Students will examine the relationship between critical points and local extrema through real-world examples. Students will zoom in on the critical points to see if the curve becomes linear to determine if the function is differentiable at the critical point. They will then discover that the sign of the slope of the tangent line to the curve changes when it passes a local minimum or maximum point.

Students will be able to:
• Identify critical points and local extrema for applications.
• Recognize that the sign of the derivative changes at a local minimum or maximum.

• critical point
• local maximum, minimum, extrema

• This lesson is a follow-up lesson to the Calculus activity Critical Points and Local Extrema and is designed to help students visualize the connections between critical points and local extrema.
• This lesson is meant to provide motivation for the first derivative test as a means to identify local extrema. Students will examine the slope of the tangent line as it approaches a critical point and connect this to an understanding of local extrema.
• This activity builds on students’ familiarity with the concept of the derivative at a point as the local slope of the function graph at that point.