Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=x/(x^2+1) on -3 , 3
on ,
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
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.1.2
Differentiate.
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Step 1.1.1.2.1
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.2
Multiply by .
Step 1.1.1.2.3
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2.4
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.6
Simplify the expression.
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Step 1.1.1.2.6.1
Add and .
Step 1.1.1.2.6.2
Multiply by .
Step 1.1.1.3
Raise to the power of .
Step 1.1.1.4
Raise to the power of .
Step 1.1.1.5
Use the power rule to combine exponents.
Step 1.1.1.6
Add and .
Step 1.1.1.7
Subtract from .
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
Set the numerator equal to zero.
Step 1.2.3
Solve the equation for .
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Step 1.2.3.1
Subtract from both sides of the equation.
Step 1.2.3.2
Divide each term in by and simplify.
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Step 1.2.3.2.1
Divide each term in by .
Step 1.2.3.2.2
Simplify the left side.
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Step 1.2.3.2.2.1
Dividing two negative values results in a positive value.
Step 1.2.3.2.2.2
Divide by .
Step 1.2.3.2.3
Simplify the right side.
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Step 1.2.3.2.3.1
Divide by .
Step 1.2.3.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 1.2.3.4
Any root of is .
Step 1.2.3.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 1.2.3.5.1
First, use the positive value of the to find the first solution.
Step 1.2.3.5.2
Next, use the negative value of the to find the second solution.
Step 1.2.3.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 1.3
Find the values where the derivative is undefined.
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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.
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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 the denominator.
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Step 1.4.1.2.1
One to any power is one.
Step 1.4.1.2.2
Add and .
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
Simplify the denominator.
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Step 1.4.2.2.1.1
Raise to the power of .
Step 1.4.2.2.1.2
Add and .
Step 1.4.2.2.2
Move the negative in front of the fraction.
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
Simplify the denominator.
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Step 2.1.2.1.1
Raise to the power of .
Step 2.1.2.1.2
Add and .
Step 2.1.2.2
Move the negative in front of the fraction.
Step 2.2
Evaluate at .
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Step 2.2.1
Substitute for .
Step 2.2.2
Simplify the denominator.
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Step 2.2.2.1
Raise to the power of .
Step 2.2.2.2
Add and .
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