Understanding the First Derivative Test

The video tutorial explains how to use the first derivative test to find relative extrema of a function on the interval from 0 to 2π. It begins by finding the derivative of the function and identifying critical numbers where the derivative is zero or undefined. The tutorial then divides the interval into smaller sections using these critical numbers and analyzes the sign of the derivative in each section to determine where the function is increasing or decreasing. Finally, it identifies the relative minimum and maximum points and evaluates the function at these points to find their values.

What is the purpose of using the first derivative test?

• To find the relative extrema of a function

• To solve differential equations

• To determine the concavity of a function

• To find the absolute extrema of a function

What are critical numbers?

• Values where the function is undefined

• Values where the first derivative is zero or undefined

• Values where the second derivative is zero

• Values where the function has a maximum

What is the derivative of f(x) = 2cos(x) - sin(2x)?

• -2sin(x) - 2cos(2x)

• -2cos(x) - 2sin(2x)

• -2sin(x) - 2cos(x)

• -2sin(x) - 2sin(2x)

How do you solve for x when the derivative is set to zero?

• By using integration

• By graphing the function

• By factoring and solving trigonometric equations

• By using the quadratic formula

What substitution is made for cos(2x) in the trigonometric equation?

• sin^2(x) - 1

• 1 - cos^2(x)

• 1 - sin^2(x)

• 2sin(x)cos(x)

What is the significance of determining the sign of the first derivative in intervals?

• To identify intervals of increase and decrease

• To find the absolute extrema

• To determine the function's concavity

• To solve for x

Which test value is used between 0 and π/2?

• π/5

• π/4

• π/6

• π/3

What does a negative sign of the derivative indicate about the function?

• The function is decreasing

• The function is constant

• The function is increasing

• The function is undefined

At which x-value is there a relative minimum?

• 2π

• 11π/6

• 7π/6

• π/2

What is the relative maximum value of the function?

• √3/2

• 2√3/2

• -3√3/2

• 3√3/2