Precalculus Examples

$A (r) = - \frac{2}{3} r - 2$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = r + h$ .

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Step 2.1.1

Replace the variable $r$ with $r + h$ in the expression.

$A (r + h) = - \frac{2}{3} \cdot (r + h) - 2$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Apply the distributive property.

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

Step 2.1.2.1.2

Combine $r$ and $\frac{2}{3}$ .

$A (r + h) = - \frac{r \cdot 2}{3} - \frac{2}{3} h - 2$

Step 2.1.2.1.3

Combine $h$ and $\frac{2}{3}$ .

$A (r + h) = - \frac{r \cdot 2}{3} - \frac{h \cdot 2}{3} - 2$

Step 2.1.2.1.4

Simplify each term.

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Step 2.1.2.1.4.1

Move $2$ to the left of $r$ .

$A (r + h) = - \frac{2 \cdot r}{3} - \frac{h \cdot 2}{3} - 2$

Step 2.1.2.1.4.2

Move $2$ to the left of $h$ .

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

Step 2.1.2.2

The final answer is $- \frac{2 r}{3} - \frac{2 h}{3} - 2$ .

$- \frac{2 r}{3} - \frac{2 h}{3} - 2$

Step 2.2

Reorder $- \frac{2 r}{3}$ and $- \frac{2 h}{3}$ .

$- \frac{2 h}{3} - \frac{2 r}{3} - 2$

Step 2.3

Find the components of the definition.

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

$A (r) = - \frac{2 r}{3} - 2$

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

$A (r) = - \frac{2 r}{3} - 2$

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{- \frac{2 h}{3} - \frac{2 r}{3} - 2 - - \frac{2 r}{3} - - 2}{h}$

Step 4.1.2

Multiply $- - \frac{2 r}{3}$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{2 h}{3} - \frac{2 r}{3} - 2 + 1 \frac{2 r}{3} - - 2}{h}$

Step 4.1.2.2

Multiply $\frac{2 r}{3}$ by $1$ .

$\frac{- \frac{2 h}{3} - \frac{2 r}{3} - 2 + \frac{2 r}{3} - - 2}{h}$

Step 4.1.3

Multiply $- 1$ by $- 2$ .

$\frac{- \frac{2 h}{3} - \frac{2 r}{3} - 2 + \frac{2 r}{3} + 2}{h}$

Step 4.1.4

Add $- \frac{2 r}{3}$ and $\frac{2 r}{3}$ .

$\frac{- \frac{2 h}{3} + 0 - 2 + 2}{h}$

Step 4.1.5

Add $- \frac{2 h}{3}$ and $0$ .

$\frac{- \frac{2 h}{3} - 2 + 2}{h}$

Step 4.1.6

Add $- 2$ and $2$ .

$\frac{- \frac{2 h}{3} + 0}{h}$

Step 4.1.7

Add $- \frac{2 h}{3}$ and $0$ .

$\frac{- \frac{2 h}{3}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{2 h}{3} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{2 h}{3}$ into the numerator.

$\frac{- 2 h}{3} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- 2 h$ .

$\frac{h \cdot - 2}{3} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 2}{3} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 2}{3}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{2}{3}$

Step 5