Find the Function

Question

Find the function f given that the slope of the tangent line to the graph of f at any point (x, f(x)) is f '(x) and that the graph of f passes through the given point. (Remember to use absolute values where appropriate.)

f '(x) = e^x + 9x; (0, 2)

Answer 1

f(x) = ∫f '(x) dx

so f(x) = ∫e^x + 9x dx

To find the antiderivative, think of a function that, if you take its derivative you end up with "e^x + 9x".

That would be e^x + 9/2x² + C

You are given a data point that f(x) passes through so plug it in to solve for C:

2 = e⁰ + 9/2(0)² + C

2 = 1 + 0 + C

C = 1

So f(x) = e^x + 9/2x² + 1