Calculus Examples

Find the Average Value of the Derivative
,
Finding the derivative of .
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Find the first derivative.
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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 to get .
Since is constant with respect to , the derivative of with respect to is .
Add and to get .
The derivative of with respect to is .
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.
is continuous on .
is continuous
The average value of function over the interval is defined as .
Substitute the actual values into the formula for the average value of a function.
Since integration is linear, the integral of with respect to is .
Since is constant with respect to , the integral of with respect to is .
By the Power Rule, the integral of with respect to is .
Combine fractions.
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Write as a fraction with denominator .
Multiply and to get .
Since is constant with respect to , the integral of with respect to is .
Simplify the answer.
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Evaluate at and at .
Evaluate at and at .
Raise to the power of to get .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by to get .
Raising to any positive power yields .
Reduce the expression by cancelling the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by to get .
Multiply by to get .
Add and to get .
Multiply by to get .
Multiply by to get .
Multiply by to get .
Add and to get .
Add and to get .
Simplify the denominator.
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Multiply by to get .
Add and to get .
Cancel the common factor of .
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Write as a fraction with denominator .
Factor out the greatest common factor .
Cancel the common factor.
Rewrite the expression.
Simplify.
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Multiply and to get .
Divide by to get .
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