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Calculus Examples
Step 1
Write as a function.
Step 2
Step 2.1
Differentiate using the chain rule, which states that is where and .
Step 2.1.1
To apply the Chain Rule, set as .
Step 2.1.2
The derivative of with respect to is .
Step 2.1.3
Replace all occurrences of with .
Step 2.2
Differentiate.
Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Multiply by .
Step 2.2.3
Differentiate using the Power Rule which states that is where .
Step 2.2.4
Multiply by .
Step 3
Step 3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2
Differentiate using the chain rule, which states that is where and .
Step 3.2.1
To apply the Chain Rule, set as .
Step 3.2.2
The derivative of with respect to is .
Step 3.2.3
Replace all occurrences of with .
Step 3.3
Differentiate.
Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Multiply by .
Step 3.3.3
Differentiate using the Power Rule which states that is where .
Step 3.3.4
Multiply by .
Step 4
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 5
Step 5.1
Divide each term in by .
Step 5.2
Simplify the left side.
Step 5.2.1
Cancel the common factor of .
Step 5.2.1.1
Cancel the common factor.
Step 5.2.1.2
Divide by .
Step 5.3
Simplify the right side.
Step 5.3.1
Divide by .
Step 6
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 7
Step 7.1
The exact value of is .
Step 8
Step 8.1
Divide each term in by .
Step 8.2
Simplify the left side.
Step 8.2.1
Cancel the common factor of .
Step 8.2.1.1
Cancel the common factor.
Step 8.2.1.2
Divide by .
Step 8.3
Simplify the right side.
Step 8.3.1
Divide by .
Step 9
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 10
Step 10.1
Simplify.
Step 10.1.1
Multiply by .
Step 10.1.2
Add and .
Step 10.2
Divide each term in by and simplify.
Step 10.2.1
Divide each term in by .
Step 10.2.2
Simplify the left side.
Step 10.2.2.1
Cancel the common factor of .
Step 10.2.2.1.1
Cancel the common factor.
Step 10.2.2.1.2
Divide by .
Step 11
The solution to the equation .
Step 12
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 13
Step 13.1
Multiply by .
Step 13.2
The exact value of is .
Step 13.3
Multiply by .
Step 14
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Step 15
Step 15.1
Replace the variable with in the expression.
Step 15.2
Simplify the result.
Step 15.2.1
Multiply by .
Step 15.2.2
The exact value of is .
Step 15.2.3
The final answer is .
Step 16
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 17
Step 17.1
Cancel the common factor of .
Step 17.1.1
Cancel the common factor.
Step 17.1.2
Rewrite the expression.
Step 17.2
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 second quadrant.
Step 17.3
The exact value of is .
Step 17.4
Multiply .
Step 17.4.1
Multiply by .
Step 17.4.2
Multiply by .
Step 18
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Step 19
Step 19.1
Replace the variable with in the expression.
Step 19.2
Simplify the result.
Step 19.2.1
Cancel the common factor of .
Step 19.2.1.1
Cancel the common factor.
Step 19.2.1.2
Rewrite the expression.
Step 19.2.2
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 second quadrant.
Step 19.2.3
The exact value of is .
Step 19.2.4
Multiply by .
Step 19.2.5
The final answer is .
Step 20
These are the local extrema for .
is a local maxima
is a local minima
Step 21