The derivative of the function f is defined by f^(')(x)=x^(2)-5x-sin(3x+3). If f(-2)=-9, then use a calculator to find the value of f(4) to the nearest thousandth. Answer:

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The derivative of the function  f is defined by  f^(')(x)=x^(2)-5x-sin(3x+3). If  f(-2)=-9, then use a calculator to find the value of  f(4) to the nearest thousandth. Answer:

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Integrate f^(')(x): To find f(4), we need to integrate the derivative f^(')(x) to get the original function f(x). We will then use the initial condition f(-2) = -9 to find the constant of integration. Integrate f^(')(x) = x^(2) - 5x - sin(3x+3). The integral of x^(2) is (1/3)x^(3), the integral of -5x is (-5/2)x^(2), and the integral of -sin(3x+3) is (1/3)cos(3x+3) plus a constant C. So, f(x) = (1/3)x^(3) - (5/2)x^(2) + (1/3)cos(3x+3) + C.Use Initial Condition: Now we use the initial condition f(-2) = -9 to find the constant C. Substitute x = -2 into f(x) to get: -9 = (1/3)(-2)^(3) - (5/2)(-2)^(2) + (1/3)cos(3(-2)+3) + C. Calculate the values: -9 = (1/3)(-8) - (5/2)(4) + (1/3)cos(-3) + C. -9 = -8/3 - 10 + (1/3)cos(-3) + C.Solve for Constant C: Simplify the equation to solve for C. -9 = -8/3 - 10 + (1/3)cos(-3) + C. Multiply all terms by 3 to clear the fraction: -27 = -8 - 30 + cos(-3) + 3C. -27 = -38 + cos(-3) + 3C. Add 38 to both sides: 11 = cos(-3) + 3C.Calculate cos(-3): Use a calculator to find the value of cos(-3) to the nearest thousandth. cos(-3) ≈ 0.990 (rounded to three decimal places). Now we have: 11 = 0.990 + 3C.Find Constant C: Solve for C. 11 - 0.990 = 3C. 10.010 = 3C. C = 10.010 / 3. C ≈ 3.337 (rounded to three decimal places).Substitute x = 4: Now that we have the constant C, we can write the original function f(x) as: f(x) = (1/3)x^(3) - (5/2)x^(2) + (1/3)cos(3x+3) + 3.337. To find f(4), substitute x = 4 into f(x): f(4) = (1/3)(4)^(3) - (5/2)(4)^(2) + (1/3)cos(3(4)+3) + 3.337. Calculate the values: f(4) = (1/3)(64) - (5/2)(16) + (1/3)cos(15) + 3.337.Calculate cos(15): Simplify the equation to find f(4). f(4) = 64/3 - 40 + (1/3)cos(15) + 3.337. Use a calculator to find the value of cos(15) to the nearest thousandth. cos(15) ≈ 0.966 (rounded to three decimal places). Now we have: f(4) = 64/3 - 40 + (1/3)(0.966) + 3.337.Calculate f(4): Calculate the final value of f(4) to the nearest thousandth. f(4) = 64/3 - 40 + 0.322 + 3.337. f(4) = 21.333 - 40 + 0.322 + 3.337. f(4) = -18.667 + 0.322 + 3.337. f(4) = -18.345 + 3.337. f(4) = -15.008. Round to the nearest thousandth: f(4) ≈ -15.008.

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2}-5 x-\sin (3 x+3)$ . If $f(-2)=-9$ , then use a calculator to find the value of $f(4)$ to the nearest thousandth. $\newline$ Answer:

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The derivative of the function f f f is defined by f′(x)=x2−5x−sin⁡(3x+3) f^{\prime}(x)=x^{2}-5 x-\sin (3 x+3) f′(x)=x2−5x−sin(3x+3). If f(−2)=−9 f(-2)=-9 f(−2)=−9, then use a calculator to find the value of f(4) f(4) f(4) to the nearest thousandth.\newlineAnswer:

The derivative of the function $f$ is defined by $f^{\prime}(x)=x^{2}-5 x-\sin (3 x+3)$ . If $f(-2)=-9$ , then use a calculator to find the value of $f(4)$ to the nearest thousandth. $\newline$ Answer: