Calculus Examples

Find the Absolute Max and Min over the Interval
,
Find the 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 .
Differentiate using the Constant Rule.
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Since is constant with respect to , the derivative of with respect to is .
Add and .
Set the derivative equal to .
Solve for .
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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 .
Factor out of .
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Factor out of .
Factor out of .
Factor out of .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set equal to .
Set equal to and solve for .
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Set equal to .
Solve for .
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Add to both sides of the equation.
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
The final solution is all the values that make true.
Substitute the values of which cause the derivative to be into the original function.
Evaluate.
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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 and subtracting.
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Add and .
Subtract from .
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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Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply 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.
Find the common denominator.
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Write as a fraction with denominator .
Multiply by .
Multiply and .
Write as a fraction with denominator .
Multiply by .
Multiply and .
Combine fractions.
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Combine fractions with similar denominators.
Multiply.
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Multiply by .
Multiply by .
Simplify the numerator.
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Subtract from .
Subtract from .
Divide by .
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:
Since there are no values of where the derivative is undefined, there are no additional critical points.
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 and subtracting.
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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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Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply 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.
Find the common denominator.
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Write as a fraction with denominator .
Multiply by .
Multiply and .
Write as a fraction with denominator .
Multiply by .
Multiply and .
Combine fractions.
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Combine fractions with similar denominators.
Multiply.
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Multiply by .
Multiply by .
Simplify the numerator.
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Subtract from .
Subtract from .
Divide by .
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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