Precalculus Examples

S (t) = t^{2} + 3 t - 6

$S (t) = t^{2} + 3 t - 6$ ,

t = 0

$t = 0$ ,

t = 2

$t = 2$

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

$y$ values of the two points divided by the change in

t

$t$ values of the two points.

\frac{f (2) - f (0)}{(2) - (0)}

$\frac{f (2) - f (0)}{(2) - (0)}$

Step 1.2

Substitute the equation

y = t^{2} + 3 t - 6

$y = t^{2} + 3 t - 6$ for

f (2)

$f (2)$ and

f (0)

$f (0)$ , replacing

t

$t$ in the function with the corresponding

t

$t$ value.

\frac{({(2)}^{2} + 3 (2) - 6) - ({(0)}^{2} + 3 (0) - 6)}{(2) - (0)}

$\frac{({(2)}^{2} + 3 (2) - 6) - ({(0)}^{2} + 3 (0) - 6)}{(2) - (0)}$

\frac{({(2)}^{2} + 3 (2) - 6) - ({(0)}^{2} + 3 (0) - 6)}{(2) - (0)}

$\frac{({(2)}^{2} + 3 (2) - 6) - ({(0)}^{2} + 3 (0) - 6)}{(2) - (0)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Raise

2

$2$ to the power of

2

$2$ .

\frac{4 + 3 (2) - 6 - (0^{2} + 3 (0) - 6)}{2 - (0)}

$\frac{4 + 3 (2) - 6 - (0^{2} + 3 (0) - 6)}{2 - (0)}$

Step 2.1.2

Multiply

3

$3$ by

2

$2$ .

\frac{4 + 6 - 6 - (0^{2} + 3 (0) - 6)}{2 - (0)}

$\frac{4 + 6 - 6 - (0^{2} + 3 (0) - 6)}{2 - (0)}$

Step 2.1.3

Simplify each term.

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Step 2.1.3.1

Raising

0

$0$ to any positive power yields

0

$0$ .

\frac{4 + 6 - 6 - (0 + 3 (0) - 6)}{2 - (0)}

$\frac{4 + 6 - 6 - (0 + 3 (0) - 6)}{2 - (0)}$

Step 2.1.3.2

Multiply

3

$3$ by

0

$0$ .

\frac{4 + 6 - 6 - (0 + 0 - 6)}{2 - (0)}

$\frac{4 + 6 - 6 - (0 + 0 - 6)}{2 - (0)}$

\frac{4 + 6 - 6 - (0 + 0 - 6)}{2 - (0)}

$\frac{4 + 6 - 6 - (0 + 0 - 6)}{2 - (0)}$

Step 2.1.4

Add

0

$0$ and

0

$0$ .

\frac{4 + 6 - 6 - (0 - 6)}{2 - (0)}

$\frac{4 + 6 - 6 - (0 - 6)}{2 - (0)}$

Step 2.1.5

Subtract

6

$6$ from

0

$0$ .

\frac{4 + 6 - 6 - - 6}{2 - (0)}

$\frac{4 + 6 - 6 - - 6}{2 - (0)}$

Step 2.1.6

Multiply

- 1

$- 1$ by

$- 6$ .

$\frac{4 + 6 - 6 + 6}{2 - (0)}$

Step 2.1.7

Add

$4$ and

$6$ .

$\frac{10 - 6 + 6}{2 - (0)}$

Step 2.1.8

Subtract

$6$ from

$10$ .

$\frac{4 + 6}{2 - (0)}$

Step 2.1.9

Add

$4$ and

$6$ .

$\frac{10}{2 - (0)}$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Multiply

$- 1$ by

$0$ .

$\frac{10}{2 + 0}$

Step 2.2.2

Add

$2$ and

$0$ .

$\frac{10}{2}$

Step 2.3

Divide

$10$ by

$2$ .

$5$