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)=6(2x-7)^5 at (4, 3/2)

Answer 1

f=INT f= (2x-7)^6 + C

now find C

3/2 = (2*4-7)^6 + C

C=1/2

Answer 2

if dy/dx = 6(2x-7)^5

y = (6/6) (2x-7)^6 (1/2) + c
y = (1/2)(2x-7)^6 + c
but (4 , 3/2) lies on it, so

3/2 = (1/2)(1)^6 + c
3/2 = 1/2 + c
c = 1

f(x) = (1/2)(2x-7)^6 + 1