Precalculus Examples

$x^{2} \sqrt{x^{3} + 1}$ , $[0, 2]$

Step 1

Write $x^{2} \sqrt{x^{3} + 1}$ as an equation.

$y = x^{2} \sqrt{x^{3} + 1}$

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 $x$ values of the two points.

$\frac{f (2) - f (0)}{(2) - (0)}$

Step 2.2

Substitute the equation $y = x^{2} \sqrt{x^{3} + 1}$ for $f (2)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{({(2)}^{2} \sqrt{{(2)}^{3} + 1}) - ({(0)}^{2} \sqrt{{(0)}^{3} + 1})}{(2) - (0)}$

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Simplify the numerator.

Tap for more steps...

Step 3.1.1

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

$\frac{4 \sqrt{2^{3} + 1} - 0^{2} \sqrt{0^{3} + 1}}{2 - (0)}$

Step 3.1.2

Raise $2$ to the power of $3$ .

$\frac{4 \sqrt{8 + 1} - 0^{2} \sqrt{0^{3} + 1}}{2 - (0)}$

Step 3.1.3

Add $8$ and $1$ .

$\frac{4 \sqrt{9} - 0^{2} \sqrt{0^{3} + 1}}{2 - (0)}$

Step 3.1.4

Rewrite $9$ as $3^{2}$ .

$\frac{4 \sqrt{3^{2}} - 0^{2} \sqrt{0^{3} + 1}}{2 - (0)}$

Step 3.1.5

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

$\frac{4 \cdot 3 - 0^{2} \sqrt{0^{3} + 1}}{2 - (0)}$

Step 3.1.6

Multiply $4$ by $3$ .

$\frac{12 - 0^{2} \sqrt{0^{3} + 1}}{2 - (0)}$

Step 3.1.7

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

$\frac{12 - 0 \sqrt{0^{3} + 1}}{2 - (0)}$

Step 3.1.8

Multiply $- 1$ by $0$ .

$\frac{12 + 0 \sqrt{0^{3} + 1}}{2 - (0)}$

Step 3.1.9

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

$\frac{12 + 0 \sqrt{0 + 1}}{2 - (0)}$

Step 3.1.10

Add $0$ and $1$ .

$\frac{12 + 0 \sqrt{1}}{2 - (0)}$

Step 3.1.11

Any root of $1$ is $1$ .

$\frac{12 + 0 \cdot 1}{2 - (0)}$

Step 3.1.12

Multiply $0$ by $1$ .

$\frac{12 + 0}{2 - (0)}$

Step 3.1.13

Add $12$ and $0$ .

$\frac{12}{2 - (0)}$

Step 3.2

Simplify the denominator.

Tap for more steps...

Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{12}{2 + 0}$

Step 3.2.2

Add $2$ and $0$ .

$\frac{12}{2}$

Step 3.3

Divide $12$ by $2$ .

$6$