Given the functions f(x) = 2x2, answer the following questions.

Question

Given the functions f(x) = 2x², answer the following questions.

a) Evaluate lim(h --> 0) ((f(x+h)-f(x))/h)

b) Hence, find lim(h --> 0) ((f(-1+h)-f(-1))/h)

Answer 1

The question is asking you to find the derivative of f(x), f '(x)

First, calculate f(x+h)

f(x+h) = 2(x+h)² = 2(x² + 2xh + h²) = 2x² + 4xh + 2h²

Thus, lim (h→0) ( f(x+h) - f(x) )/h

= lim (h→0) ( (2x² + 4xh + 2h²) - (2x²) )/h

= lim (h→0) (4xh + 2h²) /h

= lim (h→0) (4x + 2h) = 4x

Part b is the same as part a except x has been replaced with -1, so the question is asking you to find f '(-1)

Plug in -1 for x in f '(x)