Calculus Examples

Find the Absolute Max and Min over the Interval y=7x+7sin(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
Evaluate .
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Step 1.1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.3
Multiply by .
Step 1.1.1.3
Evaluate .
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Step 1.1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.3.2
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
Subtract from both sides of the equation.
Step 1.2.3
Divide each term in by and simplify.
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Step 1.2.3.1
Divide each term in by .
Step 1.2.3.2
Simplify the left side.
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Step 1.2.3.2.1
Cancel the common factor of .
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Step 1.2.3.2.1.1
Cancel the common factor.
Step 1.2.3.2.1.2
Divide by .
Step 1.2.3.3
Simplify the right side.
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Step 1.2.3.3.1
Divide by .
Step 1.2.4
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 1.2.5
Simplify the right side.
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Step 1.2.5.1
The exact value of is .
Step 1.2.6
The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 1.2.7
Subtract from .
Step 1.2.8
Find the period of .
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Step 1.2.8.1
The period of the function can be calculated using .
Step 1.2.8.2
Replace with in the formula for period.
Step 1.2.8.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 1.2.8.4
Divide by .
Step 1.2.9
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
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 1.4.1.2.1.2
The exact value of is .
Step 1.4.1.2.1.3
Multiply by .
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 each term.
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Step 1.4.2.2.1.1
Multiply by .
Step 1.4.2.2.1.2
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 1.4.2.2.1.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 1.4.2.2.1.4
The exact value of is .
Step 1.4.2.2.1.5
Multiply by .
Step 1.4.2.2.2
Add and .
Step 1.4.3
Evaluate at .
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Step 1.4.3.1
Substitute for .
Step 1.4.3.2
Simplify.
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Step 1.4.3.2.1
Simplify each term.
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Step 1.4.3.2.1.1
Multiply by .
Step 1.4.3.2.1.2
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 1.4.3.2.1.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 1.4.3.2.1.4
The exact value of is .
Step 1.4.3.2.1.5
Multiply by .
Step 1.4.3.2.2
Add and .
Step 1.4.4
Evaluate at .
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Step 1.4.4.1
Substitute for .
Step 1.4.4.2
Simplify.
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Step 1.4.4.2.1
Simplify each term.
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Step 1.4.4.2.1.1
Multiply by .
Step 1.4.4.2.1.2
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 1.4.4.2.1.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 1.4.4.2.1.4
The exact value of is .
Step 1.4.4.2.1.5
Multiply by .
Step 1.4.4.2.2
Add and .
Step 1.4.5
Evaluate at .
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Step 1.4.5.1
Substitute for .
Step 1.4.5.2
Simplify.
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Step 1.4.5.2.1
Simplify each term.
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Step 1.4.5.2.1.1
Multiply by .
Step 1.4.5.2.1.2
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 1.4.5.2.1.3
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 1.4.5.2.1.4
The exact value of is .
Step 1.4.5.2.1.5
Multiply by .
Step 1.4.5.2.2
Add and .
Step 1.4.6
List all of the points.
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
Multiply by .
Step 3.1.2.1.2
The exact value of is .
Step 3.1.2.1.3
Multiply by .
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
Multiply by .
Step 3.2.2.1.2
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 3.2.2.1.3
The exact value of is .
Step 3.2.2.1.4
Multiply by .
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