Difference Quotients

Difference quotients are expressions which represent the slope of the line between two general points on the graph of a function, such as

$$\frac{f(x+h)-f(x)}{h}$$

or

$$\frac{f(t)-f(x)}{t-x}$$

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In simplifying a difference quotient, it is important to remember that $f(x+h)$ is not the same as $f(x)+h$. To find $f(x+h)$, substitute $x+h$ for $x$ in the expression for $f(x)$.

Ex 1 Find and simplify

$$\frac{f(x+h)-f(x)}{h}$$

for $f(x)=x^2-3x-5$.

Sol

$$\frac{f(x+h)-f(x)}{h} = \frac{[(x+h)^2-3(x+h)-5]-[x^2-3x-5]}{h}$$

$$= \frac{[x^2+2hx+h^2-3x-3h-5]-[x^2-3x-5]}{h}$$

$$= \frac{x^2+2hx+h^2-3x-3h-5-x^2+3x+5}{h} = \frac{2hx+h^2-3h}{h}$$

$$=\frac{h(2x+h-3)}{h}=2x+h-3$$

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Ex 2 Find and simplify

$$\frac{f(t)-f(x)}{t-x}$$

for $f(x)=x^2+5x$.

Sol

$$\frac{f(t)-f(x)}{t-x}=\frac{(t^2+5t)-(x^2+5x)}{t-x}$$

$$=\frac{t^2+5t-x^2-5x}{t-x} = \frac{(t^2-x^2)+(5t-5x)}{t-x}$$

$$=\frac{(t+x)(t-x)+5(t-x)}{t-x}=\frac{(t+x+5)(t-x)}{t-x}=t+x+5$$

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Pr 1 Find and simplify

$$\frac{f(x+h)-f(x)}{h}$$

for $f(x)=2x^2-5x-9$.

Pr 2 Find and simplify

$$\frac{f(x+h)-f(x)}{h}$$

for $f(x)=x^3$.

Pr 3 Find and simplify

$$\frac{f(x+h)-f(x)}{h}$$

for $f(x)=1/x$.

Pr 4 Find and simplify

$$\frac{f(x+h)-f(x)}{h}$$

for $f(x)=\sqrt{x}$.

Pr 5 Find and simplify

$$\frac{f(t)-f(x)}{t-x}$$

for $f(x)=\frac{1}{x^2}$.

Pr 6 Find and simplify

$$\frac{f(t)-f(x)}{t-x}$$

for $f(x)=\frac{1}{2x+5}$.

Pr 7 Find and simplify

$$\frac{f(t)-f(x)}{t-x}$$

for $f(x)=\sqrt{4x^2+5}$.

Pr 8 Find and simplify

$$\frac{f(t)-f(x)}{t-x}$$

for $f(x)=x^{2/3}$.

Go to Solutions.

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