Precalculus Examples

$f (x) = x {(x - 4)}^{2}$ ; $[0, 4]$

Step 1

Write $f (x) = x {(x - 4)}^{2}$ as an equation.

$y = x {(x - 4)}^{2}$

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.

$\frac{f (4) - f (0)}{(4) - (0)}$

Step 2.2

Substitute the equation $y = x {(x - 4)}^{2}$ for $f (4)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{((4) {((4) - 4)}^{2}) - ((0) {((0) - 4)}^{2})}{(4) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Subtract $4$ from $4$ .

$\frac{4 \cdot 0^{2} - 0 {(0 - 4)}^{2}}{4 - (0)}$

Step 3.1.2

Raising $0$ to any positive power yields $0$ .

$\frac{4 \cdot 0 - 0 {(0 - 4)}^{2}}{4 - (0)}$

Step 3.1.3

Multiply $4$ by $0$ .

$\frac{0 - 0 {(0 - 4)}^{2}}{4 - (0)}$

Step 3.1.4

Multiply $- 1$ by $0$ .

$\frac{0 + 0 {(0 - 4)}^{2}}{4 - (0)}$

Step 3.1.5

Subtract $4$ from $0$ .

$\frac{0 + 0 {(- 4)}^{2}}{4 - (0)}$

Step 3.1.6

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

$\frac{0 + 0 \cdot 16}{4 - (0)}$

Step 3.1.7

Multiply $0$ by $16$ .

$\frac{0 + 0}{4 - (0)}$

Step 3.1.8

Add $0$ and $0$ .

$\frac{0}{4 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{0}{4 + 0}$

Step 3.2.2

Add $4$ and $0$ .

$\frac{0}{4}$

Step 3.3

Divide $0$ by $4$ .

$0$