Calculus Examples

$f (x) = \sqrt{x^{2} + 49}$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the radicand in $\sqrt{x^{2} + 49}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} + 49 \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $49$ from both sides of the inequality.

$x^{2} \geq - 49$

Step 1.2.2

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

All real numbers

Step 1.3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

Since the domain is all real numbers, $\sqrt{x^{2} + 49}$ is continuous over all real numbers.

Continuous

Step 3

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