Precalculus Examples

$f (x) = \sin (x)$ , $[0, π]$

Step 1

Write $f (x) = \sin (x)$ as an equation.

$y = \sin (x)$

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 $x$ values of the two points.

$\frac{f (π) - f (0)}{(π) - (0)}$

Step 2.2

Substitute the equation $y = \sin (x)$ for $f (π)$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sin (π)) - (\sin (0))}{(π) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sin (0) - \sin (0)}{π - (0)}$

Step 3.1.2

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

$\frac{0 - \sin (0)}{π - (0)}$

Step 3.1.3

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

$\frac{0 - 0}{π - (0)}$

Step 3.1.4

Multiply $- 1$ by $0$ .

$\frac{0 + 0}{π - (0)}$

Step 3.1.5

Add $0$ and $0$ .

$\frac{0}{π - (0)}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Multiply $- 1$ by $0$ .

$\frac{0}{π + 0}$

Step 3.2.2

Add $π$ and $0$ .

$\frac{0}{π}$

Step 3.3

Divide $0$ by $π$ .

$0$