Calculus Examples

Find the Absolute Max and Min over the Interval f(t)=4 square root of t-t , [0,16]
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Step 1
Find the critical points.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Find the first derivative.
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Step 1.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2
Evaluate .
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Step 1.1.1.2.1
Use to rewrite as .
Step 1.1.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.3
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.4
To write as a fraction with a common denominator, multiply by .
Step 1.1.1.2.5
Combine and .
Step 1.1.1.2.6
Combine the numerators over the common denominator.
Step 1.1.1.2.7
Simplify the numerator.
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Step 1.1.1.2.7.1
Multiply by .
Step 1.1.1.2.7.2
Subtract from .
Step 1.1.1.2.8
Move the negative in front of the fraction.
Step 1.1.1.2.9
Combine and .
Step 1.1.1.2.10
Combine and .
Step 1.1.1.2.11
Move to the denominator using the negative exponent rule .
Step 1.1.1.2.12
Factor out of .
Step 1.1.1.2.13
Cancel the common factors.
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Step 1.1.1.2.13.1
Factor out of .
Step 1.1.1.2.13.2
Cancel the common factor.
Step 1.1.1.2.13.3
Rewrite the expression.
Step 1.1.1.3
Evaluate .
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Step 1.1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3.3
Multiply by .
Step 1.1.2
The first derivative of with respect to is .
Step 1.2
Set the first derivative equal to then solve the equation .
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Step 1.2.1
Set the first derivative equal to .
Step 1.2.2
Add to both sides of the equation.
Step 1.2.3
Find the LCD of the terms in the equation.
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Step 1.2.3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 1.2.3.2
The LCM of one and any expression is the expression.
Step 1.2.4
Multiply each term in by to eliminate the fractions.
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Step 1.2.4.1
Multiply each term in by .
Step 1.2.4.2
Simplify the left side.
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Step 1.2.4.2.1
Cancel the common factor of .
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Step 1.2.4.2.1.1
Cancel the common factor.
Step 1.2.4.2.1.2
Rewrite the expression.
Step 1.2.4.3
Simplify the right side.
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Step 1.2.4.3.1
Multiply by .
Step 1.2.5
Solve the equation.
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Step 1.2.5.1
Rewrite the equation as .
Step 1.2.5.2
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 1.2.5.3
Simplify the exponent.
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Step 1.2.5.3.1
Simplify the left side.
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Step 1.2.5.3.1.1
Simplify .
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Step 1.2.5.3.1.1.1
Multiply the exponents in .
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Step 1.2.5.3.1.1.1.1
Apply the power rule and multiply exponents, .
Step 1.2.5.3.1.1.1.2
Cancel the common factor of .
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Step 1.2.5.3.1.1.1.2.1
Cancel the common factor.
Step 1.2.5.3.1.1.1.2.2
Rewrite the expression.
Step 1.2.5.3.1.1.2
Simplify.
Step 1.2.5.3.2
Simplify the right side.
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Step 1.2.5.3.2.1
Raise to the power of .
Step 1.3
Find the values where the derivative is undefined.
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Step 1.3.1
Convert expressions with fractional exponents to radicals.
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Step 1.3.1.1
Apply the rule to rewrite the exponentiation as a radical.
Step 1.3.1.2
Anything raised to is the base itself.
Step 1.3.2
Set the denominator in equal to to find where the expression is undefined.
Step 1.3.3
Solve for .
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Step 1.3.3.1
To remove the radical on the left side of the equation, square both sides of the equation.
Step 1.3.3.2
Simplify each side of the equation.
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Step 1.3.3.2.1
Use to rewrite as .
Step 1.3.3.2.2
Simplify the left side.
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Step 1.3.3.2.2.1
Simplify .
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Step 1.3.3.2.2.1.1
Multiply the exponents in .
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Step 1.3.3.2.2.1.1.1
Apply the power rule and multiply exponents, .
Step 1.3.3.2.2.1.1.2
Cancel the common factor of .
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Step 1.3.3.2.2.1.1.2.1
Cancel the common factor.
Step 1.3.3.2.2.1.1.2.2
Rewrite the expression.
Step 1.3.3.2.2.1.2
Simplify.
Step 1.3.3.2.3
Simplify the right side.
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Step 1.3.3.2.3.1
Raising to any positive power yields .
Step 1.3.4
Set the radicand in less than to find where the expression is undefined.
Step 1.3.5
The equation is undefined where the denominator equals , the argument of a square root is less than , or the argument of a logarithm is less than or equal to .
Step 1.4
Evaluate at each value where the derivative is or undefined.
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Step 1.4.1
Evaluate at .
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Step 1.4.1.1
Substitute for .
Step 1.4.1.2
Simplify.
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Step 1.4.1.2.1
Remove parentheses.
Step 1.4.1.2.2
Simplify each term.
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Step 1.4.1.2.2.1
Rewrite as .
Step 1.4.1.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.4.1.2.2.3
Multiply by .
Step 1.4.1.2.2.4
Multiply by .
Step 1.4.1.2.3
Subtract from .
Step 1.4.2
Evaluate at .
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Step 1.4.2.1
Substitute for .
Step 1.4.2.2
Simplify.
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Step 1.4.2.2.1
Remove parentheses.
Step 1.4.2.2.2
Rewrite as .
Step 1.4.2.2.3
Pull terms out from under the radical, assuming positive real numbers.
Step 1.4.2.2.4
Multiply by .
Step 1.4.2.2.5
Subtract from .
Step 1.4.3
List all of the points.
Step 2
Evaluate at the included endpoints.
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Step 2.1
Evaluate at .
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Step 2.1.1
Substitute for .
Step 2.1.2
Simplify.
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Step 2.1.2.1
Remove parentheses.
Step 2.1.2.2
Rewrite as .
Step 2.1.2.3
Pull terms out from under the radical, assuming positive real numbers.
Step 2.1.2.4
Multiply by .
Step 2.1.2.5
Subtract from .
Step 2.2
Evaluate at .
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Step 2.2.1
Substitute for .
Step 2.2.2
Simplify.
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Step 2.2.2.1
Remove parentheses.
Step 2.2.2.2
Simplify each term.
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Step 2.2.2.2.1
Rewrite as .
Step 2.2.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2.2.2.3
Multiply by .
Step 2.2.2.2.4
Multiply by .
Step 2.2.2.3
Subtract from .
Step 2.3
List all of the points.
Step 3
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:
Step 4