# Calculus Examples

,

Step 1

Step 1.1

Find the first derivative.

Step 1.1.1

Differentiate.

Step 1.1.1.1

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

Step 1.1.1.2

Differentiate using the Power Rule which states that is where .

Step 1.1.2

Evaluate .

Step 1.1.2.1

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

Step 1.1.2.2

Differentiate using the Power Rule which states that is where .

Step 1.1.2.3

Multiply by .

Step 1.2

The first derivative of with respect to is .

Step 2

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 3

is continuous on .

is continuous

Step 4

The average value of function over the interval is defined as .

Step 5

Substitute the actual values into the formula for the average value of a function.

Step 6

Split the single integral into multiple integrals.

Step 7

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

Step 8

By the Power Rule, the integral of with respect to is .

Step 9

Combine and .

Step 10

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

Step 11

By the Power Rule, the integral of with respect to is .

Step 12

Step 12.1

Combine and .

Step 12.2

Substitute and simplify.

Step 12.2.1

Evaluate at and at .

Step 12.2.2

Evaluate at and at .

Step 12.2.3

Simplify.

Step 12.2.3.1

Raise to the power of .

Step 12.2.3.2

Cancel the common factor of and .

Step 12.2.3.2.1

Factor out of .

Step 12.2.3.2.2

Cancel the common factors.

Step 12.2.3.2.2.1

Factor out of .

Step 12.2.3.2.2.2

Cancel the common factor.

Step 12.2.3.2.2.3

Rewrite the expression.

Step 12.2.3.3

Raising to any positive power yields .

Step 12.2.3.4

Cancel the common factor of and .

Step 12.2.3.4.1

Factor out of .

Step 12.2.3.4.2

Cancel the common factors.

Step 12.2.3.4.2.1

Factor out of .

Step 12.2.3.4.2.2

Cancel the common factor.

Step 12.2.3.4.2.3

Rewrite the expression.

Step 12.2.3.4.2.4

Divide by .

Step 12.2.3.5

Multiply by .

Step 12.2.3.6

Add and .

Step 12.2.3.7

Combine and .

Step 12.2.3.8

Multiply by .

Step 12.2.3.9

Cancel the common factor of and .

Step 12.2.3.9.1

Factor out of .

Step 12.2.3.9.2

Cancel the common factors.

Step 12.2.3.9.2.1

Factor out of .

Step 12.2.3.9.2.2

Cancel the common factor.

Step 12.2.3.9.2.3

Rewrite the expression.

Step 12.2.3.9.2.4

Divide by .

Step 12.2.3.10

Raise to the power of .

Step 12.2.3.11

Cancel the common factor of and .

Step 12.2.3.11.1

Factor out of .

Step 12.2.3.11.2

Cancel the common factors.

Step 12.2.3.11.2.1

Factor out of .

Step 12.2.3.11.2.2

Cancel the common factor.

Step 12.2.3.11.2.3

Rewrite the expression.

Step 12.2.3.11.2.4

Divide by .

Step 12.2.3.12

Raising to any positive power yields .

Step 12.2.3.13

Cancel the common factor of and .

Step 12.2.3.13.1

Factor out of .

Step 12.2.3.13.2

Cancel the common factors.

Step 12.2.3.13.2.1

Factor out of .

Step 12.2.3.13.2.2

Cancel the common factor.

Step 12.2.3.13.2.3

Rewrite the expression.

Step 12.2.3.13.2.4

Divide by .

Step 12.2.3.14

Multiply by .

Step 12.2.3.15

Add and .

Step 12.2.3.16

Multiply by .

Step 12.2.3.17

Subtract from .

Step 13

Step 13.1

Multiply by .

Step 13.2

Add and .

Step 14

Step 14.1

Factor out of .

Step 14.2

Cancel the common factor.

Step 14.3

Rewrite the expression.

Step 15