# Calculus Examples

,

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:

Set-Builder Notation:

is continuous on .

The function is continuous.

The function is continuous.

Find the derivative.

Find the first derivative.

By the Sum Rule, the derivative of with respect to is .

Evaluate .

Since is constant with respect to , the derivative of with respect to is .

Differentiate using the Power Rule which states that is where .

Multiply by .

Differentiate using the Constant Rule.

Since is constant with respect to , the derivative of with respect to is .

Add and .

The first derivative of with respect to is .

Find if the derivative is continuous on .

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:

Set-Builder Notation:

is continuous on .

The function is continuous.

The function is continuous.

The function is differentiable on because the derivative is continuous on .

The function is differentiable.

The function is differentiable.

For arc length to be guaranteed, the function and its derivative must both be continuous on the closed interval .

The function and its derivative are continuous on the closed interval .

By the Sum Rule, the derivative of with respect to is .

Evaluate .

Since is constant with respect to , the derivative of with respect to is .

Differentiate using the Power Rule which states that is where .

Multiply by .

Differentiate using the Constant Rule.

Since is constant with respect to , the derivative of with respect to is .

Add and .

To find the arc length of a function, use the formula .

Since is constant with respect to , move out of the integral.

Substitute and simplify.

Evaluate at and at .

Simplify.

Move to the left of .

Multiply by .

Multiply by .

Add and .

The result can be shown in multiple forms.

Exact Form:

Decimal Form: