Calculus Examples

Find the Average Value of the Derivative
,
Step 1
Find the derivative of .
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Find the first derivative.
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Differentiate.
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By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
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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 .
Evaluate .
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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.
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Since is constant with respect to , the derivative of with respect to is .
Add and .
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
Combine and .
Step 13
Apply the constant rule.
Step 14
Substitute and simplify.
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Evaluate at and at .
Evaluate at and at .
Evaluate at and at .
Simplify.
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Raise to the power of .
Raising to any positive power yields .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Add and .
Combine and .
Multiply by .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raising to any positive power yields .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Add and .
Multiply by .
Subtract from .
Multiply by .
Multiply by .
Add and .
Add and .
Step 15
Simplify the denominator.
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Multiply by .
Add and .
Step 16
Cancel the common factor of .
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Step 17
Enter YOUR Problem
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