# Calculus Examples

Find the Absolute Max and Min over the Interval
,
Step 1
Find the critical points.
Step 1.1
Find the first derivative.
Step 1.1.1
Find the first derivative.
Step 1.1.1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.4
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 .
Step 1.2.1
Set the first derivative equal to .
Step 1.2.2
Divide each term in by and simplify.
Step 1.2.2.1
Divide each term in by .
Step 1.2.2.2
Simplify the left side.
Step 1.2.2.2.1
Cancel the common factor of .
Step 1.2.2.2.1.1
Cancel the common factor.
Step 1.2.2.2.1.2
Divide by .
Step 1.2.2.3
Simplify the right side.
Step 1.2.2.3.1
Divide by .
Step 1.3
Find the values where the derivative is undefined.
Step 1.3.1
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.
Step 1.4
Evaluate at each value where the derivative is or undefined.
Step 1.4.1
Evaluate at .
Step 1.4.1.1
Substitute for .
Step 1.4.1.2
Simplify.
Step 1.4.1.2.1
Raising to any positive power yields .
Step 1.4.1.2.2
Subtract from .
Step 1.4.2
List all of the points.
Step 2
Evaluate at the included endpoints.
Step 2.1
Evaluate at .
Step 2.1.1
Substitute for .
Step 2.1.2
Simplify.
Step 2.1.2.1
Raising to any positive power yields .
Step 2.1.2.2
Subtract from .
Step 2.2
Evaluate at .
Step 2.2.1
Substitute for .
Step 2.2.2
Simplify.
Step 2.2.2.1
Raise to the power of .
Step 2.2.2.2
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