Precalculus Examples

$N (t) = - 0.023 {(t - 33.12)}^{2} + 131$ , $0 \leq t \leq 14$

Step 1

Write $N (t) = - 0.023 {(t - 33.12)}^{2} + 131$ as an equation.

$y = - 0.023 {(t - 33.12)}^{2} + 131$

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 $t$ values of the two points.

$\frac{f (14) - f (0)}{(14) - (0)}$

Step 2.2

Substitute the equation $y = - 0.023 {(t - 33.12)}^{2} + 131$ for $f (14)$ and $f (0)$ , replacing $t$ in the function with the corresponding $t$ value.

$\frac{(- 0.023 {((14) - 33.12)}^{2} + 131) - (- 0.023 {((0) - 33.12)}^{2} + 131)}{(14) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Subtract $33.12$ from $14$ .

$\frac{- 0.023 {(- 19.12)}^{2} + 131 - (- 0.023 {(0 - 33.12)}^{2} + 131)}{14 - (0)}$

Step 3.1.2

Raise $- 19.12$ to the power of $2$ .

$\frac{- 0.023 \cdot 365.5744 + 131 - (- 0.023 {(0 - 33.12)}^{2} + 131)}{14 - (0)}$

Step 3.1.3

Multiply $- 0.023$ by $365.5744$ .

$\frac{- 8.4082112 + 131 - (- 0.023 {(0 - 33.12)}^{2} + 131)}{14 - (0)}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Subtract $33.12$ from $0$ .

$\frac{- 8.4082112 + 131 - (- 0.023 {(- 33.12)}^{2} + 131)}{14 - (0)}$

Step 3.1.4.2

Raise $- 33.12$ to the power of $2$ .

$\frac{- 8.4082112 + 131 - (- 0.023 \cdot 1096.9344 + 131)}{14 - (0)}$

Step 3.1.4.3

Multiply $- 0.023$ by $1096.9344$ .

$\frac{- 8.4082112 + 131 - (- 25.2294912 + 131)}{14 - (0)}$

Step 3.1.5

Add $- 25.2294912$ and $131$ .

$\frac{- 8.4082112 + 131 - 1 \cdot 105.7705088}{14 - (0)}$

Step 3.1.6

Multiply $- 1$ by $105.7705088$ .

$\frac{- 8.4082112 + 131 - 105.7705088}{14 - (0)}$

Step 3.1.7

Add $- 8.4082112$ and $131$ .

$\frac{122.5917888 - 105.7705088}{14 - (0)}$

Step 3.1.8

Subtract $105.7705088$ from $122.5917888$ .

$\frac{16.82128}{14 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{16.82128}{14 + 0}$

Step 3.2.2

Add $14$ and $0$ .

$\frac{16.82128}{14}$

Step 3.3

Divide $16.82128$ by $14$ .

$1.20152$