Precalculus Examples

$f (x) = 3 x^{2} - x^{3}$ , $(- 1, 4)$

Step 1

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

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

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.

$\frac{f (4) - f (- 1)}{(4) - (- 1)}$

Step 2.2

Substitute the equation $y = 3 x^{2} - x^{3}$ for $f (4)$ and $f (- 1)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $4$ to the power of $2$ .

$\frac{3 \cdot 16 - 4^{3} - (3 {(- 1)}^{2} - {(- 1)}^{3})}{4 - (- 1)}$

Step 3.1.2

Multiply $3$ by $16$ .

$\frac{48 - 4^{3} - (3 {(- 1)}^{2} - {(- 1)}^{3})}{4 - (- 1)}$

Step 3.1.3

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

$\frac{48 - 1 \cdot 64 - (3 {(- 1)}^{2} - {(- 1)}^{3})}{4 - (- 1)}$

Step 3.1.4

Multiply $- 1$ by $64$ .

$\frac{48 - 64 - (3 {(- 1)}^{2} - {(- 1)}^{3})}{4 - (- 1)}$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

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

$\frac{48 - 64 - (3 \cdot 1 - {(- 1)}^{3})}{4 - (- 1)}$

Step 3.1.5.2

Multiply $3$ by $1$ .

$\frac{48 - 64 - (3 - {(- 1)}^{3})}{4 - (- 1)}$

Step 3.1.5.3

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 3.1.5.3.1

Multiply $- 1$ by ${(- 1)}^{3}$ .

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Step 3.1.5.3.1.1

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

$\frac{48 - 64 - (3 + {(- 1)}^{1} {(- 1)}^{3})}{4 - (- 1)}$

Step 3.1.5.3.1.2

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

$\frac{48 - 64 - (3 + {(- 1)}^{1 + 3})}{4 - (- 1)}$

Step 3.1.5.3.2

Add $1$ and $3$ .

$\frac{48 - 64 - (3 + {(- 1)}^{4})}{4 - (- 1)}$

Step 3.1.5.4

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

$\frac{48 - 64 - (3 + 1)}{4 - (- 1)}$

Step 3.1.6

Add $3$ and $1$ .

$\frac{48 - 64 - 1 \cdot 4}{4 - (- 1)}$

Step 3.1.7

Multiply $- 1$ by $4$ .

$\frac{48 - 64 - 4}{4 - (- 1)}$

Step 3.1.8

Subtract $64$ from $48$ .

$\frac{- 16 - 4}{4 - (- 1)}$

Step 3.1.9

Subtract $4$ from $- 16$ .

$\frac{- 20}{4 - (- 1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- 20}{4 + 1}$

Step 3.2.2

Add $4$ and $1$ .

$\frac{- 20}{5}$

Step 3.3

Divide $- 20$ by $5$ .

$- 4$