Calculus Examples

$v (t) = 3 t^{2} - e^{t}$ , $[0, 2]$

Step 1

Write $v (t) = 3 t^{2} - e^{t}$ as an equation.

$y = 3 t^{2} - e^{t}$

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.

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

Step 2.2

Substitute the equation $y = 3 t^{2} - e^{t}$ for $f (2)$ and $f (0)$ , replacing $t$ in the function with the corresponding $t$ value.

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

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 $2$ .

$\frac{3 \cdot 4 - e^{2} - (3 \cdot 0^{2} - e^{0})}{2 - (0)}$

Step 3.1.2

Multiply $3$ by $4$ .

$\frac{12 - e^{2} - (3 \cdot 0^{2} - e^{0})}{2 - (0)}$

Step 3.1.3

Simplify each term.

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Step 3.1.3.1

Raising $0$ to any positive power yields $0$ .

$\frac{12 - e^{2} - (3 \cdot 0 - e^{0})}{2 - (0)}$

Step 3.1.3.2

Multiply $3$ by $0$ .

$\frac{12 - e^{2} - (0 - e^{0})}{2 - (0)}$

Step 3.1.3.3

Anything raised to $0$ is $1$ .

$\frac{12 - e^{2} - (0 - 1 \cdot 1)}{2 - (0)}$

Step 3.1.3.4

Multiply $- 1$ by $1$ .

$\frac{12 - e^{2} - (0 - 1)}{2 - (0)}$

Step 3.1.4

Subtract $1$ from $0$ .

$\frac{12 - e^{2} - - 1}{2 - (0)}$

Step 3.1.5

Multiply $- 1$ by $- 1$ .

$\frac{12 - e^{2} + 1}{2 - (0)}$

Step 3.1.6

Add $12$ and $1$ .

$\frac{- e^{2} + 13}{2 - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{- e^{2} + 13}{2 + 0}$

Step 3.2.2

Add $2$ and $0$ .

$\frac{- e^{2} + 13}{2}$

Step 3.3

Simplify with factoring out.

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Step 3.3.1

Factor $- 1$ out of $- e^{2}$ .

$\frac{- (e^{2}) + 13}{2}$

Step 3.3.2

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (e^{2}) - 1 (- 13)}{2}$

Step 3.3.3

Factor $- 1$ out of $- (e^{2}) - 1 (- 13)$ .

$\frac{- (e^{2} - 13)}{2}$

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Rewrite $- (e^{2} - 13)$ as $- 1 (e^{2} - 13)$ .

$\frac{- 1 (e^{2} - 13)}{2}$

Step 3.3.4.2

Move the negative in front of the fraction.

$- \frac{e^{2} - 13}{2}$