Question

Given f’(x)=-4x - 5 compute. F(3)- f(-1)

Answer 1

-36

Explanation:

You want F(3) -F(1) given f'(x) = -4x -5.

Integral

The desired difference is the definite integral ...

`\displaystyle \int_(-1)^3{(-4x-5)}\,dx=\left[-2x^2-5x\right]_(-1)^3=-2(9-1)-5(3+1)\\\\=\boxed{-36}`

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Answer 2

-36

Explanation:

To solve this problem, we need to integrate f'(x) to find f(x), and then evaluate f(3) and f(-1) to compute the expression f(3) - f(-1).

Integrating f'(x)=-4x - 5 with respect to x, we get:

f(x) = -2x^2 - 5x + C

Where C is the constant of integration.

To find the value of C, we can use the initial condition f(0) = 1. Substituting x = 0 and f(x) = 1 into the equation above, we get:

1 = -2(0)^2 - 5(0) + C

1 = C

So, we have:

f(x) = -2x^2 - 5x + 1

Now, we can evaluate f(3) and f(-1) and compute f(3) - f(-1):

f(3) = -2(3)^2 - 5(3) + 1 = -32

f(-1) = -2(-1)^2 - 5(-1) + 1 = 4

f(3) - f(-1) = (-32) - 4 = -36

Therefore, f(3) - f(-1) = -36.