Differentiability and Slope Analysis

The video tutorial explores the concept of differentiability for a function defined over all real numbers. It focuses on visualizing the derivative and analyzing the slope of the function at various points. Key points of interest are where the slope changes abruptly, specifically at x equals negative 2 and x equals 3, where the function is not differentiable. The tutorial explains how the slope of the tangent line and the limit of the secant line differ at these points, leading to a lack of a defined derivative.

What is the main focus of the problem discussed in the video?

• Finding the maximum value of f(x)

• Calculating the integral of f(x)

• Determining where f(x) is not differentiable

• Identifying the zeros of f(x)

What is the slope of f(x) between x = -8.5 and x = -2?

• Positive 3.5

• Zero

• Negative 2

• Positive 2

What happens to the slope of f(x) at x = -2?

• It remains constant

• It becomes zero

• It becomes undefined

• It changes from negative to positive

What is the estimated slope of the tangent line at x = -2?

• Negative 1

• Positive 3.5

• Zero

• Negative 2

How does the slope behave as x approaches 3?

• It oscillates

• It becomes a flat line

• It decreases steadily

• It increases steadily

What is the slope of f(x) right after x crosses 3?

• Positive 1

• Zero

• Negative 2

• Positive 3.5

At which points is f(x) not differentiable?

• x = 1 and x = 2

• x = 0 and x = 1

• x = -2 and x = 3

• x = -8.5 and x = 2

Why is f(x) not differentiable at x = -2?

• The slope is positive

• The slope is constant

• The slope is zero

• The slope is undefined

What is the slope from the left as x approaches 3?

• Negative 1

• Zero

• Positive 3.5

• Positive 2

What is the slope from the right as x approaches 3?

• Zero

• Positive 2

• Positive 3.5

• Negative 1