Calculus Examples

Find the Average Value of the Derivative y=25-x^2 , [-5,5]
,
Step 1
Write as a function.
Step 2
Find the derivative of .
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Step 2.1
Find the first derivative.
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Step 2.1.1
Differentiate.
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Step 2.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.2
Evaluate .
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Step 2.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.2.2
Differentiate using the Power Rule which states that is where .
Step 2.1.2.3
Multiply by .
Step 2.1.3
Subtract from .
Step 2.2
The first derivative of with respect to is .
Step 3
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 4
is continuous on .
is continuous
Step 5
The average value of function over the interval is defined as .
Step 6
Substitute the actual values into the formula for the average value of a function.
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
Simplify the answer.
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Step 9.1
Combine and .
Step 9.2
Substitute and simplify.
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Step 9.2.1
Evaluate at and at .
Step 9.2.2
Simplify.
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Step 9.2.2.1
Raise to the power of .
Step 9.2.2.2
Raise to the power of .
Step 9.2.2.3
Combine the numerators over the common denominator.
Step 9.2.2.4
Subtract from .
Step 9.2.2.5
Cancel the common factor of and .
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Step 9.2.2.5.1
Factor out of .
Step 9.2.2.5.2
Cancel the common factors.
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Step 9.2.2.5.2.1
Factor out of .
Step 9.2.2.5.2.2
Cancel the common factor.
Step 9.2.2.5.2.3
Rewrite the expression.
Step 9.2.2.5.2.4
Divide by .
Step 9.2.2.6
Multiply by .
Step 10
Add and .
Step 11
Multiply by .
Step 12