an's questions at Yahoo! Answers regarding difference quotients for linear and quadratic functions

MarkFL

Staff member
Here are the questions:

Evaluate the difference quotient for?

Evaluate the difference quotient for:

1) f(x)=2x-3
2) f(x)=2x^2-3x

PLEASE SHOW ALL WORK
I have posted a link there to this topic so the OP may see my work.

MarkFL

Staff member
Hello an,

Rather than work these specific problems, let's use general functions to develop theorems which we can then use to answer the questions.

i) Linear functions.

Let $$\displaystyle f(x)=ax+b$$

and so the difference quotient is:

$$\displaystyle \lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0}\frac{(a(x+h)+b)-(ax+b)}{h}=\lim_{h\to0}\frac{ax+ah+b-ax-b}{h}=\lim_{h\to0}\frac{ah}{h}=\lim_{h\to0}a=a$$

Let $$\displaystyle f(x)=ax^2+bc+c$$

and so the difference quotient is:

$$\displaystyle \lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0}\frac{\left(a(x+h)^2+b(x+h)+c \right)-\left(ax^2+bx+c \right)}{h}=$$

$$\displaystyle \lim_{h\to0}\frac{ax^2+2ahx+ah^2+bx+bh+c-ax^2-bx-c}{h}=\lim_{h\to0}\frac{2ahx+ah^2+bh}{h}= \lim_{h\to0}(2ax+ah+b)=2ax+b$$

Now we may answer the questions:

1.) This is a linear function. We identity $a=2,\,b=-3$ and so:

$$\displaystyle \lim_{h\to0}\frac{f(x+h)-f(x)}{h}=a=2$$

2.) This is a quadratic function. We identity $a=2,\,b=-3,\,c=0$ and so:

$$\displaystyle \lim_{h\to0}\frac{f(x+h)-f(x)}{h}=2ax+b=4x-3$$