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 .
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.
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 .
Cancel the common factor of .
Divide by .
Set equal to and solve for .
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Set the factor equal to .
Add to both sides of the equation.
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 .
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 .
Simplify the right side.
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Simplify each term.
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Raising to any positive power yields .
Multiply by .
Raising to any positive power yields .
Multiply by .
Simplify by adding zeros.
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Add and .
Subtract from .
Evaluate the function at .
Simplify the right side.
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Simplify each term.
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Apply the product rule to .
Raise to the power of .
Raise to the power of .
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 .
Multiply by .
Divide by .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Multiply .
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Combine and .
Multiply by .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Multiply by .
Combine the numerators over the common denominator.
Multiply by .
Subtract from .
Divide by .
Subtract from .
Evaluate the function at .
Simplify the right side.
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Simplify each term.
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Raise to the power of .
Multiply by .
Raise to the power of .
Multiply by .
Simplify by subtracting numbers.
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Subtract from .
Subtract from .
Evaluate the function at .
Simplify the right side.
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Simplify each term.
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Raise to the power of .
Multiply by .
Raise to the power of .
Multiply by .
Simplify by subtracting numbers.
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Subtract from .
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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