Calculus Examples

Find the Absolute Max and Min over the Interval
,
Find the first derivative.
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Differentiate.
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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 .
Cancel the common factor of .
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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 .
Add to both sides of the equation.
Take the root of both sides of the 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 .
Simplify the right side.
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Simplify each term.
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Raising to any positive power yields .
Raising to any positive power yields .
Multiply by .
Add and .
Evaluate the function at .
Simplify the right side.
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Simplify each term.
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Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply by .
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 .
Multiply by .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Raise to the power of .
Apply the product rule to .
Raise to the power of .
Multiply by .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Multiply 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 .
Raise to the power of .
Multiply 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 .
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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