Final answer:
To find the original function f(x), given the difference quotient f'(x) = -(x+h)-(-x)/h, simplify the quotient and integrate the resulting expression -2x to get f(x) = -x^2 + C.
Step-by-step explanation:
The student is given the quotient f'(x) = -(x+h)-(-x)/h and needs to find the original function f(x). The given expression is a difference quotient used to find the derivative of a function as h approaches 0. To find f(x), integrate the expression f'(x).
First, simplify the given difference quotient:
f'(x) = -((x + h) - (-x))/h = -((x + h + x))/h = -((2x + h))/h
As h approaches 0, the expression simplifies to:
f'(x) = -2x
Now, integrate f'(x) to get f(x):
f(x) = ∫ -2x dx = -x^2 + C
Here, C is the constant of integration.