Finite Math Examples

$f (x) = \frac{x}{3 x^{2} - 5 x}$ , $[10, 11]$

Step 1

Write $f (x) = \frac{x}{3 x^{2} - 5 x}$ as an equation.

$y = \frac{x}{3 x^{2} - 5 x}$

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.

$\frac{f (11) - f (10)}{(11) - (10)}$

Step 2.2

Substitute the equation $y = \frac{x}{3 x^{2} - 5 x}$ for $f (11)$ and $f (10)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\frac{11}{3 {(11)}^{2} - 5 \cdot 11}) - (\frac{10}{3 {(10)}^{2} - 5 \cdot 10})}{(11) - (10)}$

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 $11$ and $3 {(11)}^{2} - 5 \cdot 11$ .

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Step 3.1.1.1

Factor $11$ out of $11$ .

$\frac{\frac{11 \cdot 1}{3 \cdot 11^{2} - 5 \cdot 11} - (\frac{10}{3 {(10)}^{2} - 5 \cdot 10})}{(11) - (10)}$

Step 3.1.1.2

Cancel the common factors.

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Step 3.1.1.2.1

Factor $11$ out of $3 \cdot 11^{2}$ .

$\frac{\frac{11 \cdot 1}{11 (3 \cdot 11) - 5 \cdot 11} - (\frac{10}{3 {(10)}^{2} - 5 \cdot 10})}{(11) - (10)}$

Step 3.1.1.2.2

Factor $11$ out of $- 5 \cdot 11$ .

$\frac{\frac{11 \cdot 1}{11 (3 \cdot 11) + 11 \cdot - 5} - (\frac{10}{3 {(10)}^{2} - 5 \cdot 10})}{(11) - (10)}$

Step 3.1.1.2.3

Factor $11$ out of $11 (3 \cdot 11) + 11 \cdot - 5$ .

$\frac{\frac{11 \cdot 1}{11 (3 \cdot 11 - 5)} - (\frac{10}{3 {(10)}^{2} - 5 \cdot 10})}{(11) - (10)}$

Step 3.1.1.2.4

Cancel the common factor.

$\frac{\frac{11 \cdot 1}{11 (3 \cdot 11 - 5)} - (\frac{10}{3 {(10)}^{2} - 5 \cdot 10})}{(11) - (10)}$

Step 3.1.1.2.5

Rewrite the expression.

$\frac{\frac{1}{3 \cdot 11 - 5} - (\frac{10}{3 {(10)}^{2} - 5 \cdot 10})}{(11) - (10)}$

Step 3.1.2

Cancel the common factor of $10$ and $3 {(10)}^{2} - 5 \cdot 10$ .

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Step 3.1.2.1

Factor $10$ out of $10$ .

$\frac{\frac{1}{3 \cdot 11 - 5} - \frac{10 \cdot 1}{3 \cdot 10^{2} - 5 \cdot 10}}{(11) - (10)}$

Step 3.1.2.2

Cancel the common factors.

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Step 3.1.2.2.1

Factor $10$ out of $3 \cdot 10^{2}$ .

$\frac{\frac{1}{3 \cdot 11 - 5} - \frac{10 \cdot 1}{10 (3 \cdot 10) - 5 \cdot 10}}{(11) - (10)}$

Step 3.1.2.2.2

Factor $10$ out of $- 5 \cdot 10$ .

$\frac{\frac{1}{3 \cdot 11 - 5} - \frac{10 \cdot 1}{10 (3 \cdot 10) + 10 \cdot - 5}}{(11) - (10)}$

Step 3.1.2.2.3

Factor $10$ out of $10 (3 \cdot 10) + 10 \cdot - 5$ .

$\frac{\frac{1}{3 \cdot 11 - 5} - \frac{10 \cdot 1}{10 (3 \cdot 10 - 5)}}{(11) - (10)}$

Step 3.1.2.2.4

Cancel the common factor.

$\frac{\frac{1}{3 \cdot 11 - 5} - \frac{10 \cdot 1}{10 (3 \cdot 10 - 5)}}{(11) - (10)}$

Step 3.1.2.2.5

Rewrite the expression.

$\frac{\frac{1}{3 \cdot 11 - 5} - \frac{1}{3 \cdot 10 - 5}}{(11) - (10)}$

Step 3.2

Multiply the numerator and denominator of the fraction by $(3 \cdot 11 - 5) (3 \cdot 10 - 5)$ .

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Step 3.2.1

Multiply $\frac{\frac{1}{3 \cdot 11 - 5} - \frac{1}{3 \cdot 10 - 5}}{11 - (10)}$ by $\frac{(3 \cdot 11 - 5) (3 \cdot 10 - 5)}{(3 \cdot 11 - 5) (3 \cdot 10 - 5)}$ .

$\frac{(3 \cdot 11 - 5) (3 \cdot 10 - 5)}{(3 \cdot 11 - 5) (3 \cdot 10 - 5)} \cdot \frac{\frac{1}{3 \cdot 11 - 5} - \frac{1}{3 \cdot 10 - 5}}{11 - (10)}$

Step 3.2.2

Combine.

$\frac{(3 \cdot 11 - 5) (3 \cdot 10 - 5) (\frac{1}{3 \cdot 11 - 5} - \frac{1}{3 \cdot 10 - 5})}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) (11 - (10))}$

Step 3.3

Apply the distributive property.

$\frac{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \frac{1}{3 \cdot 11 - 5} + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- \frac{1}{3 \cdot 10 - 5})}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.4

Simplify by cancelling.

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Step 3.4.1

Cancel the common factor of $3 \cdot 11 - 5$ .

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Step 3.4.1.1

Cancel the common factor.

Step 3.4.1.2

Rewrite the expression.

$\frac{3 \cdot 10 - 5 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- \frac{1}{3 \cdot 10 - 5})}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.4.2

Cancel the common factor of $3 \cdot 10 - 5$ .

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Step 3.4.2.1

Move the leading negative in $- \frac{1}{3 \cdot 10 - 5}$ into the numerator.

$\frac{3 \cdot 10 - 5 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) \frac{- 1}{3 \cdot 10 - 5}}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.4.2.2

Factor $3 \cdot 10 - 5$ out of $(3 \cdot 11 - 5) (3 \cdot 10 - 5)$ .

$\frac{3 \cdot 10 - 5 + (3 \cdot 10 - 5) (3 \cdot 11 - 5) \frac{- 1}{3 \cdot 10 - 5}}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.4.2.3

Cancel the common factor.

$\frac{3 \cdot 10 - 5 + (3 \cdot 10 - 5) (3 \cdot 11 - 5) \frac{- 1}{3 \cdot 10 - 5}}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.4.2.4

Rewrite the expression.

$\frac{3 \cdot 10 - 5 + (3 \cdot 11 - 5) \cdot - 1}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.4.3

Multiply $3$ by $11$ .

$\frac{3 \cdot 10 - 5 + (33 - 5) \cdot - 1}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $3$ by $10$ .

$\frac{30 - 5 + (33 - 5) \cdot - 1}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.5.2

Subtract $5$ from $33$ .

$\frac{30 - 5 + 28 \cdot - 1}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.5.3

Multiply $28$ by $- 1$ .

$\frac{30 - 5 - 28}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.5.4

Subtract $5$ from $30$ .

$\frac{25 - 28}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.5.5

Subtract $28$ from $25$ .

$\frac{- 3}{(3 \cdot 11 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Multiply $3$ by $11$ .

$\frac{- 3}{(33 - 5) (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.6.2

Subtract $5$ from $33$ .

$\frac{- 3}{28 (3 \cdot 10 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.6.3

Multiply $3$ by $10$ .

$\frac{- 3}{28 (30 - 5) \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.6.4

Subtract $5$ from $30$ .

$\frac{- 3}{28 \cdot 25 \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.6.5

Multiply $28 \cdot 25 \cdot 11$ .

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Step 3.6.5.1

Multiply $28$ by $25$ .

$\frac{- 3}{700 \cdot 11 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.6.5.2

Multiply $700$ by $11$ .

$\frac{- 3}{7700 + (3 \cdot 11 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.6.6

Multiply $3$ by $11$ .

$\frac{- 3}{7700 + (33 - 5) (3 \cdot 10 - 5) (- (10))}$

Step 3.6.7

Subtract $5$ from $33$ .

$\frac{- 3}{7700 + 28 (3 \cdot 10 - 5) (- (10))}$

Step 3.6.8

Multiply $3$ by $10$ .

$\frac{- 3}{7700 + 28 (30 - 5) (- (10))}$

Step 3.6.9

Subtract $5$ from $30$ .

$\frac{- 3}{7700 + 28 \cdot 25 (- (10))}$

Step 3.6.10

Multiply $28 \cdot 25 (- (10))$ .

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Step 3.6.10.1

Multiply $28$ by $25$ .

$\frac{- 3}{7700 + 700 (- (10))}$

Step 3.6.10.2

Multiply $- 1$ by $10$ .

$\frac{- 3}{7700 + 700 \cdot - 10}$

Step 3.6.10.3

Multiply $700$ by $- 10$ .

$\frac{- 3}{7700 - 7000}$

Step 3.6.11

Subtract $7000$ from $7700$ .

$\frac{- 3}{700}$

Step 3.7

Move the negative in front of the fraction.

$- \frac{3}{700}$