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 .
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 .
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 .
Factor out of .
Factor out of .
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 .
Set equal to and solve for .
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Set the factor equal to .
Move to the right side of the equation by subtracting from both sides of the equation.
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
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First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
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 .
Remove parentheses around .
Raising to any positive power yields .
Multiply by .
Add and .
Evaluate the function at .
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Simplify each term.
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Remove parentheses around .
Rewrite as .
Raise to the power of .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Remove parentheses around .
Rewrite as .
Multiply by .
Subtract from .
Evaluate the function at .
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Simplify each term.
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Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
Raise to the power of .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
Multiply by .
Subtract from .
Evaluate the function at .
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Simplify each term.
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Raise to the power of .
Raise to the power of .
Multiply by .
Subtract from .
Evaluate the function at .
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Simplify each term.
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Remove parentheses around .
Raise to the power of .
Remove parentheses around .
Raise to the power of .
Multiply by .
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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