The slope of y = f(x)at (x,y) is sin(2x), and f(0) = 2. Find f(x).

Question

Step by step process please?

Answer 1

What have you tried so far? Where are you getting stuck?

Answer 2

y=f(x)

slope = y' = f'(x) = sin(2x)

y = f(x) = integral [sin(2x)dx --> use u-substituion to integrate:

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Let u = 2x

du/dx = 2

du = 2dx

du/2 = dx

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y = f(x) = [sin(u)(du/2)

= 1/2[sin(u)du)

= 1/2(-cos(u)) +C

= (-1/2)cos(2x) + C

Plug in f(0) = 2 to find C:

f(x) = -1/2 cos(2x) + C

f(0) = -1/2 cos(2*0) + C = 2

-1/2 cos(0) + C = 2

-1/2 (1) + C = 2

-1/2 + C = 2

C = 2 + 1/2

C = 5/2 = 2.5

So, f(x) = -1/2 cos(2x) + 5/2