antiderivatives

Question

f''(x)=-4sin(2x) and f'(0)=4 and f(0)=-1

Find f(pi/4)

Answer 1

You can use u-substitution to figure out the first antiderivative. Let u = 2x, then du = 2dx. The values for f'(0) and f(0) are provided so that you can figure out the values of C at each step. So let's do the first one:

f'(x) = ∫-4sin(2x)dx

let u = 2x, du = 2dx

f'(x) = ∫-2sin(u)du = 2cos(u) + C = 2cos(2x) + C

Now plug in 0 to find C:

f'(0) = 2cos(0) + C = 4

2 + C = 4

C = 2

So now we know the complete equation for f'(x):

f'(x) = 2cos(2x) + 2

Can you repeat this to find f(x) and then f(π/4)?