Calculus Examples

The derivative of with respect to is .
Set the derivative equal to then solve for .
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Simplify the expression to find the first solution.
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Take the inverse of both sides of the equation to extract from inside the .
The exact value of is .
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.
Simplify the expression to find the second solution.
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To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Multiply by to get .
Combine the numerators over the common denominator.
Simplify the numerator.
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Factor out of .
Multiply by to get .
Subtract from to get .
Simplify the expression.
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Move to the left of the expression .
Multiply by to get .
Find the period.
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The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
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The absolute value is the distance between a number and zero. The distance between and is .
Divide by to get .
The period of the function is so values will repeat every radians in both directions.
Consolidate the answers under the reference angle.
Substitute the values of which cause the derivative to be into the original function.
Evaluate.
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The exact value of is .
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.
The exact value of is .
Multiply by to get .
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.
Since there are no values of where the derivative is undefined, there are no additional critical points.
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