Calculus Examples

$f (x) = - \cos (x)$ ; $[- \frac{π}{2}, \frac{π}{2}]$

Step 1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

$f (x)$ is continuous on $[- \frac{π}{2}, \frac{π}{2}]$ .

$f (x)$ is continuous

Step 3

The average value of function $f$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 4

Substitute the actual values into the formula for the average value of a function.

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (\int_{- \frac{π}{2}}^{\frac{π}{2}} - \cos (x) d x)$

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (x) d x)$

Step 6

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- (\sin (x)]_{- \frac{π}{2}}^{\frac{π}{2}}))$

Step 7

Simplify the answer.

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Step 7.1

Evaluate $\sin (x)$ at $\frac{π}{2}$ and at $- \frac{π}{2}$ .

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- (\sin (\frac{π}{2}) - \sin (- \frac{π}{2})))$

Step 7.2

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

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- (1 - \sin (- \frac{π}{2})))$

Step 7.3

Simplify.

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Step 7.3.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- (1 - \sin (\frac{3 π}{2})))$

Step 7.3.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- (1 + \sin (\frac{π}{2})))$

Step 7.3.3

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

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- (1 - (- 1 \cdot 1)))$

Step 7.3.4

Multiply $- 1$ by $1$ .

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- (1 + 1))$

Step 7.3.5

Multiply $- 1$ by $- 1$ .

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- (1 + 1))$

Step 7.3.6

Add $1$ and $1$ .

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- 1 \cdot 2)$

Step 7.3.7

Multiply $- 1$ by $2$ .

$A (x) = \frac{1}{\frac{π}{2} + \frac{π}{2}} (- 2)$

Step 8

Simplify the denominator.

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Step 8.1

Combine the numerators over the common denominator.

$A (x) = \frac{1}{\frac{π + π}{2}} \cdot - 2$

Step 8.2

Rewrite $\frac{π + π}{2}$ in a factored form.

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Step 8.2.1

Add $π$ and $π$ .

$A (x) = \frac{1}{\frac{2 π}{2}} \cdot - 2$

Step 8.2.2

Reduce the expression by cancelling the common factors.

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Step 8.2.2.1

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

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Step 8.2.2.1.1

Cancel the common factor.

$A (x) = \frac{1}{\frac{2 π}{2}} \cdot - 2$

Step 8.2.2.1.2

Rewrite the expression.

$A (x) = \frac{1}{\frac{π}{1}} \cdot - 2$

Step 8.2.2.2

Divide $π$ by $1$ .

$A (x) = \frac{1}{π} \cdot - 2$

Step 9

Combine fractions.

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Step 9.1

Combine $\frac{1}{π}$ and $- 2$ .

$A (x) = \frac{- 2}{π}$

Step 9.2

Move the negative in front of the fraction.

$A (x) = - \frac{2}{π}$

Step 10