Precalculus Examples

$f (x) = \sin (x)$ , $[0, \frac{π}{4}]$

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.

$\frac{f (\frac{π}{4}) - f (0)}{(\frac{π}{4}) - (0)}$

Step 2.2

Substitute the equation $y = \sin (x)$ for $f (\frac{π}{4})$ and $f (0)$ , replacing $x$ in the function with the corresponding $x$ value.

$\frac{(\sin (\frac{π}{4})) - (\sin (0))}{(\frac{π}{4}) - (0)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 3.1.1

Multiply $\frac{\sin (\frac{π}{4}) - \sin (0)}{\frac{π}{4} - (0)}$ by $\frac{4}{4}$ .

$\frac{4}{4} \cdot \frac{\sin (\frac{π}{4}) - \sin (0)}{\frac{π}{4} - (0)}$

Step 3.1.2

Combine.

$\frac{4 (\sin (\frac{π}{4}) - \sin (0))}{4 (\frac{π}{4} - (0))}$

Step 3.2

Apply the distributive property.

$\frac{4 \sin (\frac{π}{4}) + 4 (- \sin (0))}{4 \frac{π}{4} + 4 (- (0))}$

Step 3.3

Cancel the common factor of $4$ .

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Step 3.3.1

Cancel the common factor.

$\frac{4 \sin (\frac{π}{4}) + 4 (- \sin (0))}{4 \frac{π}{4} + 4 (- (0))}$

Step 3.3.2

Rewrite the expression.

$\frac{4 \sin (\frac{π}{4}) + 4 (- \sin (0))}{π + 4 (- (0))}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Factor $4$ out of $4 \sin (\frac{π}{4}) + 4 (- \sin (0))$ .

$\frac{4 (\sin (\frac{π}{4}) - \sin (0))}{π + 4 (- (0))}$

Step 3.4.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{4 (\frac{\sqrt{2}}{2} - \sin (0))}{π + 4 (- (0))}$

Step 3.4.3

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

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

Step 3.4.4

Multiply $- 1$ by $0$ .

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

Step 3.4.5

Add $\frac{\sqrt{2}}{2}$ and $0$ .

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

Step 3.5

Simplify the denominator.

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Step 3.5.1

Multiply $4 (- (0))$ .

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Step 3.5.1.1

Multiply $- 1$ by $0$ .

$\frac{4 \frac{\sqrt{2}}{2}}{π + 4 \cdot 0}$

Step 3.5.1.2

Multiply $4$ by $0$ .

$\frac{4 \frac{\sqrt{2}}{2}}{π + 0}$

Step 3.5.2

Add $π$ and $0$ .

$\frac{4 \frac{\sqrt{2}}{2}}{π}$

Step 3.6

Combine $4$ and $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{4 \sqrt{2}}{2}}{π}$

Step 3.7

Reduce the expression by cancelling the common factors.

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Step 3.7.1

Reduce the expression $\frac{4 \sqrt{2}}{2}$ by cancelling the common factors.

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Step 3.7.1.1

Factor $2$ out of $4 \sqrt{2}$ .

$\frac{\frac{2 (2 \sqrt{2})}{2}}{π}$

Step 3.7.1.2

Factor $2$ out of $2$ .

$\frac{\frac{2 (2 \sqrt{2})}{2 (1)}}{π}$

Step 3.7.1.3

Cancel the common factor.

$\frac{\frac{2 (2 \sqrt{2})}{2 \cdot 1}}{π}$

Step 3.7.1.4

Rewrite the expression.

$\frac{\frac{2 \sqrt{2}}{1}}{π}$

Step 3.7.2

Divide $2 \sqrt{2}$ by $1$ .

$\frac{2 \sqrt{2}}{π}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{2}}{π}$

Decimal Form:

$0.90031631 \dots$