Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=sin(x)+cos(x) , 0<=x<=2pi
,
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
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2
The derivative of with respect to is .
Step 1.1.1.3
The derivative of with respect to is .
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
Divide each term in the equation by .
Step 1.2.3
Cancel the common factor of .
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Step 1.2.3.1
Cancel the common factor.
Step 1.2.3.2
Rewrite the expression.
Step 1.2.4
Separate fractions.
Step 1.2.5
Convert from to .
Step 1.2.6
Divide by .
Step 1.2.7
Separate fractions.
Step 1.2.8
Convert from to .
Step 1.2.9
Divide by .
Step 1.2.10
Multiply by .
Step 1.2.11
Subtract from both sides of the equation.
Step 1.2.12
Divide each term in by and simplify.
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Step 1.2.12.1
Divide each term in by .
Step 1.2.12.2
Simplify the left side.
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Step 1.2.12.2.1
Dividing two negative values results in a positive value.
Step 1.2.12.2.2
Divide by .
Step 1.2.12.3
Simplify the right side.
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Step 1.2.12.3.1
Divide by .
Step 1.2.13
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 1.2.14
Simplify the right side.
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Step 1.2.14.1
The exact value of is .
Step 1.2.15
The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Step 1.2.16
Simplify .
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Step 1.2.16.1
To write as a fraction with a common denominator, multiply by .
Step 1.2.16.2
Combine fractions.
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Step 1.2.16.2.1
Combine and .
Step 1.2.16.2.2
Combine the numerators over the common denominator.
Step 1.2.16.3
Simplify the numerator.
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Step 1.2.16.3.1
Move to the left of .
Step 1.2.16.3.2
Add and .
Step 1.2.17
Find the period of .
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Step 1.2.17.1
The period of the function can be calculated using .
Step 1.2.17.2
Replace with in the formula for period.
Step 1.2.17.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 1.2.17.4
Divide by .
Step 1.2.18
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
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.
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Step 1.4.1.2.1
Simplify each term.
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Step 1.4.1.2.1.1
The exact value of is .
Step 1.4.1.2.1.2
The exact value of is .
Step 1.4.1.2.2
Simplify terms.
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Step 1.4.1.2.2.1
Combine the numerators over the common denominator.
Step 1.4.1.2.2.2
Add and .
Step 1.4.1.2.2.3
Cancel the common factor of .
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Step 1.4.1.2.2.3.1
Cancel the common factor.
Step 1.4.1.2.2.3.2
Divide by .
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 each term.
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Step 1.4.2.2.1.1
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 third quadrant.
Step 1.4.2.2.1.2
The exact value of is .
Step 1.4.2.2.1.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.
Step 1.4.2.2.1.4
The exact value of is .
Step 1.4.2.2.2
Simplify terms.
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Step 1.4.2.2.2.1
Combine the numerators over the common denominator.
Step 1.4.2.2.2.2
Subtract from .
Step 1.4.2.2.2.3
Cancel the common factor of and .
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Step 1.4.2.2.2.3.1
Factor out of .
Step 1.4.2.2.2.3.2
Cancel the common factors.
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Step 1.4.2.2.2.3.2.1
Factor out of .
Step 1.4.2.2.2.3.2.2
Cancel the common factor.
Step 1.4.2.2.2.3.2.3
Rewrite the expression.
Step 1.4.2.2.2.3.2.4
Divide by .
Step 1.4.3
List all of the points.
, for any integer
, for any integer
, for any integer
Step 2
Exclude the points that are not on the interval.
Step 3
Evaluate at the included endpoints.
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Step 3.1
Evaluate at .
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Step 3.1.1
Substitute for .
Step 3.1.2
Simplify.
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Step 3.1.2.1
Simplify each term.
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Step 3.1.2.1.1
The exact value of is .
Step 3.1.2.1.2
The exact value of is .
Step 3.1.2.2
Add and .
Step 3.2
Evaluate at .
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Step 3.2.1
Substitute for .
Step 3.2.2
Simplify.
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Step 3.2.2.1
Simplify each term.
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Step 3.2.2.1.1
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 3.2.2.1.2
The exact value of is .
Step 3.2.2.1.3
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 3.2.2.1.4
The exact value of is .
Step 3.2.2.2
Add and .
Step 3.3
List all of the points.
Step 4
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 5