Precalculus Examples

$f (x) = x^{3} + 5 x^{2} - 2$ ; $[- 9, - 5]$

Step 1

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

$y = x^{3} + 5 x^{2} - 2$

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.

$\frac{f (- 5) - f (- 9)}{(- 5) - (- 9)}$

Step 2.2

Substitute the equation $y = x^{3} + 5 x^{2} - 2$ for $f (- 5)$ and $f (- 9)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(- 5)}^{3} + 5 {(- 5)}^{2} - 2) - ({(- 9)}^{3} + 5 {(- 9)}^{2} - 2)}{(- 5) - (- 9)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 5$ to the power of $3$ .

$\frac{- 125 + 5 {(- 5)}^{2} - 2 - ({(- 9)}^{3} + 5 {(- 9)}^{2} - 2)}{- 5 - (- 9)}$

Step 3.1.2

Raise $- 5$ to the power of $2$ .

$\frac{- 125 + 5 \cdot 25 - 2 - ({(- 9)}^{3} + 5 {(- 9)}^{2} - 2)}{- 5 - (- 9)}$

Step 3.1.3

Multiply $5$ by $25$ .

$\frac{- 125 + 125 - 2 - ({(- 9)}^{3} + 5 {(- 9)}^{2} - 2)}{- 5 - (- 9)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

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

$\frac{- 125 + 125 - 2 - (- 729 + 5 {(- 9)}^{2} - 2)}{- 5 - (- 9)}$

Step 3.1.4.2

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

$\frac{- 125 + 125 - 2 - (- 729 + 5 \cdot 81 - 2)}{- 5 - (- 9)}$

Step 3.1.4.3

Multiply $5$ by $81$ .

$\frac{- 125 + 125 - 2 - (- 729 + 405 - 2)}{- 5 - (- 9)}$

Step 3.1.5

Add $- 729$ and $405$ .

$\frac{- 125 + 125 - 2 - (- 324 - 2)}{- 5 - (- 9)}$

Step 3.1.6

Subtract $2$ from $- 324$ .

$\frac{- 125 + 125 - 2 - - 326}{- 5 - (- 9)}$

Step 3.1.7

Multiply $- 1$ by $- 326$ .

$\frac{- 125 + 125 - 2 + 326}{- 5 - (- 9)}$

Step 3.1.8

Add $- 125$ and $125$ .

$\frac{0 - 2 + 326}{- 5 - (- 9)}$

Step 3.1.9

Subtract $2$ from $0$ .

$\frac{- 2 + 326}{- 5 - (- 9)}$

Step 3.1.10

Add $- 2$ and $326$ .

$\frac{324}{- 5 - (- 9)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 9$ .

$\frac{324}{- 5 + 9}$

Step 3.2.2

Add $- 5$ and $9$ .

$\frac{324}{4}$

Step 3.3

Divide $324$ by $4$ .

$81$