# Calculus Examples

,

Step 1

Step 1.1

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:

Set-Builder Notation:

Step 1.2

is continuous on .

The function is continuous.

The function is continuous.

Step 2

Step 2.1

Find the derivative.

Step 2.1.1

Find the first derivative.

Step 2.1.1.1

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

Step 2.1.1.2

Evaluate .

Step 2.1.1.2.1

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

Step 2.1.1.2.2

Differentiate using the Power Rule which states that is where .

Step 2.1.1.2.3

Multiply by .

Step 2.1.1.3

Differentiate using the Constant Rule.

Step 2.1.1.3.1

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

Step 2.1.1.3.2

Add and .

Step 2.1.2

The first derivative of with respect to is .

Step 2.2

Find if the derivative is continuous on .

Step 2.2.1

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:

Set-Builder Notation:

Step 2.2.2

is continuous on .

The function is continuous.

The function is continuous.

Step 2.3

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

The function is differentiable.

The function is differentiable.

Step 3

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 .

Step 4

Step 4.1

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

Step 4.2

Evaluate .

Step 4.2.1

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

Step 4.2.2

Differentiate using the Power Rule which states that is where .

Step 4.2.3

Multiply by .

Step 4.3

Differentiate using the Constant Rule.

Step 4.3.1

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

Step 4.3.2

Add and .

Step 5

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

Step 6

Step 6.1

Apply the constant rule.

Step 6.2

Substitute and simplify.

Step 6.2.1

Evaluate at and at .

Step 6.2.2

Simplify.

Step 6.2.2.1

Move to the left of .

Step 6.2.2.2

Multiply by .

Step 6.2.2.3

Subtract from .

Step 7

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Step 8