Understanding Absolute Extrema

This video tutorial covers the concept of absolute extrema, providing examples using transcendental functions. It outlines the procedure for finding absolute extrema on a closed interval, including finding critical numbers and evaluating the function at these points and endpoints. The video includes three examples: using derivatives to find critical numbers, applying the double angle identity, and using the product rule. Each example is verified graphically to ensure accuracy.

What is the first step in finding the absolute extrema of a function on a closed interval?

• Calculate the second derivative

• Find the critical numbers in the open interval

• Graph the function

• Evaluate the function at the endpoints

In the first example, what is the function used to find the critical numbers?

• sin(x) + cos(x)

• sin(x)^2 + cos(x)

• tan(x) + sin(x)

• cos(x)^2 + sin(x)

What is the maximum value found in the first example, and at which point does it occur?

• 0 at x = pi/2

• 1 at x = pi

• 5/4 at x = pi/3

• 5/4 at x = 0

In the second example, which trigonometric identity is used to simplify the expression?

• Half angle identity

• Double angle identity

• Sum of angles identity

• Pythagorean identity

What is the absolute minimum value found in the second example, and at which point does it occur?

• -1.5 at x = 7pi/6

• 0 at x = 0

• -1 at x = pi

• 3 at x = pi/2

Which tool is used to verify the results graphically in the third section?

• Scientific calculator

• Spreadsheet software

• Graphing calculator

• Mathematical software

In the final example, why are only the endpoints tested for function values?

• The function is not continuous

• There are no critical numbers in the interval

• The derivative is undefined

• The function is linear

What is the maximum value found in the final example, and at which point does it occur?

• 3 at x = 1

• 0 at x = 0

• 2 at x = 2

• 5.38 at x = 3

Which mathematical rule is applied in the final example to find the derivative?

• Power rule

• Product rule

• Quotient rule

• Chain rule

What is the minimum value found in the final example, and at which point does it occur?

• 0 at x = 0

• 5.38 at x = 3

• 2 at x = 2

• 3 at x = 1