Understanding Continuous Functions and Derivatives

The video tutorial explains how to find the smallest possible value of f(5) for a continuous function f(x) given that f(-2) = 4 and the derivative f'(x) is at least 2.5 over the interval [-2, 5]. By assuming the minimum derivative value of 2.5, the function is shown to be linear with a slope of 2.5. The change in x over the interval is calculated as 7, leading to a change in y of 17.5. This results in f(5) being 21.5, which is the smallest possible value given the conditions.

What is the value of f(-2) given in the problem?

• 7

• 5

• 4

• 2

What is the minimum value of the derivative f'(x) over the interval?

• 1.5

• 2.0

• 2.5

• 3.0

Why is the smallest possible value of f(5) calculated using the smallest derivative value?

• Because it is the average function value

• Because it has no effect on the function value

• Because it minimizes the function value

• Because it maximizes the function value

What does a constant derivative of 2.5 imply about the graph of f(x)?

• The graph is a curve

• The graph is a line with a slope of 2.5

• The graph is a horizontal line

• The graph is a vertical line

What is the change in x over the interval from -2 to 5?

• 5

• 6

• 8

• 7

How much does the function value increase when x increases by 7?

• 16.5

• 17.5

• 15.5

• 14.5

What is the y-coordinate of the point when x is 5?

• 21.5

• 22.5

• 19.5

• 20.5

What is the smallest possible value of f(5)?

• 19.5

• 21.5

• 22.5

• 20.5

What does the point (5, 21.5) represent in the context of the problem?

• The average value of f(x)

• The maximum value of f(x)

• The minimum value of f(x)

• The smallest possible value of f(5)

Why is the derivative value of 2.5 significant in this problem?

• It is the most common value

• It is the average value

• It is the maximum possible value

• It is the least possible value