Precalculus Examples

$f (x) = \frac{1}{x} - x^{2} - 2$ for $3 \leq x \leq 6$

Step 1

Write $f (x) = \frac{1}{x} - x^{2} - 2$ as an equation.

$y = \frac{1}{x} - x^{2} - 2$

Step 2

Substitute using the average rate of change formula.

Tap for more steps...

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.

$\frac{f (6) - f (3)}{(6) - (3)}$

Step 2.2

Substitute the equation $y = \frac{1}{x} - x^{2} - 2$ for $f (6)$ and $f (3)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{1}{6} - {(6)}^{2} - 2) - (\frac{1}{3} - {(3)}^{2} - 2)}{(6) - (3)}$

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Multiply the numerator and denominator of the fraction by $6$ .

Tap for more steps...

Step 3.1.1

Multiply $\frac{\frac{1}{6} - 6^{2} - 2 - (\frac{1}{3} - 3^{2} - 2)}{6 - (3)}$ by $\frac{6}{6}$ .

$\frac{6}{6} \cdot \frac{\frac{1}{6} - 6^{2} - 2 - (\frac{1}{3} - 3^{2} - 2)}{6 - (3)}$

Step 3.1.2

Combine.

$\frac{6 (\frac{1}{6} - 6^{2} - 2 - (\frac{1}{3} - 3^{2} - 2))}{6 (6 - (3))}$

Step 3.2

Apply the distributive property.

$\frac{6 (\frac{1}{6}) + 6 (- 6^{2}) + 6 \cdot - 2 + 6 (- (\frac{1}{3} - 3^{2} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.3

Cancel the common factor of $6$ .

Tap for more steps...

Step 3.3.1

Cancel the common factor.

$\frac{6 (\frac{1}{6}) + 6 (- 6^{2}) + 6 \cdot - 2 + 6 (- (\frac{1}{3} - 3^{2} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.3.2

Rewrite the expression.

$\frac{1 + 6 (- 6^{2}) + 6 \cdot - 2 + 6 (- (\frac{1}{3} - 3^{2} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4

Simplify the numerator.

Tap for more steps...

Step 3.4.1

Multiply $6$ by $6^{2}$ by adding the exponents.

Tap for more steps...

Step 3.4.1.1

Move $6^{2}$ .

$\frac{1 + 6^{2} \cdot 6 \cdot - 1 + 6 \cdot - 2 + 6 (- (\frac{1}{3} - 3^{2} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.1.2

Multiply $6^{2}$ by $6$ .

Tap for more steps...

Step 3.4.1.2.1

Raise $6$ to the power of $1$ .

$\frac{1 + 6^{2} \cdot 6^{1} \cdot - 1 + 6 \cdot - 2 + 6 (- (\frac{1}{3} - 3^{2} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1 + 6^{2 + 1} \cdot - 1 + 6 \cdot - 2 + 6 (- (\frac{1}{3} - 3^{2} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.1.3

Add $2$ and $1$ .

$\frac{1 + 6^{3} \cdot - 1 + 6 \cdot - 2 + 6 (- (\frac{1}{3} - 3^{2} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.2

Raise $6$ to the power of $3$ .

$\frac{1 + 216 \cdot - 1 + 6 \cdot - 2 + 6 (- (\frac{1}{3} - 3^{2} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.3

Multiply $216$ by $- 1$ .

$\frac{1 - 216 + 6 \cdot - 2 + 6 (- (\frac{1}{3} - 3^{2} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.4

Multiply $6$ by $- 2$ .

$\frac{1 - 216 - 12 + 6 (- (\frac{1}{3} - 3^{2} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.5

Simplify each term.

Tap for more steps...

Step 3.4.5.1

Raise $3$ to the power of $2$ .

$\frac{1 - 216 - 12 + 6 (- (\frac{1}{3} - 1 \cdot 9 - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.5.2

Multiply $- 1$ by $9$ .

$\frac{1 - 216 - 12 + 6 (- (\frac{1}{3} - 9 - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.6

Find the common denominator.

Tap for more steps...

Step 3.4.6.1

Write $- 9$ as a fraction with denominator $1$ .

$\frac{1 - 216 - 12 + 6 (- (\frac{1}{3} + \frac{- 9}{1} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.6.2

Multiply $\frac{- 9}{1}$ by $\frac{3}{3}$ .

$\frac{1 - 216 - 12 + 6 (- (\frac{1}{3} + \frac{- 9}{1} \cdot \frac{3}{3} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.6.3

Multiply $\frac{- 9}{1}$ by $\frac{3}{3}$ .

$\frac{1 - 216 - 12 + 6 (- (\frac{1}{3} + \frac{- 9 \cdot 3}{3} - 2))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.6.4

Write $- 2$ as a fraction with denominator $1$ .

$\frac{1 - 216 - 12 + 6 (- (\frac{1}{3} + \frac{- 9 \cdot 3}{3} + \frac{- 2}{1}))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.6.5

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

$\frac{1 - 216 - 12 + 6 (- (\frac{1}{3} + \frac{- 9 \cdot 3}{3} + \frac{- 2}{1} \cdot \frac{3}{3}))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.6.6

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

$\frac{1 - 216 - 12 + 6 (- (\frac{1}{3} + \frac{- 9 \cdot 3}{3} + \frac{- 2 \cdot 3}{3}))}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.7

Combine the numerators over the common denominator.

$\frac{1 - 216 - 12 + 6 (- \frac{1 - 9 \cdot 3 - 2 \cdot 3}{3})}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.8

Simplify each term.

Tap for more steps...

Step 3.4.8.1

Multiply $- 9$ by $3$ .

$\frac{1 - 216 - 12 + 6 (- \frac{1 - 27 - 2 \cdot 3}{3})}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.8.2

Multiply $- 2$ by $3$ .

$\frac{1 - 216 - 12 + 6 (- \frac{1 - 27 - 6}{3})}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.9

Subtract $27$ from $1$ .

$\frac{1 - 216 - 12 + 6 (- \frac{- 26 - 6}{3})}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.10

Subtract $6$ from $- 26$ .

$\frac{1 - 216 - 12 + 6 (- \frac{- 32}{3})}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.11

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.4.11.1

Move the leading negative in $- \frac{- 32}{3}$ into the numerator.

$\frac{1 - 216 - 12 + 6 (\frac{32}{3})}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.11.2

Factor $3$ out of $6$ .

$\frac{1 - 216 - 12 + 3 (2) \frac{32}{3}}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.11.3

Cancel the common factor.

$\frac{1 - 216 - 12 + 3 \cdot 2 \frac{32}{3}}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.11.4

Rewrite the expression.

$\frac{1 - 216 - 12 + 2 \cdot 32}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.12

Multiply $2$ by $32$ .

$\frac{1 - 216 - 12 + 64}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.13

Subtract $216$ from $1$ .

$\frac{- 215 - 12 + 64}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.14

Subtract $12$ from $- 215$ .

$\frac{- 227 + 64}{6 \cdot 6 + 6 (- (3))}$

Step 3.4.15

Add $- 227$ and $64$ .

$\frac{- 163}{6 \cdot 6 + 6 (- (3))}$

Step 3.5

Simplify the denominator.

Tap for more steps...

Step 3.5.1

Multiply $6$ by $6$ .

$\frac{- 163}{36 + 6 (- (3))}$

Step 3.5.2

Multiply $6 (- (3))$ .

Tap for more steps...

Step 3.5.2.1

Multiply $- 1$ by $3$ .

$\frac{- 163}{36 + 6 \cdot - 3}$

Step 3.5.2.2

Multiply $6$ by $- 3$ .

$\frac{- 163}{36 - 18}$

Step 3.5.3

Subtract $18$ from $36$ .

$\frac{- 163}{18}$

Step 3.6

Move the negative in front of the fraction.

$- \frac{163}{18}$