Precalculus Examples

$f (x) = 1 - x^{2}$ over $[1, 2]$

Step 1

Write $f (x) = 1 - x^{2}$ as an equation.

$y = 1 - x^{2}$

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.

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

Step 2.2

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

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

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

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

Step 3.1.2

Multiply $- 1$ by $4$ .

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

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

One to any power is one.

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

Step 3.1.3.2

Multiply $- 1$ by $1$ .

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

Step 3.1.4

Subtract $1$ from $1$ .

$\frac{1 - 4 - 0}{2 - (1)}$

Step 3.1.5

Multiply $- 1$ by $0$ .

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

Step 3.1.6

Add $1 - 4$ and $0$ .

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

Step 3.1.7

Subtract $4$ from $1$ .

$\frac{- 3}{2 - (1)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $1$ .

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

Step 3.2.2

Subtract $1$ from $2$ .

$\frac{- 3}{1}$

Step 3.3

Divide $- 3$ by $1$ .

$- 3$