Finite Math Examples

$f (x) = x^{3} - 12 x^{2} + 21 x + 14$ , $[- 1, 9]$

Step 1

Write $f (x) = x^{3} - 12 x^{2} + 21 x + 14$ as an equation.

$y = x^{3} - 12 x^{2} + 21 x + 14$

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.

$\frac{f (9) - f (- 1)}{(9) - (- 1)}$

Step 2.2

Substitute the equation $y = x^{3} - 12 x^{2} + 21 x + 14$ for $f (9)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(9)}^{3} - 12 {(9)}^{2} + 21 (9) + 14) - ({(- 1)}^{3} - 12 {(- 1)}^{2} + 21 (- 1) + 14)}{(9) - (- 1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $9$ to the power of $3$ .

$\frac{729 - 12 \cdot 9^{2} + 21 (9) + 14 - ({(- 1)}^{3} - 12 {(- 1)}^{2} + 21 (- 1) + 14)}{9 - (- 1)}$

Step 3.1.2

Raise $9$ to the power of $2$ .

$\frac{729 - 12 \cdot 81 + 21 (9) + 14 - ({(- 1)}^{3} - 12 {(- 1)}^{2} + 21 (- 1) + 14)}{9 - (- 1)}$

Step 3.1.3

Multiply $- 12$ by $81$ .

$\frac{729 - 972 + 21 (9) + 14 - ({(- 1)}^{3} - 12 {(- 1)}^{2} + 21 (- 1) + 14)}{9 - (- 1)}$

Step 3.1.4

Multiply $21$ by $9$ .

$\frac{729 - 972 + 189 + 14 - ({(- 1)}^{3} - 12 {(- 1)}^{2} + 21 (- 1) + 14)}{9 - (- 1)}$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

Raise $- 1$ to the power of $3$ .

$\frac{729 - 972 + 189 + 14 - (- 1 - 12 {(- 1)}^{2} + 21 (- 1) + 14)}{9 - (- 1)}$

Step 3.1.5.2

Raise $- 1$ to the power of $2$ .

$\frac{729 - 972 + 189 + 14 - (- 1 - 12 \cdot 1 + 21 (- 1) + 14)}{9 - (- 1)}$

Step 3.1.5.3

Multiply $- 12$ by $1$ .

$\frac{729 - 972 + 189 + 14 - (- 1 - 12 + 21 (- 1) + 14)}{9 - (- 1)}$

Step 3.1.5.4

Multiply $21$ by $- 1$ .

$\frac{729 - 972 + 189 + 14 - (- 1 - 12 - 21 + 14)}{9 - (- 1)}$

Step 3.1.6

Subtract $12$ from $- 1$ .

$\frac{729 - 972 + 189 + 14 - (- 13 - 21 + 14)}{9 - (- 1)}$

Step 3.1.7

Subtract $21$ from $- 13$ .

$\frac{729 - 972 + 189 + 14 - (- 34 + 14)}{9 - (- 1)}$

Step 3.1.8

Add $- 34$ and $14$ .

$\frac{729 - 972 + 189 + 14 - - 20}{9 - (- 1)}$

Step 3.1.9

Multiply $- 1$ by $- 20$ .

$\frac{729 - 972 + 189 + 14 + 20}{9 - (- 1)}$

Step 3.1.10

Subtract $972$ from $729$ .

$\frac{- 243 + 189 + 14 + 20}{9 - (- 1)}$

Step 3.1.11

Add $- 243$ and $189$ .

$\frac{- 54 + 14 + 20}{9 - (- 1)}$

Step 3.1.12

Add $- 54$ and $14$ .

$\frac{- 40 + 20}{9 - (- 1)}$

Step 3.1.13

Add $- 40$ and $20$ .

$\frac{- 20}{9 - (- 1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- 20}{9 + 1}$

Step 3.2.2

Add $9$ and $1$ .

$\frac{- 20}{10}$

Step 3.3

Divide $- 20$ by $10$ .

$- 2$