Precalculus Examples

$f (t) = e^{\sin (t)} \cos (t)$ , $[0, \frac{π}{2}]$

Step 1

Write $f (t) = e^{\sin (t)} \cos (t)$ as an equation.

$y = e^{\sin (t)} \cos (t)$

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.

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

Step 2.2

Substitute the equation $y = e^{\sin (t)} \cos (t)$ for $f (\frac{π}{2})$ and $f (0)$ , replacing $t$ in the function with the corresponding $t$ value.

$\frac{(e^{\sin (\frac{π}{2})} \cos (\frac{π}{2})) - (e^{\sin (0)} \cos (0))}{(\frac{π}{2}) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 3.1.1

Multiply $\frac{e^{\sin (\frac{π}{2})} \cos (\frac{π}{2}) - e^{\sin (0)} \cos (0)}{\frac{π}{2} - (0)}$ by $\frac{2}{2}$ .

$\frac{2}{2} \cdot \frac{e^{\sin (\frac{π}{2})} \cos (\frac{π}{2}) - e^{\sin (0)} \cos (0)}{\frac{π}{2} - (0)}$

Step 3.1.2

Combine.

$\frac{2 (e^{\sin (\frac{π}{2})} \cos (\frac{π}{2}) - e^{\sin (0)} \cos (0))}{2 (\frac{π}{2} - (0))}$

Step 3.2

Apply the distributive property.

$\frac{2 (e^{\sin (\frac{π}{2})} \cos (\frac{π}{2})) + 2 (- e^{\sin (0)} \cos (0))}{2 \frac{π}{2} + 2 (- (0))}$

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

Cancel the common factor.

$\frac{2 (e^{\sin (\frac{π}{2})} \cos (\frac{π}{2})) + 2 (- e^{\sin (0)} \cos (0))}{2 \frac{π}{2} + 2 (- (0))}$

Step 3.3.2

Rewrite the expression.

$\frac{2 (e^{\sin (\frac{π}{2})} \cos (\frac{π}{2})) + 2 (- e^{\sin (0)} \cos (0))}{π + 2 (- (0))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{2 e^{1} \cos (\frac{π}{2}) + 2 (- e^{\sin (0)} \cos (0))}{π + 2 (- (0))}$

Step 3.4.2

Simplify.

$\frac{2 e \cos (\frac{π}{2}) + 2 (- e^{\sin (0)} \cos (0))}{π + 2 (- (0))}$

Step 3.4.3

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{2 e \cdot 0 + 2 (- e^{\sin (0)} \cos (0))}{π + 2 (- (0))}$

Step 3.4.4

Multiply $2 e \cdot 0$ .

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Step 3.4.4.1

Multiply $0$ by $2$ .

$\frac{0 e + 2 (- e^{\sin (0)} \cos (0))}{π + 2 (- (0))}$

Step 3.4.4.2

Multiply $0$ by $e$ .

$\frac{0 + 2 (- e^{\sin (0)} \cos (0))}{π + 2 (- (0))}$

Step 3.4.5

The exact value of $\sin (0)$ is $0$ .

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

Step 3.4.6

Anything raised to $0$ is $1$ .

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

Step 3.4.7

Multiply $- 1$ by $1$ .

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

Step 3.4.8

The exact value of $\cos (0)$ is $1$ .

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

Step 3.4.9

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

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Step 3.4.9.1

Multiply $- 1$ by $1$ .

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

Step 3.4.9.2

Multiply $2$ by $- 1$ .

$\frac{0 - 2}{π + 2 (- (0))}$

Step 3.4.10

Subtract $2$ from $0$ .

$\frac{- 2}{π + 2 (- (0))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $2 (- (0))$ .

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Step 3.5.1.1

Multiply $- 1$ by $0$ .

$\frac{- 2}{π + 2 \cdot 0}$

Step 3.5.1.2

Multiply $2$ by $0$ .

$\frac{- 2}{π + 0}$

Step 3.5.2

Add $π$ and $0$ .

$\frac{- 2}{π}$

Step 3.6

Move the negative in front of the fraction.

$- \frac{2}{π}$