Calculus Examples

$f (x) = 2$ , $[1, 4]$

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 $[1, 4]$ .

$f (x)$ 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.

$A (x) = \frac{1}{4 - 1} (\int_{1}^{4} 2 d x)$

Step 5

Apply the constant rule.

$A (x) = \frac{1}{4 - 1} (2 x]_{1}^{4})$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $2 x$ at $4$ and at $1$ .

$A (x) = \frac{1}{4 - 1} ((2 \cdot 4) - 2 \cdot 1)$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $2$ by $4$ .

$A (x) = \frac{1}{4 - 1} (8 - 2 \cdot 1)$

Step 6.2.2

Multiply $- 2$ by $1$ .

$A (x) = \frac{1}{4 - 1} (8 - 2)$

Step 6.2.3

Subtract $2$ from $8$ .

$A (x) = \frac{1}{4 - 1} (6)$

Step 7

Subtract $1$ from $4$ .

$A (x) = \frac{1}{3} \cdot 6$

Step 8

Cancel the common factor of $3$ .

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Step 8.1

Factor $3$ out of $6$ .

$A (x) = \frac{1}{3} \cdot (3 (2))$

Step 8.2

Cancel the common factor.

$A (x) = \frac{1}{3} \cdot (3 \cdot 2)$

Step 8.3

Rewrite the expression.

$A (x) = 2$

Step 9