Graph Behavior and Derivatives

The video tutorial explains how graphs behave based on first and second derivatives. It covers the concepts of gradient and concavity, and how these relate to graph shapes. The tutorial discusses decreasing, increasing, and stationary points, as well as points of inflection. It also explores graph patterns, exceptions, and the behavior of even powers in graphing.

What does a negative first derivative indicate about the graph's behavior?

• The graph is stationary.

• The graph is decreasing.

• The graph is increasing.

• The graph is concave up.

When the second derivative is positive, what can be said about the graph's concavity?

• The graph is stationary.

• The graph is linear.

• The graph is concave up.

• The graph is concave down.

What does a positive second derivative imply about the graph's curvature?

• The graph is concave up.

• The graph is concave down.

• The graph is linear.

• The graph is stationary.

What does it mean when a graph is increasing with no concavity?

• The graph is a straight line.

• The graph has a point of inflection.

• The graph is concave down.

• The graph is decreasing.

How is a point of inflection characterized in terms of derivatives?

• First derivative is positive, second derivative is zero.

• First derivative is zero, second derivative is zero.

• First derivative is negative, second derivative is zero.

• First derivative is zero, second derivative is positive.

Which of the following is true for a graph with a negative first derivative and zero second derivative?

• The graph is increasing and concave up.

• The graph is stationary and concave down.

• The graph is increasing and concave down.

• The graph is decreasing and linear.

What does a stationary point with concave down indicate?

• A local minimum.

• A straight line.

• A local maximum.

• A point of inflection.

What happens to the graph when the first derivative is zero and the second derivative is positive?

• The graph has a local maximum.

• The graph is concave down.

• The graph has a local minimum.

• The graph is decreasing.

What is the significance of a horizontal point of inflection?

• It indicates a local maximum.

• It indicates a change in concavity.

• It indicates a local minimum.

• It indicates a constant function.

How do even power functions behave as they increase in power?

• They become wider but pass through the same points.

• They pass through different points.

• They become narrower.

• They become linear.