Calculus Examples

Find the Absolute Max and Min over the Interval
,
Find the first derivative.
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By the Sum Rule, the derivative of with respect to is .
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 .
Since is constant with respect to , the derivative of with respect to is .
Add and .
Set the first derivative equal to zero.
Divide each term by and simplify.
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Divide each term in by .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by .
Divide by .
Use the endpoints and all critical points on the interval to test for any absolute extrema over the given interval.
Evaluate the function at .
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Simplify each term.
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Remove parentheses around .
One to any power is one.
Multiply by .
Subtract from .
Evaluate the function at .
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Simplify each term.
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Multiply by by adding the exponents.
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Combine and
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Raise to the power of .
Use the power rule to combine exponents.
Add and .
Raise to the power of .
Subtract from .
Compare the values found for each value of in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest value and the minimum will occur at the lowest value.
Absolute Maximum:
Absolute Minimum:
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