Precalculus Examples

$f (x) = 3 x^{2} - x^{3}$ , $(1, 5)$

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.

$\frac{f (5) - f (1)}{(5) - (1)}$

Step 2.2

Substitute the equation $y = 3 x^{2} - x^{3}$ for $f (5)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$\frac{3 \cdot 25 - 5^{3} - (3 \cdot 1^{2} - 1^{3})}{5 - (1)}$

Step 3.1.2

Multiply $3$ by $25$ .

$\frac{75 - 5^{3} - (3 \cdot 1^{2} - 1^{3})}{5 - (1)}$

Step 3.1.3

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

$\frac{75 - 1 \cdot 125 - (3 \cdot 1^{2} - 1^{3})}{5 - (1)}$

Step 3.1.4

Multiply $- 1$ by $125$ .

$\frac{75 - 125 - (3 \cdot 1^{2} - 1^{3})}{5 - (1)}$

Step 3.1.5

Simplify each term.

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Step 3.1.5.1

One to any power is one.

$\frac{75 - 125 - (3 \cdot 1 - 1^{3})}{5 - (1)}$

Step 3.1.5.2

Multiply $3$ by $1$ .

$\frac{75 - 125 - (3 - 1^{3})}{5 - (1)}$

Step 3.1.5.3

One to any power is one.

$\frac{75 - 125 - (3 - 1 \cdot 1)}{5 - (1)}$

Step 3.1.5.4

Multiply $- 1$ by $1$ .

$\frac{75 - 125 - (3 - 1)}{5 - (1)}$

Step 3.1.6

Subtract $1$ from $3$ .

$\frac{75 - 125 - 1 \cdot 2}{5 - (1)}$

Step 3.1.7

Multiply $- 1$ by $2$ .

$\frac{75 - 125 - 2}{5 - (1)}$

Step 3.1.8

Subtract $125$ from $75$ .

$\frac{- 50 - 2}{5 - (1)}$

Step 3.1.9

Subtract $2$ from $- 50$ .

$\frac{- 52}{5 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{- 52}{5 - 1}$

Step 3.2.2

Subtract $1$ from $5$ .

$\frac{- 52}{4}$

Step 3.3

Divide $- 52$ by $4$ .

$- 13$