Precalculus Examples

$s (t) = \frac{22}{t + 1}$ , $[0, 1]$

Step 1

Write $s (t) = \frac{22}{t + 1}$ as an equation.

$y = \frac{22}{t + 1}$

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 $t$ values of the two points.

$\frac{f (1) - f (0)}{(1) - (0)}$

Step 2.2

Substitute the equation $y = \frac{22}{t + 1}$ for $f (1)$ and $f (0)$ , replacing $t$ in the function with the corresponding $t$ value.

$\frac{(\frac{22}{(1) + 1}) - (\frac{22}{(0) + 1})}{(1) - (0)}$

Step 3

Simplify the expression.

Tap for more steps...

Step 3.1

Multiply the numerator and denominator of the fraction by $(1 + 1) (0 + 1)$ .

Tap for more steps...

Step 3.1.1

Multiply $\frac{\frac{22}{1 + 1} - \frac{22}{0 + 1}}{1 - (0)}$ by $\frac{(1 + 1) (0 + 1)}{(1 + 1) (0 + 1)}$ .

$\frac{(1 + 1) (0 + 1)}{(1 + 1) (0 + 1)} \cdot \frac{\frac{22}{1 + 1} - \frac{22}{0 + 1}}{1 - (0)}$

Step 3.1.2

Combine.

$\frac{(1 + 1) (0 + 1) (\frac{22}{1 + 1} - \frac{22}{0 + 1})}{(1 + 1) (0 + 1) (1 - (0))}$

Step 3.2

Apply the distributive property.

$\frac{(1 + 1) (0 + 1) \frac{22}{1 + 1} + (1 + 1) (0 + 1) (- \frac{22}{0 + 1})}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.3

Simplify by cancelling.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $1 + 1$ .

Tap for more steps...

Step 3.3.1.1

Cancel the common factor.

$\frac{(1 + 1) (0 + 1) \frac{22}{1 + 1} + (1 + 1) (0 + 1) (- \frac{22}{0 + 1})}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.3.1.2

Rewrite the expression.

$\frac{(0 + 1) \cdot 22 + (1 + 1) (0 + 1) (- \frac{22}{0 + 1})}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.3.2

Cancel the common factor of $0 + 1$ .

Tap for more steps...

Step 3.3.2.1

Move the leading negative in $- \frac{22}{0 + 1}$ into the numerator.

$\frac{(0 + 1) \cdot 22 + (1 + 1) (0 + 1) \frac{- 22}{0 + 1}}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.3.2.2

Factor $0 + 1$ out of $(1 + 1) (0 + 1)$ .

$\frac{(0 + 1) \cdot 22 + (0 + 1) (1 + 1) \frac{- 22}{0 + 1}}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.3.2.3

Cancel the common factor.

$\frac{(0 + 1) \cdot 22 + (0 + 1) (1 + 1) \frac{- 22}{0 + 1}}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.3.2.4

Rewrite the expression.

$\frac{(0 + 1) \cdot 22 + (1 + 1) \cdot - 22}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.4

Simplify the numerator.

Tap for more steps...

Step 3.4.1

Add $0$ and $1$ .

$\frac{1 \cdot 22 + (1 + 1) \cdot - 22}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.4.2

Multiply $22$ by $1$ .

$\frac{22 + (1 + 1) \cdot - 22}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.4.3

Add $1$ and $1$ .

$\frac{22 + 2 \cdot - 22}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.4.4

Multiply $2$ by $- 22$ .

$\frac{22 - 44}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.4.5

Subtract $44$ from $22$ .

$\frac{- 22}{(1 + 1) (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.5

Simplify the denominator.

Tap for more steps...

Step 3.5.1

Add $1$ and $1$ .

$\frac{- 22}{2 (0 + 1) \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.5.2

Add $0$ and $1$ .

$\frac{- 22}{2 \cdot 1 \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.5.3

Multiply $2 \cdot 1 \cdot 1$ .

Tap for more steps...

Step 3.5.3.1

Multiply $2$ by $1$ .

$\frac{- 22}{2 \cdot 1 + (1 + 1) (0 + 1) (- (0))}$

Step 3.5.3.2

Multiply $2$ by $1$ .

$\frac{- 22}{2 + (1 + 1) (0 + 1) (- (0))}$

Step 3.5.4

Add $1$ and $1$ .

$\frac{- 22}{2 + 2 (0 + 1) (- (0))}$

Step 3.5.5

Add $0$ and $1$ .

$\frac{- 22}{2 + 2 \cdot 1 (- (0))}$

Step 3.5.6

Multiply $2 \cdot 1 (- (0))$ .

Tap for more steps...

Step 3.5.6.1

Multiply $2$ by $1$ .

$\frac{- 22}{2 + 2 (- (0))}$

Step 3.5.6.2

Multiply $- 1$ by $0$ .

$\frac{- 22}{2 + 2 \cdot 0}$

Step 3.5.6.3

Multiply $2$ by $0$ .

$\frac{- 22}{2 + 0}$

Step 3.5.7

Add $2$ and $0$ .

$\frac{- 22}{2}$

Step 3.6

Divide $- 22$ by $2$ .

$- 11$