f(0) to the result of the integration.Integrate each term: First, we set up the integral of f'(x) from 0 to 6: ∫(from 0 to 6) (x^2 - 4x + 2cos(2x-4)) dxEvaluate integrals: We integrate each term separately: ∫(from 0 to 6) x^2 dx = [x^3/3] (from 0 to 6) ∫(from 0 to 6) -4x dx = [-2x^2] (from 0 to 6) ∫(from 0 to 6) 2cos(2x-4) dx = [sin(2x-4)] (from 0 to 6) (Note: We need to use substitution for this integral)Substitute and integrate: We evaluate each integral at the bounds 0 and 6: [x^3/3] (from 0 to 6) = (6^3/3) - (0^3/3) = 72 [-2x^2] (from 0 to 6) = (-2*6^2) - (-2*0^2) = -72 For the cosine integral, we use substitution: let u = 2x-4, then du = 2dx, so (1/2)du = dx.Evaluate cosine integral: Now we integrate 2cos(u) with respect to u: ∫2cos(u) du = 2sin(u) We need to change the limits of integration according to the substitution: When x = 0, u = 2*0 - 4 = -4 When x = 6, u = 2*6 - 4 = 8 So we evaluate 2sin(u) from u = -4 to u = 8.Add integrals: We find the value of the integral of the cosine term: 2sin(u) (from u = -4 to u = 8) = 2sin(8) - 2sin(-4) Using a calculator, we find: 2sin(8) ≈ 2*0.989 = 1.978 2sin(-4) ≈ 2*(-0.756) = -1.512 So the integral from -4 to 8 is approximately 1.978 - (-1.512) = 3.490Add initial value: We add the results of the integrals: 72 - 72 + 3.490 = 3.490Finally, we add the initial value f(0) = 1 to the result of the integration to find f(6): f(6) = 1 + 3.490 = 4.490

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2}-4 x+2 \cos (2 x-4)$ . If $f(0)=1$ , then use a calculator to find the value of $f(6)$ to the nearest thousandth. $\newline$ Answer:

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The derivative of the function f f f is defined by f′(x)=x2−4x+2cos⁡(2x−4) f^{\prime}(x)=x^{2}-4 x+2 \cos (2 x-4) f′(x)=x2−4x+2cos(2x−4). If f(0)=1 f(0)=1 f(0)=1, then use a calculator to find the value of f(6) f(6) f(6) to the nearest thousandth.\newlineAnswer:

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2}-4 x+2 \cos (2 x-4)$ . If $f(0)=1$ , then use a calculator to find the value of $f(6)$ to the nearest thousandth. $\newline$ Answer: