Find f by solving the initial-value problem.

Question

f '(x) = 9x² − 8x, f(1) = 1

Answer 1

First here we find f(x) by integrating f´(x) with respect to x, then use the initial value to set the integration constant C.

Integrating xⁿ gives xⁿ⁺¹/ (n+1) + C, where C is a constant.

So applying this to f´(x) gives 9. x³/3 - 8 x ²/2 + C, which simplified is f(x) = 3x³ - 4x² + C

f(1)=1, and to find f(1) we just substitute x=1 into f(x).

f(1) = 3.1 - 4.1 + C = 3-4+C = C-1. C-1=1 so C=2

So f(x) = 3x ³-4x²+2.

I recommend checking this by finding f´(x). I leave that as an exercise for you, but if you do so you will find it corresponds to the expression given.

Mike

Answer 2

f(x) = ∫f'(x)dx = 3x³ - 4x² + C

Since f(1) = 1, we have 3 - 4 + C = 1. So, C = 2

Therefore, f(x) = 3x³ - 4x² + 2