Calculus Examples

Find the Critical Points cos(x)^2
Step 1
Find the first derivative.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Differentiate using the chain rule, which states that is where and .
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Step 1.1.1.1
To apply the Chain Rule, set as .
Step 1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3
Replace all occurrences of with .
Step 1.1.2
The derivative of with respect to is .
Step 1.1.3
Multiply by .
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
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Step 2.1
Set the first derivative equal to .
Step 2.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.3
Set equal to and solve for .
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Step 2.3.1
Set equal to .
Step 2.3.2
Solve for .
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Step 2.3.2.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2.3.2.2
Simplify the right side.
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Step 2.3.2.2.1
The exact value of is .
Step 2.3.2.3
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 2.3.2.4
Simplify .
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Step 2.3.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 2.3.2.4.2
Combine fractions.
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Step 2.3.2.4.2.1
Combine and .
Step 2.3.2.4.2.2
Combine the numerators over the common denominator.
Step 2.3.2.4.3
Simplify the numerator.
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Step 2.3.2.4.3.1
Multiply by .
Step 2.3.2.4.3.2
Subtract from .
Step 2.3.2.5
Find the period of .
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Step 2.3.2.5.1
The period of the function can be calculated using .
Step 2.3.2.5.2
Replace with in the formula for period.
Step 2.3.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.3.2.5.4
Divide by .
Step 2.3.2.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 2.4
Set equal to and solve for .
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Step 2.4.1
Set equal to .
Step 2.4.2
Solve for .
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Step 2.4.2.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2.4.2.2
Simplify the right side.
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Step 2.4.2.2.1
The exact value of is .
Step 2.4.2.3
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 2.4.2.4
Subtract from .
Step 2.4.2.5
Find the period of .
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Step 2.4.2.5.1
The period of the function can be calculated using .
Step 2.4.2.5.2
Replace with in the formula for period.
Step 2.4.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.4.2.5.4
Divide by .
Step 2.4.2.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 2.5
The final solution is all the values that make true.
, for any integer
Step 2.6
Consolidate the answers.
, for any integer
, for any integer
Step 3
Find the values where the derivative is undefined.
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Step 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 4
Evaluate at each value where the derivative is or undefined.
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Step 4.1
Evaluate at .
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Step 4.1.1
Substitute for .
Step 4.1.2
Simplify.
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Step 4.1.2.1
The exact value of is .
Step 4.1.2.2
One to any power is one.
Step 4.2
Evaluate at .
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Step 4.2.1
Substitute for .
Step 4.2.2
Simplify.
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Step 4.2.2.1
The exact value of is .
Step 4.2.2.2
Raising to any positive power yields .
Step 4.3
List all of the points.
, for any integer
, for any integer
Step 5