The derivative of the function f is defined by f^(')(x)=(x^(2)-3x)cos(2x). If f(0)=5, 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)-3x)cos(2x). If  f(0)=5, then use a calculator to find the value of  f(6) to the nearest thousandth. Answer:

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Integrate f'(x) for f(x): To find f(6), we need to integrate the derivative f'(x) to get the original function f(x). We will then use the initial condition f(0) = 5 to find the constant of integration.Use Integration by Parts: The integral of f'(x) = (x^2 - 3x)cos(2x) is not straightforward due to the product of a polynomial and a trigonometric function. We will use integration by parts or a computer algebra system to find the integral.Find du and v: Let's denote u = x^2 - 3x and dv = cos(2x)dx. Then we need to find du and v: du = (2x - 3)dx and v = (1/2)sin(2x).Integrate by Parts Again: Applying integration by parts, we get: ∫f'(x)dx = ∫u dv = uv - ∫v du = (x^2 - 3x)(1/2)sin(2x) - ∫(1/2)sin(2x)(2x - 3)dx.Evaluate Initial Condition: We need to integrate (1/2)sin(2x)(2x - 3)dx, which again requires integration by parts or a computer algebra system.Calculate Constant of Integration: Let's denote u = 2x - 3 and dv = (1/2)sin(2x)dx. Then we need to find du and v: du = 2dx and v = -(1/4)cos(2x).Final Function f(x): Applying integration by parts again, we get: ∫(1/2)sin(2x)(2x - 3)dx = uv - ∫v du = -(1/4)(2x - 3)cos(2x) - ∫-(1/4)cos(2x)(2)dx.Evaluate f(6): The integral of -(1/4)cos(2x)(2)dx is -(1/4)(2)(1/2)sin(2x) = -sin(2x)/4.Combining all parts, we get: f(x) = (x^2 - 3x)(1/2)sin(2x) + (1/4)(2x - 3)cos(2x) + sin(2x)/4 + C, where C is the constant of integration.Using the initial condition f(0) = 5, we find C: f(0) = (0^2 - 3*0)(1/2)sin(2*0) + (1/4)(2*0 - 3)cos(2*0) + sin(2*0)/4 + C = 5. Since sin(0) = 0 and cos(0) = 1, this simplifies to: C = 5 + (3/4).Now we have the function f(x) = (x^2 - 3x)(1/2)sin(2x) + (1/4)(2x - 3)cos(2x) + sin(2x)/4 + 5 + (3/4).We can now evaluate f(6) using a calculator: f(6) = (6^2 - 3*6)(1/2)sin(2*6) + (1/4)(2*6 - 3)cos(2*6) + sin(2*6)/4 + 5 + (3/4).After calculating the above expression using a calculator, we find the value of f(6) to the nearest thousandth.

The derivative of the function $f$ is defined by $f^{\prime}(x)=\left(x^{2}-3 x\right) \cos (2 x)$ . If $f(0)=5$ , 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 $f$ is defined by $f^{\prime}(x)=\left(x^{2}-3 x\right) \cos (2 x)$ . If $f(0)=5$ , then use a calculator to find the value of $f(6)$ to the nearest thousandth. $\newline$ Answer: