In some cases, we are asked to find the difference quotient of a given function. Additionally, there are specific measures we need to take in order to find the difference quotient with fractions and with square roots.

Test Objectives
• Demonstrate the ability to find the difference quotient
• Demonstrate the ability to rationalize the numerator
Difference Quotient Practice Test:

#1:

Instructions: find the difference quotient.

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

$$a)\hspace{.2em}f(x)=5x - 3$$

$$b)\hspace{.2em}f(x)=3x - 7$$

#2:

Instructions: find the difference quotient.

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

$$a)\hspace{.2em}f(x)=3x^2 - 5x - 1$$

$$b)\hspace{.2em}f(x)=x^2 - 25$$

#3:

Instructions: find the difference quotient.

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

$$a)\hspace{.2em}f(x)=x^3 + 1$$

$$b)\hspace{.2em}f(x)=-2x^3 - x - 7$$

#4:

Instructions: find the difference quotient.

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

$$a)\hspace{.2em}f(x)=\frac{7}{x - 1}$$

$$b)\hspace{.2em}f(x)=\frac{4x}{x - 5}$$

#5:

Instructions: find the difference quotient.

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

$$a)\hspace{.2em}f(x)=\sqrt{x - 4}$$

$$b)\hspace{.2em}f(x)=\sqrt{7 - x^2}$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}5$$

$$b)\hspace{.2em}3$$

#2:

Solutions:

$$a)\hspace{.2em}6x + 3h - 5$$

$$b)\hspace{.2em}2x + h$$

#3:

Solutions:

$$a)\hspace{.2em}3x^2 + 3xh + h^2$$

$$b)\hspace{.2em}{-}6x^2 - 6xh - 2h^2 - 1$$

#4:

Solutions:

$$a)\hspace{.2em}\frac{-7}{(x + h - 1)(x - 1)}$$

$$b)\hspace{.2em}\frac{-20}{(x + h - 5)(x - 5)}$$

#5:

Solutions:

$$a)\hspace{.2em}\frac{1}{\sqrt{x + h - 4}+ \sqrt{x - 4}}$$

$$b)\hspace{.2em}\frac{-2x - h}{\sqrt{7 - (x + h)^2} + \sqrt{7 - x^2}}$$