Limits of Average Rates of Change

[Alyosha]

I'm not looking for an answer to a specific question, but I want to know in general how to evaluate the limit of average rates of change.

Homework Statement

lim$$_{}h \rightarrow0$$ f (x + h) - f (x) / h

Homework Equations

f(x) = x^2 , x = 1

The Attempt at a Solution

I really don't know what to do. Obviously we need the denominator not equal to 0. An example in my book showed them multiply by 1 by multiplying the numerator and denominator by the conjugate since the numerator had roots...but this has no roots.

 

[nicksauce]

I don't want to do your homework, so I'll do f(x) = x^3.
$$\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$

$$\lim_{h\rightarrow 0} \frac{(x+h)^3-x^3}{h}$$

$$\lim_{h\rightarrow 0} \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h}$$

$$\lim_{h\rightarrow 0} \frac{3x^2h + 3xh^2 + h^3}{h}$$

$$\lim_{h\rightarrow 0} 3x^2 + 3xh + h^2$$

$$=3x^2$$

 

[Alyosha]

If it says for example x = 1, all you do f (1) and evaluate?

 

[nicksauce]

Right, so in the example I did, f ' (1) = 3, f ' (2) = 12 etc.

 

[Gib Z]

Well Since you know that you must evaluate at x=1, you can do two things. Either do as nicksauce did, and sub in x=1 at the end, or simply evaluate

$$\lim_{h\to 0} \frac{(1+h)^2 - 1^2}{h}$$ Directly.