The derivative of the function f is defined by f′(x)=x2cos(x−3). If f(−2)=−4, then use a calculator to find the value of f(6) to the nearest thousandth.Answer:
Q. The derivative of the function f is defined by f′(x)=x2cos(x−3). If f(−2)=−4, then use a calculator to find the value of f(6) to the nearest thousandth.Answer:
Integrate f′(x): To find the value of 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(−2)=−4 to find the constant of integration.
Apply integration by parts: Integrate f′(x)=x2cos(x−3) with respect to x. This requires integration by parts or a special technique since it is a product of a polynomial and a trigonometric function.
Simplify and integrate: Let u=x2, which means du=2xdx. Let dv=cos(x−3)dx, which means v=∫cos(x−3)dx=sin(x−3). Now we can apply integration by parts: ∫udv=uv−∫vdu.
Use initial condition: Using integration by parts, we get f(x)=x2sin(x−3)−∫2xsin(x−3)dx. The second integral also requires integration by parts.
Calculate trigonometric values: Let u=2x, which means du=2dx. Let dv=sin(x−3)dx, which means v=−cos(x−3). Apply integration by parts again: ∫udv=uv−∫vdu.
Find constant of integration: We get f(x)=x2sin(x−3)−(2x(−cos(x−3))−∫−2cos(x−3)dx). Simplify and integrate the remaining term.
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3).
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3).Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration.
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3).Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration.Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C.
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3). Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration. Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C. Calculate the trigonometric values and simplify: −4=4sin(−5)−4cos(−5)+2sin(−5)+C. Use a calculator to find the values of f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx0 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx1.
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3). Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration. Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C. Calculate the trigonometric values and simplify: −4=4sin(−5)−4cos(−5)+2sin(−5)+C. Use a calculator to find the values of f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx0 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx1. After calculating, we find that f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx2 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx3. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx4.
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3).Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration.Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C.Calculate the trigonometric values and simplify: −4=4sin(−5)−4cos(−5)+2sin(−5)+C. Use a calculator to find the values of f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx0 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx1.After calculating, we find that f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx2 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx3. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx4.Simplify the equation to find C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx6. Now solve for C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx8.
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3). Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration. Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C. Calculate the trigonometric values and simplify: −4=4sin(−5)−4cos(−5)+2sin(−5)+C. Use a calculator to find the values of f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx0 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx1. After calculating, we find that f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx2 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx3. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx4. Simplify the equation to find C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx6. Now solve for C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx8. Calculate the value of C: ∫−2cos(x−3)dx=−2sin(x−3)0. This is the constant of integration.
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3). Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration. Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C. Calculate the trigonometric values and simplify: −4=4sin(−5)−4cos(−5)+2sin(−5)+C. Use a calculator to find the values of f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx0 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx1. After calculating, we find that f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx2 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx3. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx4. Simplify the equation to find C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx6. Now solve for C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx8. Calculate the value of C: ∫−2cos(x−3)dx=−2sin(x−3)0. This is the constant of integration. Now that we have C, we can find f(6) by plugging ∫−2cos(x−3)dx=−2sin(x−3)3 into the integrated function: ∫−2cos(x−3)dx=−2sin(x−3)4.
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3). Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration. Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C. Calculate the trigonometric values and simplify: −4=4sin(−5)−4cos(−5)+2sin(−5)+C. Use a calculator to find the values of f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx0 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx1. After calculating, we find that f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx2 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx3. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx4. Simplify the equation to find C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx6. Now solve for C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx8. Calculate the value of C: ∫−2cos(x−3)dx=−2sin(x−3)0. This is the constant of integration. Now that we have C, we can find f(6) by plugging ∫−2cos(x−3)dx=−2sin(x−3)3 into the integrated function: ∫−2cos(x−3)dx=−2sin(x−3)4. Calculate the trigonometric values for ∫−2cos(x−3)dx=−2sin(x−3)5 and ∫−2cos(x−3)dx=−2sin(x−3)6 using a calculator, then substitute them into the equation to find f(6).
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3). Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration. Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C. Calculate the trigonometric values and simplify: −4=4sin(−5)−4cos(−5)+2sin(−5)+C. Use a calculator to find the values of f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx0 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx1. After calculating, we find that f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx2 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx3. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx4. Simplify the equation to find C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx6. Now solve for C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx8. Calculate the value of C: ∫−2cos(x−3)dx=−2sin(x−3)0. This is the constant of integration. Now that we have C, we can find f(6) by plugging ∫−2cos(x−3)dx=−2sin(x−3)3 into the integrated function: ∫−2cos(x−3)dx=−2sin(x−3)4. Calculate the trigonometric values for ∫−2cos(x−3)dx=−2sin(x−3)5 and ∫−2cos(x−3)dx=−2sin(x−3)6 using a calculator, then substitute them into the equation to find f(6). After calculating, we find that ∫−2cos(x−3)dx=−2sin(x−3)8 and ∫−2cos(x−3)dx=−2sin(x−3)9. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C0.
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3). Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration. Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C. Calculate the trigonometric values and simplify: −4=4sin(−5)−4cos(−5)+2sin(−5)+C. Use a calculator to find the values of f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx0 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx1. After calculating, we find that f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx2 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx3. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx4. Simplify the equation to find C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx6. Now solve for C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx8. Calculate the value of C: ∫−2cos(x−3)dx=−2sin(x−3)0. This is the constant of integration. Now that we have C, we can find f(6) by plugging ∫−2cos(x−3)dx=−2sin(x−3)3 into the integrated function: ∫−2cos(x−3)dx=−2sin(x−3)4. Calculate the trigonometric values for ∫−2cos(x−3)dx=−2sin(x−3)5 and ∫−2cos(x−3)dx=−2sin(x−3)6 using a calculator, then substitute them into the equation to find f(6). After calculating, we find that ∫−2cos(x−3)dx=−2sin(x−3)8 and ∫−2cos(x−3)dx=−2sin(x−3)9. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C0. Simplify the equation to find f(6): f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C2. Now calculate the sum to get the value of f(6).
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3). Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration. Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C. Calculate the trigonometric values and simplify: −4=4sin(−5)−4cos(−5)+2sin(−5)+C. Use a calculator to find the values of f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx0 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx1. After calculating, we find that f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx2 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx3. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx4. Simplify the equation to find C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx6. Now solve for C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx8. Calculate the value of C: ∫−2cos(x−3)dx=−2sin(x−3)0. This is the constant of integration. Now that we have C, we can find f(6) by plugging ∫−2cos(x−3)dx=−2sin(x−3)3 into the integrated function: ∫−2cos(x−3)dx=−2sin(x−3)4. Calculate the trigonometric values for ∫−2cos(x−3)dx=−2sin(x−3)5 and ∫−2cos(x−3)dx=−2sin(x−3)6 using a calculator, then substitute them into the equation to find f(6). After calculating, we find that ∫−2cos(x−3)dx=−2sin(x−3)8 and ∫−2cos(x−3)dx=−2sin(x−3)9. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C0. Simplify the equation to find f(6): f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C2. Now calculate the sum to get the value of f(6). Calculate the value of f(6): f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C5. Round to the nearest thousandth.
Find f(6): Simplifying, we have f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx. The last integral is straightforward: ∫−2cos(x−3)dx=−2sin(x−3).Putting it all together, we have f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C, where C is the constant of integration.Now we use the initial condition f(−2)=−4 to solve for C. Plug in x=−2 into the equation: −4=(−2)2sin(−2−3)+2(−2)cos(−2−3)+2sin(−2−3)+C.Calculate the trigonometric values and simplify: −4=4sin(−5)−4cos(−5)+2sin(−5)+C. Use a calculator to find the values of f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx0 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx1.After calculating, we find that f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx2 and f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx3. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx4.Simplify the equation to find C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx6. Now solve for C: f(x)=x2sin(x−3)+2xcos(x−3)−∫−2cos(x−3)dx8.Calculate the value of C: ∫−2cos(x−3)dx=−2sin(x−3)0. This is the constant of integration.Now that we have C, we can find f(6) by plugging ∫−2cos(x−3)dx=−2sin(x−3)3 into the integrated function: ∫−2cos(x−3)dx=−2sin(x−3)4.Calculate the trigonometric values for ∫−2cos(x−3)dx=−2sin(x−3)5 and ∫−2cos(x−3)dx=−2sin(x−3)6 using a calculator, then substitute them into the equation to find f(6).After calculating, we find that ∫−2cos(x−3)dx=−2sin(x−3)8 and ∫−2cos(x−3)dx=−2sin(x−3)9. Now substitute these values into the equation: f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C0.Simplify the equation to find f(6): f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C2. Now calculate the sum to get the value of f(6).Calculate the value of f(6): f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C5. Round to the nearest thousandth.The value of f(6) to the nearest thousandth is approximately f(x)=x2sin(x−3)+2xcos(x−3)+2sin(x−3)+C7.