Understanding the Monotonicity Theorem

The video tutorial explains the monotonicity theorem, which states that a function is increasing when its derivative is positive and decreasing when its derivative is negative. The instructor uses two examples to illustrate the theorem: a linear function and a polynomial function. The video emphasizes that the concept is straightforward and relates to the slope of a function, a concept introduced in Algebra 1. The tutorial encourages viewers to understand the application of the theorem through practical examples.

What does the monotonicity theorem state about a function's behavior when its derivative is positive?

• The function is increasing.

• The function is constant.

• The function is decreasing.

• The function is oscillating.

In the context of the monotonicity theorem, what does a negative derivative indicate about a function?

• The function is increasing.

• The function is undefined.

• The function is decreasing.

• The function is constant.

For the function f(x) = 3x + 3, what can be said about its behavior based on its derivative?

• It is increasing for all x.

• It is decreasing for all x.

• It is constant for all x.

• It is undefined for all x.

What is the derivative of the function f(x) = 3x + 3?

• f'(x) = 0

• f'(x) = 3x

• f'(x) = 3

• f'(x) = x

What is the first step in analyzing the function H(z) = z^4/4 - 4z^3/6 to determine its intervals of increase and decrease?

• Find the second derivative.

• Set the function equal to zero.

• Graph the function.

• Find the derivative and set it equal to zero.

What are the critical points of the function H(z) = z^4/4 - 4z^3/6?

• z = 2 and z = 4

• z = -1 and z = 1

• z = 1 and z = 3

• z = 0 and z = 2

When is the function H(z) = z^4/4 - 4z^3/6 increasing?

• For all z

• For z < 0

• For z > 2

• For 0 < z < 2

What is the behavior of the function H(z) = z^4/4 - 4z^3/6 when z is between 0 and 2?

• The function is constant.

• The function is undefined.

• The function is decreasing.

• The function is increasing.

What does it mean for a function to be monotonic?

• It is always decreasing.

• It is always increasing.

• It is either always increasing or always decreasing.

• It is neither increasing nor decreasing.

What should you do if you feel something was missed in the video?

• Search online for answers.

• Let the instructor know.

• Rewatch the video.

• Ignore it.