Question

# Find the function f given that the slope of the tangent line at any point (x, f(x)) is f '(x) and that the graph of f passes through the given point. f '(x)=5(2x − 1)^{4}, (1, 9)

Functions
Find the function f given that the slope of the tangent line at any point (x, f(x)) is f '(x) and that the graph of f passes through the given point. $$f '(x)=5(2x − 1)^{4}, (1, 9)$$
Start by finding the slope of f(x). To do this, integrate f'(x). The integral comes out to $$\frac{(2x-1)^{5}}{2}+C$$. Now solve for C by solving $$f(1)=\frac{(2\times1−1)^{5}}{2}+C=9$$, which gives us a C value of $$\frac{17}{2}$$, thus the function $$f(x)=\frac{(2x−1)^{5}}{2}+\frac{17}{2}$$