Precalculus Examples

$f (x) = x^{2} - \frac{1}{x}$ , $[2, 4]$

Step 1

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

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

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.

$\frac{f (4) - f (2)}{(4) - (2)}$

Step 2.2

Substitute the equation $y = x^{2} - \frac{1}{x}$ for $f (4)$ and $f (2)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(4)}^{2} - \frac{1}{4}) - ({(2)}^{2} - \frac{1}{2})}{(4) - (2)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 3.1.1

Multiply $\frac{4^{2} - \frac{1}{4} - (2^{2} - \frac{1}{2})}{4 - (2)}$ by $\frac{4}{4}$ .

$\frac{4}{4} \cdot \frac{4^{2} - \frac{1}{4} - (2^{2} - \frac{1}{2})}{4 - (2)}$

Step 3.1.2

Combine.

$\frac{4 (4^{2} - \frac{1}{4} - (2^{2} - \frac{1}{2}))}{4 (4 - (2))}$

Step 3.2

Apply the distributive property.

$\frac{4 \cdot 4^{2} + 4 (- \frac{1}{4}) + 4 (- (2^{2} - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.3

Cancel the common factor of $4$ .

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Step 3.3.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$\frac{4 \cdot 4^{2} + 4 (\frac{- 1}{4}) + 4 (- (2^{2} - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.3.2

Cancel the common factor.

$\frac{4 \cdot 4^{2} + 4 (\frac{- 1}{4}) + 4 (- (2^{2} - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.3.3

Rewrite the expression.

$\frac{4 \cdot 4^{2} - 1 + 4 (- (2^{2} - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Multiply $4$ by $4^{2}$ by adding the exponents.

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Step 3.4.1.1

Multiply $4$ by $4^{2}$ .

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Step 3.4.1.1.1

Raise $4$ to the power of $1$ .

$\frac{4^{1} \cdot 4^{2} - 1 + 4 (- (2^{2} - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.1.1.2

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

$\frac{4^{1 + 2} - 1 + 4 (- (2^{2} - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.1.2

Add $1$ and $2$ .

$\frac{4^{3} - 1 + 4 (- (2^{2} - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.2

Raise $4$ to the power of $3$ .

$\frac{64 - 1 + 4 (- (2^{2} - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.3

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

$\frac{64 - 1 + 4 (- (4 - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{64 - 1 + 4 (- (4 \cdot \frac{2}{2} - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.5

Combine $4$ and $\frac{2}{2}$ .

$\frac{64 - 1 + 4 (- (\frac{4 \cdot 2}{2} - \frac{1}{2}))}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.6

Combine the numerators over the common denominator.

$\frac{64 - 1 + 4 (- \frac{4 \cdot 2 - 1}{2})}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.7

Simplify the numerator.

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Step 3.4.7.1

Multiply $4$ by $2$ .

$\frac{64 - 1 + 4 (- \frac{8 - 1}{2})}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.7.2

Subtract $1$ from $8$ .

$\frac{64 - 1 + 4 (- \frac{7}{2})}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.8

Cancel the common factor of $2$ .

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Step 3.4.8.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$\frac{64 - 1 + 4 (\frac{- 7}{2})}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.8.2

Factor $2$ out of $4$ .

$\frac{64 - 1 + 2 (2) \frac{- 7}{2}}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.8.3

Cancel the common factor.

$\frac{64 - 1 + 2 \cdot 2 \frac{- 7}{2}}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.8.4

Rewrite the expression.

$\frac{64 - 1 + 2 \cdot - 7}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.9

Multiply $2$ by $- 7$ .

$\frac{64 - 1 - 14}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.10

Subtract $1$ from $64$ .

$\frac{63 - 14}{4 \cdot 4 + 4 (- (2))}$

Step 3.4.11

Subtract $14$ from $63$ .

$\frac{49}{4 \cdot 4 + 4 (- (2))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $4$ by $4$ .

$\frac{49}{16 + 4 (- (2))}$

Step 3.5.2

Multiply $4 (- (2))$ .

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Step 3.5.2.1

Multiply $- 1$ by $2$ .

$\frac{49}{16 + 4 \cdot - 2}$

Step 3.5.2.2

Multiply $4$ by $- 2$ .

$\frac{49}{16 - 8}$

Step 3.5.3

Subtract $8$ from $16$ .

$\frac{49}{8}$