Exploring Average Rate of Change in Polynomials

The video tutorial explores how to determine the interval over which a function has a positive average rate of change. It begins with an introduction to the problem and a reminder of what an average rate of change is. The instructor explains how to determine if the rate is positive by comparing function values at endpoints. Various intervals are analyzed, with calculations and graphing used to visualize the results. The tutorial identifies the interval with a positive rate and concludes with a verification of the findings.

What is the main problem we are trying to solve in this video?

• Finding the roots of h(x)

• Calculating the derivative of h(x)

• Determining the interval where h(x) has a positive average rate of change

• Finding the maximum value of h(x)

What does the average rate of change represent?

• The change in the value of the function for a given change in x

• The maximum value of the function

• The integral of the function

• The minimum value of the function

How can we determine if the average rate of change is positive?

• If the function is always decreasing

• If the function is always increasing

• If the value of the function at the lower endpoint is larger than at the higher endpoint

• If the value of the function at the higher endpoint is larger than at the lower endpoint

What is the value of h(0)?

• 2

• -1

• 1

• 0

What is the value of h(2)?

• 1

• 0

• -3

• 2

What is the average rate of change for the interval [0, 2]?

• Undefined

• Positive

• Negative

• Zero

What is the value of h(8)?

• 0

• -64

• 8

• 64

What is the average rate of change for the interval [0, 8]?

• Positive

• Negative

• Zero

• Undefined

What is the value of h(6)?

• 9

• -9

• 0

• 36

Which interval has a positive average rate of change?

• [6, 8]

• [0, 8]

• [0, 2]

• [0, 6]