Question

Consider the following initial-value problem. f′(x)=5eˣ−8x;f(0)=9 Integrate the function f′(x). (Remember the constant of integration.) ∫f′(x)dx= Find the value of C using the condition f(0)=9. C= State the function f(x) found by solving the given initial-value problem. f(x)= [−11 Points ] TANAPMATH7 11.1.070.MI. C′(x)=0.004x+95 The fixed costs incurred by Carlota are $2,000/ month. Find the total monthly cost C(x) incurred by Carlota in manufacturing x guitars/month. C(x)=

Answer 1

To integrate the function f'(x) = 5e^x - 8x and solve the initial-value problem, we can use the power rule for integration and the condition f(0) = 9. The function f(x) = 5e^x - 4x^2 + 4 represents the solution to the initial-value problem.

Step-by-step explanation:

To integrate the function f'(x) = 5e^x - 8x, we can use the power rule for integration and the constant of integration. The integral of 5e^x with respect to x is 5e^x, and the integral of 8x with respect to x is 4x^2. Therefore, the integral of f'(x) = 5e^x - 8x is f(x) = 5e^x - 4x^2 + C, where C is the constant of integration.

To find the value of C using the condition f(0) = 9, we substitute x = 0 into the function f(x). f(0) = 5e^0 - 4(0)^2 + C = 5 - 0 + C = 9. Therefore, C = 4.

The function f(x) found by solving the given initial-value problem is f(x) = 5e^x - 4x^2 + 4.