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 to get .
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 .
Set the first derivative equal to zero.
Solve to find the critical points.
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Factor out of .
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Factor out of .
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Factor out of .
Move .
Multiply by to get .
Reorder and .
Factor out of .
Set equal to and solve for .
Set equal to and solve for .
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Set the factor equal to .
Since does not contain the variable to solve for, move it to the right side of the equation by adding to both sides.
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
Divide by to get .
The solution is the result of and .
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 .
Raising to any positive power yields .
Multiply by to get .
Remove parentheses around .
Raising to any positive power yields .
Multiply by to get .
Simplify by adding zeros.
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Add and to get .
Subtract from to get .
Evaluate the function at .
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Simplify each term.
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Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Simplify by subtracting numbers.
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Subtract from to get .
Subtract from to get .
Evaluate the function at .
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Simplify each term.
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Raise to the power of to get .
Multiply by to get .
Raise to the power of to get .
Multiply by to get .
Simplify by subtracting numbers.
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Subtract from to get .
Subtract from to get .
Evaluate the function at .
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Simplify each term.
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Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Simplify by subtracting numbers.
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Subtract from to get .
Subtract from to get .
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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