Calculus Examples

Find the Average Value of the Derivative
,
Step 1
Find the derivative of .
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
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
Step 12.2.3.16
Multiply by .
Step 12.2.3.17
Subtract from .
Step 13
Simplify the denominator.
Step 13.1
Multiply by .
Step 13.2