How do you find the difference quotient of f, that is, find ((f(x+h)-f(x))/h), h != 0 for f(x)=x^2-5x+7?

1 Answer
Dec 14, 2017

see explanation

Explanation:

If I understand the question correctly, you have to start by substituting (x+h) wherever you see x in your original function definition:

(x+h)^2 - 5(x+h) + 7

multiplying out, you have: x^2+ 2xh + h^2 - 5(x+h) + 7

= x^2 + 2xh + h^2 - 5x - 5h + 7

So, substitute this as the value of f(x+h) in the definition of the difference quotient.

(f(x+h) - f(x))/h

= ((x^2 + 2xh + h^2 - 5x - 5h + 7) - (x^2 - 5x + 7))/h

...this simplifies to:

(2xh + h^2 - 5h)/h

...now, since this is calculus, the inevitable next step is to find the limit of this function as h -> 0. For this, you can't have h in the denominator. But we can do a little factoring kung-fu. The equation above can be re-written:

(h(2x + h - 5))/h = 2x - 5 + h

...and the limit of this function, as h approaches 0, is :

2x - 5

...which is the derivative of the original function x^2 - 5x + 7

GOOD LUCK