Precalculus Examples

$s (t) = \sqrt{2 t}$ , $[2, 8]$

Step 1

Write $s (t) = \sqrt{2 t}$ as an equation.

$y = \sqrt{2 t}$

Step 2

Substitute using the average rate of change formula.

Tap for more steps...

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 $t$ values of the two points.

$\frac{f (8) - f (2)}{(8) - (2)}$

Step 2.2

Substitute the equation $y = \sqrt{2 t}$ for $f (8)$ and $f (2)$ , replacing $t$ in the function with the corresponding $t$ value.

$\frac{(\sqrt{2 (8)}) - (\sqrt{2 (2)})}{(8) - (2)}$

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

Multiply $2$ by $8$ .

$\frac{\sqrt{16} - \sqrt{2 (2)}}{8 - (2)}$

Step 3.1.2

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2}} - \sqrt{2 (2)}}{8 - (2)}$

Step 3.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4 - \sqrt{2 (2)}}{8 - (2)}$

Step 3.1.4

Multiply $2$ by $2$ .

$\frac{4 - \sqrt{4}}{8 - (2)}$

Step 3.1.5

Rewrite $4$ as $2^{2}$ .

$\frac{4 - \sqrt{2^{2}}}{8 - (2)}$

Step 3.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4 - 1 \cdot 2}{8 - (2)}$

Step 3.1.7

Multiply $- 1$ by $2$ .

$\frac{4 - 2}{8 - (2)}$

Step 3.1.8

Subtract $2$ from $4$ .

$\frac{2}{8 - (2)}$

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

Multiply $- 1$ by $2$ .

$\frac{2}{8 - 2}$

Step 3.2.2

Subtract $2$ from $8$ .

$\frac{2}{6}$

Step 3.3

Cancel the common factor of $2$ and $6$ .

Tap for more steps...

Step 3.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6}$

Step 3.3.2

Cancel the common factors.

Tap for more steps...

Step 3.3.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3}$

Step 3.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3}$

Step 3.3.2.3

Rewrite the expression.

$\frac{1}{3}$