Precalculus Examples

$y = \sqrt{x^{3} + 1}$ at $(2, 3)$

Step 1

Substitute using the average rate of change formula.

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Step 1.1

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.

$\frac{f (3) - f (2)}{(3) - (2)}$

Step 1.2

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

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

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

$\frac{\sqrt{27 + 1} - \sqrt{2^{3} + 1}}{3 - (2)}$

Step 2.1.2

Add $27$ and $1$ .

$\frac{\sqrt{28} - \sqrt{2^{3} + 1}}{3 - (2)}$

Step 2.1.3

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 2.1.3.1

Factor $4$ out of $28$ .

$\frac{\sqrt{4 (7)} - \sqrt{2^{3} + 1}}{3 - (2)}$

Step 2.1.3.2

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

$\frac{\sqrt{2^{2} \cdot 7} - \sqrt{2^{3} + 1}}{3 - (2)}$

Step 2.1.4

Pull terms out from under the radical.

$\frac{2 \sqrt{7} - \sqrt{2^{3} + 1}}{3 - (2)}$

Step 2.1.5

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

$\frac{2 \sqrt{7} - \sqrt{8 + 1}}{3 - (2)}$

Step 2.1.6

Add $8$ and $1$ .

$\frac{2 \sqrt{7} - \sqrt{9}}{3 - (2)}$

Step 2.1.7

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

$\frac{2 \sqrt{7} - \sqrt{3^{2}}}{3 - (2)}$

Step 2.1.8

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

$\frac{2 \sqrt{7} - 1 \cdot 3}{3 - (2)}$

Step 2.1.9

Multiply $- 1$ by $3$ .

$\frac{2 \sqrt{7} - 3}{3 - (2)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 1$ by $2$ .

$\frac{2 \sqrt{7} - 3}{3 - 2}$

Step 2.2.2

Subtract $2$ from $3$ .

$\frac{2 \sqrt{7} - 3}{1}$

Step 2.3

Divide $2 \sqrt{7} - 3$ by $1$ .

$2 \sqrt{7} - 3$

Step 3

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{7} - 3$

Decimal Form:

$2.29150262 \dots$