SWBAT compute integrals with variable upper limits of integration.

Why do integrals with variable upper limits of integration produce functions? Students answer this question with multiple representations.

40 minutes

As usual class begins with a set of Warmup problems. As I circulate the room, I plan to watch for students who approach Warm-up #2 graphically and come up with only one value for *x*, typically the solution greater than 3. Students who approach this problem algebraically will more readily obtain both solutions since the algebraic approach involves solving a quadratic equation. I will point out the existence of two solutions to students who only find one solution, perhaps just by piquing their curiosity by saying, “I’ve noticed some other students found two solutions. How is that possible?” Then, I will let students wrestle with this possibility.

When going over Accumulate_This!_Warmup as a class, I will be sure to connect the algebraic and graphical approaches to the problems. I want my students to understand how switching the limits of integration switches the magnitude of positive and negative areas under the function. This concept tends to be an area of weakness, so I will also look for evidence in last night’s homework problems.

I will also be on the watch for students who solve algebraically by factoring expressions and misapplying the Zero Product Property (e.g., setting each factor equal to 14 after failing to move all terms to one side before factoring). Rather than reteaching the **Zero Product Property**, I will ask students to resubstitute solutions into the original equation. This will make the problem evident. I expect my students to be able to debug their original process.

Warm-up #3 reminds students that critical points occur where the first derivative equals zero or is undefined. I want my students to investigate these critical points graphically to improve student understanding and retention.

Students usually struggle with some of the tasks on last night’s AP question set for homework. So, I allow alot of time to review these questions, including the errors behind the most popular distractors. Last night’s homework solutions appear in the In The Classroom file.

10 minutes

15 minutes

2 minutes