Visualize the connections between the first derivative of a function, critical points, and local extrema.

• Relate the first derivative of a function to its critical points and identify which of these critical points are local extrema.
• Visualize why the first derivative test works and how it is used to determine local minima and maxima.

• first derivative
• critical point
• local maximum, local minimum, extrema

This lesson involves visualizing the connections between the first derivative of a function, critical points, and local extrema. As a result, students will:

• Develop an understanding of the first derivative test.
• Explore a sequence of functions, some by moving a tangent line along the function graph and noting changes in the first derivative of a function near its critical points.
• Build on their familiarity with the concept of the derivative at a point as the local slope of the function graph at that point.