Motion and Optimization

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SWBAT use derivatives to solve optimization problems, and model linear motion using multiple derivatives and the Fundamental Theorem of Calculus.

Big Idea

Spring break is here, but Calculus does not need a break. Today we finish optimization and motion, then preview the spring break assignment.

Video Narrative - Lesson Overview

Warm-Up + Homework Review

25 minutes


Today, we will check last night’s homework right away…

From last night’s homework, the two questions that give students the most difficulty are from 1988, Questions #16 and #45. 

  • Question #16 requires students to identify the time(s) when an object is at rest.  Most students recognize that the language “at rest” means “velocity is 0” and can differentiate and solve from there.  For some reason answer (e) tends to be a common distractor, but this answer seems to follow from minor +/– errors when students factor, and not necessarily a conceptual misunderstanding. 
  • Question #45 is a straight-forward optimization problem, but the most common distractors (a, b, c) involve different powers of 2.  For this problem it is easiest to substitute out the variable h and then solve for r, but the question asks for the height h and therefore requires resubstitution into the constraint to solve for h.  Review or reteach exponent and radical operations with students as necessary, but most likely students will have made a minor algebraic error somewhere along the way in obtaining the most common distractors.

 The questions are available from the AP Calculus Release Tasks


10 minutes

Depending on how much time remains after going over homework, the In The Classroom file contains several practice optimization problems – the two optimization problems for minimizing distance from a point to a function are represented in the Minimize Distance to Function GeoGebra file. I like to use this applet to help students model (SMP #4) each optimization problem and provide appropriate support for struggling students to gain entry into these problems. 

If time is short, I may pick one of the four optimization questions that I know students need to practice, and then spend the full 10 minutes (or however much time you have) solving and discussing this one problem.  I find that with these problems it is more important for students to solve, discuss, summarize the solution process, and truly master a smaller number of problems, rather than treating these problems as rapid-fire exercises where attaining the solution is the end of the time students spend with each problem.  Alternatively, I might present all four problems to students and then discuss only the setups for each without spending class time to work out the specific solutions. I will make this decision based on my assessment of my students’ needs at this time (and their energy).

DIFFERENTIATION:  As a scaffold for students' work today, I may direct students to review their solutions to other optimization problems and follow the same process with a current problem. This approach helps strengthen students’ connections among different optimization problems. When they see similar processes used, it helps them make connections that will be useful in the context of an exam. 


Closure + Homework

25 minutes

The Quiz Optimization and Motion will take 20-25 minutes, and should occupy students through the end of the period.  There are several versions of the quiz in the resources. Each version asks detailed interpretative questions about motion and have one optimization question.

Tonight’s Homework: AP Test – MC 1997.  Submit your “preliminary answers” by Wednesday and I will e-mail you a scores report.  Make corrections then resubmit your “final answers”, due Sunday night.  Look up every question in the textbook, notes, homework examples, etc. to make sure you solve every question correctly.  Score 100%!  Do not wait until the night before each deadline to complete this assignment.

To learn more about my strategies for wrapping up a lesson, watch my Closing Lessons video where I discuss my common strategies for closing lessons.