Question

How do you find an equation for the function `f'(x)=sec^2(2x)` whose graph passes through the point (pi/2,2)?

Answer 1

I take this to be the challenge:

Given `f'(x) = sec^2(2x)` and `f(pi/2) = 2`, we are to find an explicit expression for `f(x)`.

From `f'(x) = sec^2(2x)`, we can find

`f(x) = int sec^2(2x) dx`

Integrate (antidifferentiate) by substitution to get

`f(x) = 1/2 tan(2x) +C`

Use the point we were given to find `C`.

`f(pi/2) = 1/2 tan(2(pi/2)) +C = 2` so

`1/2tan(pi)+C = 2`

Since `tan(pi)=0`, we get `C = 2`.

We finish with

`f(x) = 1/2 tan(2x) +2`