Precalculus Examples

What is the average rate of change of $h (x) = \frac{1}{10 - x}$ over the interval $5 \leq x \leq 8$ ?

Step 1

Write $h (x) = \frac{1}{10 - x}$ as an equation.

$y = \frac{1}{10 - x}$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $x$ values of the two points.

$\frac{f (8) - f (5)}{(8) - (5)}$

Step 2.2

Substitute the equation $y = \frac{1}{10 - x}$ for $f (8)$ and $f (5)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{1}{10 - (8)}) - (\frac{1}{10 - (5)})}{(8) - (5)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(10 - (8)) (10 - (5))$ .

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Step 3.1.1

Multiply $\frac{\frac{1}{10 - (8)} - \frac{1}{10 - (5)}}{8 - (5)}$ by $\frac{(10 - 1 \cdot 8) (10 - 1 \cdot 5)}{(10 - 1 \cdot 8) (10 - 1 \cdot 5)}$ .

$\frac{(10 - 1 \cdot 8) (10 - 1 \cdot 5)}{(10 - 1 \cdot 8) (10 - 1 \cdot 5)} \cdot \frac{\frac{1}{10 - (8)} - \frac{1}{10 - (5)}}{8 - (5)}$

Step 3.1.2

Combine.

$\frac{(10 - 1 \cdot 8) (10 - 1 \cdot 5) (\frac{1}{10 - (8)} - \frac{1}{10 - (5)})}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) (8 - (5))}$

Step 3.2

Apply the distributive property.

$\frac{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \frac{1}{10 - (8)} + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- \frac{1}{10 - (5)})}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $10 - 1 \cdot 8$ .

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Step 3.3.1.1

Multiply by $1$ .

$\frac{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \frac{1}{(10 - (8)) \cdot 1} + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- \frac{1}{10 - (5)})}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.3.1.2

Cancel the common factor.

Step 3.3.1.3

Rewrite the expression.

$\frac{10 - 1 \cdot 5 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- \frac{1}{10 - (5)})}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.3.2

Cancel the common factor of $10 - 1 \cdot 5$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{1}{10 - (5)}$ into the numerator.

$\frac{10 - 1 \cdot 5 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) \frac{- 1}{10 - (5)}}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.3.2.2

Factor $10 - 1 \cdot 5$ out of $(10 - 1 \cdot 8) (10 - 1 \cdot 5)$ .

$\frac{10 - 1 \cdot 5 + (10 - 1 \cdot 5) (10 - 1 \cdot 8) \frac{- 1}{10 - (5)}}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.3.2.3

Multiply by $1$ .

$\frac{10 - 1 \cdot 5 + (10 - 1 \cdot 5) (10 - 1 \cdot 8) \frac{- 1}{(10 - (5)) \cdot 1}}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.3.2.4

Cancel the common factor.

$\frac{10 - 1 \cdot 5 + (10 - 1 \cdot 5) (10 - 1 \cdot 8) \frac{- 1}{(10 - (5)) \cdot 1}}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.3.2.5

Rewrite the expression.

$\frac{10 - 1 \cdot 5 + (10 - 1 \cdot 8) \cdot - 1}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.3.3

Multiply $- 1$ by $8$ .

$\frac{10 - 1 \cdot 5 + (10 - 8) \cdot - 1}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $- 1$ by $5$ .

$\frac{10 - 5 + (10 - 8) \cdot - 1}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.4.2

Subtract $8$ from $10$ .

$\frac{10 - 5 + 2 \cdot - 1}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.4.3

Multiply $2$ by $- 1$ .

$\frac{10 - 5 - 2}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.4.4

Subtract $5$ from $10$ .

$\frac{5 - 2}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.4.5

Subtract $2$ from $5$ .

$\frac{3}{(10 - 1 \cdot 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $- 1$ by $8$ .

$\frac{3}{(10 - 8) (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.5.2

Subtract $8$ from $10$ .

$\frac{3}{2 (10 - 1 \cdot 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.5.3

Multiply $- 1$ by $5$ .

$\frac{3}{2 (10 - 5) \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.5.4

Subtract $5$ from $10$ .

$\frac{3}{2 \cdot 5 \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.5.5

Multiply $2 \cdot 5 \cdot 8$ .

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Step 3.5.5.1

Multiply $2$ by $5$ .

$\frac{3}{10 \cdot 8 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.5.5.2

Multiply $10$ by $8$ .

$\frac{3}{80 + (10 - 1 \cdot 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.5.6

Multiply $- 1$ by $8$ .

$\frac{3}{80 + (10 - 8) (10 - 1 \cdot 5) (- (5))}$

Step 3.5.7

Subtract $8$ from $10$ .

$\frac{3}{80 + 2 (10 - 1 \cdot 5) (- (5))}$

Step 3.5.8

Multiply $- 1$ by $5$ .

$\frac{3}{80 + 2 (10 - 5) (- (5))}$

Step 3.5.9

Subtract $5$ from $10$ .

$\frac{3}{80 + 2 \cdot 5 (- (5))}$

Step 3.5.10

Multiply $2 \cdot 5 (- (5))$ .

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Step 3.5.10.1

Multiply $2$ by $5$ .

$\frac{3}{80 + 10 (- (5))}$

Step 3.5.10.2

Multiply $- 1$ by $5$ .

$\frac{3}{80 + 10 \cdot - 5}$

Step 3.5.10.3

Multiply $10$ by $- 5$ .

$\frac{3}{80 - 50}$

Step 3.5.11

Subtract $50$ from $80$ .

$\frac{3}{30}$

Step 3.6

Cancel the common factor of $3$ and $30$ .

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Step 3.6.1

Factor $3$ out of $3$ .

$\frac{3 (1)}{30}$

Step 3.6.2

Cancel the common factors.

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Step 3.6.2.1

Factor $3$ out of $30$ .

$\frac{3 \cdot 1}{3 \cdot 10}$

Step 3.6.2.2

Cancel the common factor.

$\frac{3 \cdot 1}{3 \cdot 10}$

Step 3.6.2.3

Rewrite the expression.

$\frac{1}{10}$