Calculus Examples

Find the Critical Points sin(2x)
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
The derivative of with respect to is .
Step 1.1.1.3
Replace all occurrences of with .
Step 1.1.2
Differentiate.
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Step 1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3
Simplify the expression.
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Step 1.1.2.3.1
Multiply by .
Step 1.1.2.3.2
Move to the left of .
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
Divide each term in by and simplify.
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Step 2.2.1
Divide each term in by .
Step 2.2.2
Simplify the left side.
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Step 2.2.2.1
Cancel the common factor of .
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Step 2.2.2.1.1
Cancel the common factor.
Step 2.2.2.1.2
Divide by .
Step 2.2.3
Simplify the right side.
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Step 2.2.3.1
Divide by .
Step 2.3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2.4
Simplify the right side.
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Step 2.4.1
The exact value of is .
Step 2.5
Divide each term in by and simplify.
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Step 2.5.1
Divide each term in by .
Step 2.5.2
Simplify the left side.
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Step 2.5.2.1
Cancel the common factor of .
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Step 2.5.2.1.1
Cancel the common factor.
Step 2.5.2.1.2
Divide by .
Step 2.5.3
Simplify the right side.
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Step 2.5.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.5.3.2
Multiply .
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Step 2.5.3.2.1
Multiply by .
Step 2.5.3.2.2
Multiply by .
Step 2.6
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 2.7
Solve for .
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Step 2.7.1
Simplify.
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Step 2.7.1.1
To write as a fraction with a common denominator, multiply by .
Step 2.7.1.2
Combine and .
Step 2.7.1.3
Combine the numerators over the common denominator.
Step 2.7.1.4
Multiply by .
Step 2.7.1.5
Subtract from .
Step 2.7.2
Divide each term in by and simplify.
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Step 2.7.2.1
Divide each term in by .
Step 2.7.2.2
Simplify the left side.
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Step 2.7.2.2.1
Cancel the common factor of .
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Step 2.7.2.2.1.1
Cancel the common factor.
Step 2.7.2.2.1.2
Divide by .
Step 2.7.2.3
Simplify the right side.
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Step 2.7.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.7.2.3.2
Multiply .
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Step 2.7.2.3.2.1
Multiply by .
Step 2.7.2.3.2.2
Multiply by .
Step 2.8
Find the period of .
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Step 2.8.1
The period of the function can be calculated using .
Step 2.8.2
Replace with in the formula for period.
Step 2.8.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.8.4
Cancel the common factor of .
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Step 2.8.4.1
Cancel the common factor.
Step 2.8.4.2
Divide by .
Step 2.9
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 2.10
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
Cancel the common factor of .
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Step 4.1.2.1.1
Factor out of .
Step 4.1.2.1.2
Cancel the common factor.
Step 4.1.2.1.3
Rewrite the expression.
Step 4.1.2.2
The exact value of is .
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
Cancel the common factor of .
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Step 4.2.2.1.1
Factor out of .
Step 4.2.2.1.2
Cancel the common factor.
Step 4.2.2.1.3
Rewrite the expression.
Step 4.2.2.2
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.
Step 4.2.2.3
The exact value of is .
Step 4.2.2.4
Multiply by .
Step 4.3
List all of the points.
, for any integer
, for any integer
Step 5