Precalculus Examples

$f (x) = x - 2 \sqrt{x}$ on the interval $[1, 9]$

Step 1

Write $f (x) = x - 2 \sqrt{x}$ as an equation.

$y = x - 2 \sqrt{x}$

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.

$\frac{f (9) - f (1)}{(9) - (1)}$

Step 2.2

Substitute the equation $y = x - 2 \sqrt{x}$ for $f (9)$ and $f (1)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{((9) - 2 \sqrt{9}) - ((1) - 2 \sqrt{1})}{(9) - (1)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

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

Step 3.1.2

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

$\frac{9 - 2 \cdot 3 - (1 - 2 \sqrt{1})}{9 - (1)}$

Step 3.1.3

Multiply $- 2$ by $3$ .

$\frac{9 - 6 - (1 - 2 \sqrt{1})}{9 - (1)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Any root of $1$ is $1$ .

$\frac{9 - 6 - (1 - 2 \cdot 1)}{9 - (1)}$

Step 3.1.4.2

Multiply $- 2$ by $1$ .

$\frac{9 - 6 - (1 - 2)}{9 - (1)}$

Step 3.1.5

Subtract $2$ from $1$ .

$\frac{9 - 6 - - 1}{9 - (1)}$

Step 3.1.6

Multiply $- 1$ by $- 1$ .

$\frac{9 - 6 + 1}{9 - (1)}$

Step 3.1.7

Subtract $6$ from $9$ .

$\frac{3 + 1}{9 - (1)}$

Step 3.1.8

Add $3$ and $1$ .

$\frac{4}{9 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

$\frac{4}{9 - 1}$

Step 3.2.2

Subtract $1$ from $9$ .

$\frac{4}{8}$

Step 3.3

Cancel the common factor of $4$ and $8$ .

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Step 3.3.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{8}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 1}{4 \cdot 2}$

Step 3.3.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 2}$

Step 3.3.2.3

Rewrite the expression.

$\frac{1}{2}$