Precalculus Examples

$y = \sqrt{x}$ , $[1, 4]$

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.

$\frac{f (4) - f (1)}{(4) - (1)}$

Step 1.2

Substitute the equation $y = \sqrt{x}$ for $f (4)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sqrt{4}) - (\sqrt{1})}{(4) - (1)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

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

Step 2.1.2

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

$\frac{2 - \sqrt{1}}{4 - (1)}$

Step 2.1.3

Any root of $1$ is $1$ .

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

Step 2.1.4

Multiply $- 1$ by $1$ .

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

Step 2.1.5

Subtract $1$ from $2$ .

$\frac{1}{4 - (1)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply $- 1$ by $1$ .

$\frac{1}{4 - 1}$

Step 2.2.2

Subtract $1$ from $4$ .

$\frac{1}{3}$