##### Autumn 2013 MA111

Below are brief summaries of each day with supplements. These supplements are not intended to be substitutes for lectures. Instead, they are opportunities to outline and/or provide details that were skipped in class for the sake of brevity, to provide more involved examples, and to pose questions (some straightforward, some challenging) to help solidify your understanding of the material.

#### September 5

We briefly discussed the history of calculus. This naturally lead to a discussion about models for continual change, i.e., functions.

###### Supplement - .pdf

#### September 6

We talked more about the plots of functions. Specifically, we discussed how to derive the plot of a graph if it's a simple transformation of a given plot.

###### Supplement - .pdf

#### September 9

Inspired by our

childhood , we considered a fixed point on a circle carving out a plane curve as the circle rolls down the x-axis. This lead us to discover the angle measure ``radians'' and the trigonometric functions ``sine'' and ``cosine''.

###### Supplement - .pdf

#### September 10

We introduced the computer algebra system

Maple .

###### Supplement - .mw

#### September 11

We introduced exponential functions, and the so-called natural number e.

###### Supplement - .pdf

#### September 12

We introduced inverse functions wrapping up our pre-calc review of functions.

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#### September 13

We asked the question: can we derive velocity from position? We determined that (under many circumstances) that the average velocity over small time intervals approximated the velocity over that small time interval. What's more, the approximation gets better as the time interval gets smaller.

###### Supplement - .pdf

#### September 16, 17

If average velocity over the interval [t,t+h] gets closer to v as h gets smaller and smaller, we realized that this means that v must, in fact, be velocity at time t. So we focused our attention on trying to determine if the average velocity approached some value, and thus we discovered limits.

###### Supplement - .pdf

#### September 18

Continuing our discussion about limits, we talked about how to determine limits rigorously.

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#### September 19

How do you find the speed of a particle whose position is sin(t)? We answer this question, and along the way discover one-sided limits and the squeeze theorem.

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#### September 20

We wanted to know precisely which functions had the property that taking the limit of f(x) as x approached c was equal to f(c). We decided to call these functions continuous since it appeared their graphs had something of a connected look, and we made a list of continuous functions which we could take for granted.

###### Supplement - .pdf

#### September 23,24

Limits should generally tell us about the behavior of a function. Along those lines, how can you get information about the behavior of a function f(x) as x grows in magnitude? This question lead to a discussion about infinite limits.

###### Supplement - .pdf

#### September 26 - October 1

In an effort to be computationally economical, we've derived the so-called rules of differentiation, as well as, computing the derivative (by hand) of the exponential function. Using the small list of functions we've differentiated by hand and the list of general differentiation rules, we can easily differentiate many functions.

###### Supplement - .pdf

#### October 21

We've derived a method for finding the global maximum and minimum of a continuous function on a closed and bounded interval. See the supplement for example problems.

###### Supplement - .pdf

#### October 23

We discovered how to derive information about a function f(x) from its first and second derivatives. This allowed us to accurately sketch the graphs of functions. See the supplement for an example problem.

###### Supplement - .pdf

#### November 4

The supplement below describes how to tell how far one needs to take Newton's method to approximate the square-root of an integer up to a desirable accuracy. In this particular case, the speed of convergence of Newton's method is ridiculously fast.

###### Supplement - .pdf