Question

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`?

Answer 1

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