Precalculus Examples

$f (x) = - x^{2} + \frac{9}{x}$ , $[1, 6]$

Step 1

Write $f (x) = - x^{2} + \frac{9}{x}$ as an equation.

$y = - x^{2} + \frac{9}{x}$

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.

$\frac{f (6) - f (1)}{(6) - (1)}$

Step 2.2

Substitute the equation $y = - x^{2} + \frac{9}{x}$ for $f (6)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(- {(6)}^{2} + \frac{9}{6}) - (- {(1)}^{2} + \frac{9}{1})}{(6) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Reduce the expression by cancelling the common factors.

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Step 3.1.1

Cancel the common factor of $9$ and $6$ .

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Step 3.1.1.1

Factor $3$ out of $9$ .

$\frac{- {(6)}^{2} + \frac{3 (3)}{6} - (- {(1)}^{2} + \frac{9}{1})}{(6) - (1)}$

Step 3.1.1.2

Cancel the common factors.

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Step 3.1.1.2.1

Factor $3$ out of $6$ .

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

Step 3.1.1.2.2

Cancel the common factor.

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

Step 3.1.1.2.3

Rewrite the expression.

$\frac{- {(6)}^{2} + \frac{3}{2} - (- {(1)}^{2} + \frac{9}{1})}{(6) - (1)}$

Step 3.1.2

Divide $9$ by $1$ .

$\frac{- {(6)}^{2} + \frac{3}{2} - (- {(1)}^{2} + 9)}{(6) - (1)}$

Step 3.2

Multiply the numerator and denominator of the fraction by $2$ .

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Step 3.2.1

Multiply $\frac{- 6^{2} + \frac{3}{2} - (- 1^{2} + 9)}{6 - (1)}$ by $\frac{2}{2}$ .

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

Step 3.2.2

Combine.

$\frac{2 (- 6^{2} + \frac{3}{2} - (- 1^{2} + 9))}{2 (6 - (1))}$

Step 3.3

Apply the distributive property.

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

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Raise $6$ to the power of $2$ .

$\frac{2 (- 1 \cdot 36) + 3 + 2 (- (- 1^{2} + 9))}{2 \cdot 6 + 2 (- (1))}$

Step 3.5.2

Multiply $2 (- 1 \cdot 36)$ .

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Step 3.5.2.1

Multiply $- 1$ by $36$ .

$\frac{2 \cdot - 36 + 3 + 2 (- (- 1^{2} + 9))}{2 \cdot 6 + 2 (- (1))}$

Step 3.5.2.2

Multiply $2$ by $- 36$ .

$\frac{- 72 + 3 + 2 (- (- 1^{2} + 9))}{2 \cdot 6 + 2 (- (1))}$

Step 3.5.3

Simplify each term.

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Step 3.5.3.1

One to any power is one.

$\frac{- 72 + 3 + 2 (- (- 1 \cdot 1 + 9))}{2 \cdot 6 + 2 (- (1))}$

Step 3.5.3.2

Multiply $- 1$ by $1$ .

$\frac{- 72 + 3 + 2 (- (- 1 + 9))}{2 \cdot 6 + 2 (- (1))}$

Step 3.5.4

Add $- 1$ and $9$ .

$\frac{- 72 + 3 + 2 (- 1 \cdot 8)}{2 \cdot 6 + 2 (- (1))}$

Step 3.5.5

Multiply $2 (- 1 \cdot 8)$ .

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Step 3.5.5.1

Multiply $- 1$ by $8$ .

$\frac{- 72 + 3 + 2 \cdot - 8}{2 \cdot 6 + 2 (- (1))}$

Step 3.5.5.2

Multiply $2$ by $- 8$ .

$\frac{- 72 + 3 - 16}{2 \cdot 6 + 2 (- (1))}$

Step 3.5.6

Add $- 72$ and $3$ .

$\frac{- 69 - 16}{2 \cdot 6 + 2 (- (1))}$

Step 3.5.7

Subtract $16$ from $- 69$ .

$\frac{- 85}{2 \cdot 6 + 2 (- (1))}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $2$ by $6$ .

$\frac{- 85}{12 + 2 (- (1))}$

Step 3.6.2

Multiply $2 (- (1))$ .

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Step 3.6.2.1

Multiply $- 1$ by $1$ .

$\frac{- 85}{12 + 2 \cdot - 1}$

Step 3.6.2.2

Multiply $2$ by $- 1$ .

$\frac{- 85}{12 - 2}$

Step 3.6.3

Subtract $2$ from $12$ .

$\frac{- 85}{10}$

Step 3.7

Reduce the expression by cancelling the common factors.

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Step 3.7.1

Cancel the common factor of $- 85$ and $10$ .

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Step 3.7.1.1

Factor $5$ out of $- 85$ .

$\frac{5 (- 17)}{10}$

Step 3.7.1.2

Cancel the common factors.

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Step 3.7.1.2.1

Factor $5$ out of $10$ .

$\frac{5 \cdot - 17}{5 \cdot 2}$

Step 3.7.1.2.2

Cancel the common factor.

$\frac{5 \cdot - 17}{5 \cdot 2}$

Step 3.7.1.2.3

Rewrite the expression.

$\frac{- 17}{2}$

Step 3.7.2

Move the negative in front of the fraction.

$- \frac{17}{2}$