# The Labs

## Technical Overview

Our labs are designed to support a Calculus II curriculum that includes:

• Application and techniques of integration
• Exponential growth/decay and separable differential equations
• Sequences and series
• Calculus of curves: arc length, parametrized curves, calculus in polar coordinates

The labs are meant to take place every other week during a 50-minute long recitation. Students are expected to engage with the pre-lab, which are given both as a PDF document and as a supplementary/complementary short video prior to the recitation. The pre-lab provides the basic context for the lab exercise. During the recitation the students are expected to work in groups of 3–4 students. The labs themselves are presented as interactive programming documents using MATLAB LiveScripts. Students modify and execute the provided code, using the results of the simulations to complete a worksheet with both factual and interpretive questions.

On this page you will find short descriptions of each lab, together with PDF copies of both the pre-lab and the lab worksheets, links to the pre-lab videos, and the MATLAB LiveScript file. We recommend using MATLAB version 2018b or later to open the lab files; alternatively one can also use MATLAB Online.

# Intro to MATLAB LiveScripts (optional lab)

Lab 0

Lab 0 is intended to be a self-directed lab familiarizing students to MATLAB LiveScripts and introducing students to the basic MATLAB functionality that will be used in the subsequent labs

### Lab Material

Lab Livescript

Matlab Cheat Sheet

# Baseball Rocketry

Lab 2

Lab 2 showcases the power of “approximating discrete processes by continuous ones”, through a slightly absurd model of rocket science. Along the way we also touch on the concept of the natural logarithm function.

Pre-lab video:

Lab LiveScript

Lab Worksheet

# Trials and Tribble-ations

Lab 4

In Lab 4 we explore algorithm design and convergent sequences through a variant of the old number guessing game.

Pre-lab video:

### Lab Material

Lab LiveScript

Lab Worksheet

Lab Worksheet Supplement

# Boom Bust Butterfly

Lab 5

Lab 5 explores the concept of a bounded divergent sequence, showcasing it in the context of a chaotic toy model that is typical to economics or biological sciences.

Pre-lab video:

Lab LiveScript

Lab Worksheet

# Etch-a-sketch Time

Lab 7

Lab 7 introduces students to concepts related to applications of calculus to the analysis of parametrized curves, through the analogy with the Etch-a-sketch toy. The students then use those concepts to analyze real life data from a Corvette racing around a race-track.

Pre-lab video:

Lab LiveScript

Lab Data File

Lab Worksheet

### Acknowledgements

The Lab Data File comes from the REVS Vehicle Dynamics Database, made available by JC Kegelman and JC Gerdes. The original data has been released under the Open Data Commons Attribution License.

# Numerical Integration

Lab 1

Lab 1 gives a quick introduction to numerical integration. Emphasis is placed on the connection between summing and integration (including the analogy between sums and cumulative sums against definite and indefinite integration), through the defining concept of the Riemann integral.

Pre-lab video:

Lab LiveScript

Lab WorkSheet

# Zombie Attack

Lab 3

In Lab 3 we examine the logistics equation through a discrete time population model modelling disease propagation.

Pre-lab video:

Lab LiveScript

Lab Worksheet

# Leibniz’s Wheel

Lab 6

Lab 6 addresses the question of “how does a calculator know what the value of sin(1) is?” (Replace sin by your favourite transcendental function.) One method of answering this question is to use the Taylor polynomial approximation to compute values of transcendental functions to a desired degree of accuracy.