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# The Calculus of Motion

Lesson 5 of 17

## Objective: SWBAT model basic linear motion using multiple derivative relationships.

## Big Idea: Derivatives and integrals are fundamentally related to each other. What better way to illuminate those relationships than applied motion?

*60 minutes*

#### Setting the Stage

*8 min*

In this section we introduce linear motion as an intuitive context for modeling with differential and integral calculus. **Play the first part of the Intro Moving Man – Motion Context video**, which shows a man moving back-and-forth along a line and plotting out his position, velocity and acceleration functions as he moves. Pause the video after this first part, and ask students several questions about the multiple derivative relationships that exist among the graphs of the position, velocity and acceleration functions, sticking to the mathematical relationships and not yet connecting them to the man’s movements.

**Then play the second part of the video**, which indicates the time on each function as the man moves along the line. **Pause the video at several points of interest to ask questions like “The man changed directions here. How do the position, velocity, and/or acceleration functions show this?”**, so students have to interpret their multiple derivative flashcards in this motion context **(SMP #1 Make sense of problems, SMP #2 Reason abstractly and quantitatively, SMP #4 Model with mathematics).** The purpose here is to develop intuition in students for modeling calculus concepts with motion, so when they are given information about an object’s motion symbolically students will have prior experience with visual representations of motion to help them represent and make sense of those symbolic representations.

**Resources**: Intro Moving Man – Motion Context video

#### Resources

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#### Investigation

*20 min*

**Distribute the p201 Example 1 handout, which is an example I took from our Stewart textbook but contains only the questions, not the full answers as appears in the textbook.** Several years ago I would give students this handout and work through it with them in class, but this wasn’t very engaging for students, it jumped too quickly to the abstraction of linear motion, and I realized that many students were unable to visualize and contextualize the particle’s movement very well when they were immediately given the position function symbolically. But actually, the position function on this handout is precisely the one we just encountered in the Moving Man video!

Because we just engaged students in representing linear motion through that video, students are more equipped to make connections between the calculus concepts and the man’s movements in answering the questions on this handout. Part (f) asks students to “Find the total distance traveled by the man during the first five seconds” and is students’ first encounter with the distinction between distance and displacement. We merely introduce the total distance concept today, and in upcoming lessons we will do much more work with distance vs. displacement and their respective integral representations. Part (i) asks “When is the man speeding up? When is he slowing down?”. Although we have touched on the distinction between velocity and speed on various occasions in our prior coursework, this distinction will still be fairly unfamiliar to some students. Tonight’s homework file online has students read more about the distinction between velocity and speed. We will also attend to this distinction further in tomorrow’s lesson.

To conclude this section, **replay the Intro Moving Man – Motion Context video and pause where applicable for students to check the reasonableness of their answers from the handout.** It is important to revisit this video to validate students’ calculus work and convince them that the calculus we have been learning makes sense in this linear motion context. (SMP #4 Model with mathematics).

**Resources**: p201 Example 1 Handout, Intro Moving Man – Motion Context video (same video from Setting the Stage section)

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#### Closure + Homework

*2 min*

Closure

Describe an object’s speed if its velocity and acceleration are both negative. Why does this make sense?

**Watch my closure video** discussing strategies for closing lessons.

Tonight’s Homework

F – Find the area of the region bound by *y* = –2, *y* = *e ^{x}* – 3, and

*y*= –2

*x*+ 8.

I - Review “Optimization – Notes with Example” file online. TEXTBOOK: Read example(s), then solve optimization problems.

V - Read “Motion” file pp7-11; solve p11 #1-9 odd.

E - For part F, find the area of the region with respect to the other variable (x or y – whichever you did not use in F).

Post the “Optimization – Notes with Example” file online for students to review prior to completing part I of tonight’s homework. Students received a paper copy of this handout back in the 1^{st} semester, but if they cannot find their paper copy then they can access it online.

**Resources**: In The Classroom file, Optimization – Notes with Example (post online), PR Motion file (post online)

#### Resources

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- UNIT 1: Back to School
- UNIT 2: Limits and Derivatives
- UNIT 3: Formalizing Derivatives and Techniques for Differentiation
- UNIT 4: Applications of Differentiation, Part 1
- UNIT 5: Applications of Differentiation, Part 2
- UNIT 6: The Integral
- UNIT 7: Applications of Integration
- UNIT 8: Differential Equations
- UNIT 9: Full Course Review via Motion
- UNIT 10: The Final Stretch - Preparing for the AP Exam

- LESSON 1: Limits and l'Hospital
- LESSON 2: Know Your Limits
- LESSON 3: Local Linearization, 1st and 2nd Derivative Tests, and Computing Derivatives
- LESSON 4: Derivatives Algebraically and Graphically
- LESSON 5: The Calculus of Motion
- LESSON 6: Motion - Velocity on Intervals
- LESSON 7: Motion - Distance vs Displacement
- LESSON 8: Motion - With Multiple Derivatives
- LESSON 9: Motion and Optimization
- LESSON 10: Calculus and My Car's Dashboard
- LESSON 11: Rockin' Related Rates
- LESSON 12: Meet My Friend Riemann
- LESSON 13: Cookies and Pi
- LESSON 14: Accumulate This!
- LESSON 15: Wait, the Interval Width Varies?
- LESSON 16: Integrating Areas to Get Volumes
- LESSON 17: More Areas and Volumes