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Section 1.3 Functions as models

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Worksheet
A4US

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Part of the innovation that makes calculus so useful is that it is a collection of general tools that can be applied in many different situations - rather than solving each individual problem as it comes along, we can translate a problem into a generalized type of problem that calculus can solve.

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Part of this translation is to take the concrete description of a problem and create more abstract representations of the various pieces. In many cases, our first step is to create a functional representation of the information in the problem at hand.

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Much of what drove the creation of calculus was the desire to understand objects in motion, so in this exercise we will focus on modeling motion using functions.

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1.

Let's consider a small rodent called a jerboa that hops a little bit like a kangaroo. To the right, you'll see a gif of one of these animals hopping around an enclosure.

For our first exercise, have your group brainstorm about this little video. What are the different ways we could describe the motion shown?

A Jerboa hopping around an enclosure. Image credit: Talia Moore, University of Michigan
Figure 1.3.1. A Jerboa hopping around an enclosure. Image credit: Talia Moore, University of Michigan
Hint

What are the different components of the jerboa's movement? Which do you want to focus on? (it is ok to pick only one component)

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2.
What are the variables we can define to help describe the motion more precisely? Can you draw a sketch of a function showing the relationship between some of the variables? Can you write down a definition of the same function mathematically? If not, why not?
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3.

Repeat the previous exercises for some of the video clips and other descriptions of motion below. Were any of them harder or easier than others? Why was this? What are the similarities and differences between the functions that you find?

Anthony Edwards of the Georgia Bulldogs making a spectacular underhanded shot against the opposing basketball team, Ole Miss, in the SEC tournament. Source: Minnesota Timberwolves
A stick figure running through teleportation portal in the video game 'Portal' from left to right. Source: IGN
Simone Biles landing part of a triple double in a gymnastics floor routine. Source: USA Today
A SpaceX booster landing on a remote pad in the ocean. Source: Bloomberg
Appa, a six legged sky bison from Avatar: the last airbender, doing a belly-flop in the snow. Source: Nickelodeon
Grogu, one of the main characters in The Madalorian, eating a cookie. Source: Disney