MHB Difference Quotient for Linear and Quadratic Functions

AI Thread Summary
The discussion focuses on evaluating the difference quotient for linear and quadratic functions. For the linear function f(x) = 2x - 3, the difference quotient simplifies to a constant value of 2. In contrast, for the quadratic function f(x) = 2x^2 - 3x, the difference quotient results in 4x - 3, which varies with x. The thread emphasizes deriving general formulas for the difference quotient before applying them to specific functions. Overall, the difference quotient provides insights into the slope of linear and quadratic functions at any point.
MarkFL
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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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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 $$f(x)=ax+b$$

and so the difference quotient is:

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

ii) Quadratic functions.

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

and so the difference quotient is:

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

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

Now we may answer the questions:

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

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

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

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