Calculus Examples

$f (x) = 2 \sqrt{x}$ , $[0, 25]$

Step 1

To find the average value of a function, the function should be continuous on the closed interval $[a, b]$ . To find whether $f (x)$ is continuous on $[0, 25]$ or not, find the domain of $f (x) = 2 \sqrt{x}$ .

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Step 1.1

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 1.2

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2

$f (x)$ is not continuous on $[0, 25]$ because $0$ is not in the domain of $f (x)$ . The function should be continuous to find the average value

$f (x)$ is not continuous

Step 3

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$