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

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MarkFL

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Feb 24, 2012
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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.
 
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MarkFL

Administrator
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Feb 24, 2012
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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:

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

ii) Quadratic functions.

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:

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