Calculus Examples

Find the Absolute Max and Min over the Interval f(x)=sin(x+pi/4) , 0<=x<=(7pi)/4
,
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
Differentiate using the chain rule, which states that is where and .
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Step 1.1.1.1.1
To apply the Chain Rule, set as .
Step 1.1.1.1.2
The derivative of with respect to is .
Step 1.1.1.1.3
Replace all occurrences of with .
Step 1.1.1.2
Differentiate.
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Step 1.1.1.2.1
By the Sum Rule, 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
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.4
Simplify the expression.
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Step 1.1.1.2.4.1
Add and .
Step 1.1.1.2.4.2
Multiply by .
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
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 1.2.3
Simplify the right side.
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Step 1.2.3.1
The exact value of is .
Step 1.2.4
Move all terms not containing to the right side of the equation.
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Step 1.2.4.1
Subtract from both sides of the equation.
Step 1.2.4.2
To write as a fraction with a common denominator, multiply by .
Step 1.2.4.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.2.4.3.1
Multiply by .
Step 1.2.4.3.2
Multiply by .
Step 1.2.4.4
Combine the numerators over the common denominator.
Step 1.2.4.5
Simplify the numerator.
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Step 1.2.4.5.1
Move to the left of .
Step 1.2.4.5.2
Subtract from .
Step 1.2.5
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 1.2.6
Solve for .
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Step 1.2.6.1
Simplify .
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Step 1.2.6.1.1
To write as a fraction with a common denominator, multiply by .
Step 1.2.6.1.2
Combine fractions.
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Step 1.2.6.1.2.1
Combine and .
Step 1.2.6.1.2.2
Combine the numerators over the common denominator.
Step 1.2.6.1.3
Simplify the numerator.
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Step 1.2.6.1.3.1
Multiply by .
Step 1.2.6.1.3.2
Subtract from .
Step 1.2.6.2
Move all terms not containing to the right side of the equation.
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Step 1.2.6.2.1
Subtract from both sides of the equation.
Step 1.2.6.2.2
To write as a fraction with a common denominator, multiply by .
Step 1.2.6.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.2.6.2.3.1
Multiply by .
Step 1.2.6.2.3.2
Multiply by .
Step 1.2.6.2.4
Combine the numerators over the common denominator.
Step 1.2.6.2.5
Simplify the numerator.
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Step 1.2.6.2.5.1
Multiply by .
Step 1.2.6.2.5.2
Subtract from .
Step 1.2.7
Find the period of .
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Step 1.2.7.1
The period of the function can be calculated using .
Step 1.2.7.2
Replace with in the formula for period.
Step 1.2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 1.2.7.4
Divide by .
Step 1.2.8
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 1.2.9
Consolidate the answers.
, 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
Combine the numerators over the common denominator.
Step 1.4.1.2.2
Add and .
Step 1.4.1.2.3
Cancel the common factor of and .
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Step 1.4.1.2.3.1
Factor out of .
Step 1.4.1.2.3.2
Cancel the common factors.
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Step 1.4.1.2.3.2.1
Factor out of .
Step 1.4.1.2.3.2.2
Cancel the common factor.
Step 1.4.1.2.3.2.3
Rewrite the expression.
Step 1.4.1.2.4
The exact value of is .
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
Combine the numerators over the common denominator.
Step 1.4.2.2.2
Add and .
Step 1.4.2.2.3
Cancel the common factor of and .
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Step 1.4.2.2.3.1
Factor out of .
Step 1.4.2.2.3.2
Cancel the common factors.
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Step 1.4.2.2.3.2.1
Factor out of .
Step 1.4.2.2.3.2.2
Cancel the common factor.
Step 1.4.2.2.3.2.3
Rewrite the expression.
Step 1.4.2.2.4
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 1.4.2.2.5
The exact value of is .
Step 1.4.2.2.6
Multiply 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
Add and .
Step 3.1.2.2
The exact value of is .
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
Combine the numerators over the common denominator.
Step 3.2.2.2
Add and .
Step 3.2.2.3
Cancel the common factor of and .
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Step 3.2.2.3.1
Factor out of .
Step 3.2.2.3.2
Cancel the common factors.
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Step 3.2.2.3.2.1
Factor out of .
Step 3.2.2.3.2.2
Cancel the common factor.
Step 3.2.2.3.2.3
Rewrite the expression.
Step 3.2.2.3.2.4
Divide by .
Step 3.2.2.4
Subtract full rotations of until the angle is greater than or equal to and less than .
Step 3.2.2.5
The exact value of is .
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