Below are movies (almost as good as Warren Miller's) showing partial sums for two power series we looked at in class today. In each, the graph of the partial sums are green and the graph of the limit is red. It's probably best to advance the movies one frame at a time using the movie control arrows.
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Our first goal on the calculus side is to review some ideas from the first semester of calculus, specifically limits, continuity, and derivatives. We will do this using both the type of function you are familiar with and with a new type of function. The new type of function has vector outputs. Vectors play an important role in physics.
We talked about numbers briefly today. Here's a handout with some problems on real numbers that you should begin work on. We will address any questions you have on these problems in class on Wednesday.
Homework from Section 1.2 was due today. I did not say anything about collecting it in class today. In the future, on the day a homework assignment is due, you should put it on the front table some time before you leave class. Do this with the Section 1.2 assignment in class on Wednesday.
After class, someone asked about the distinction between using rounded
brackets for an ordered pair and angle brackets. For example, what is the
difference between (1,2) and <1,2>? The difference comes in how we think
about these geometrically. Geometrically, we think about the pair (1,2) as
the cartesian coordinates of a point. We think about the pair <1,2> as the
components of a vector which we picture as an arrow. If we interpret <1,2>
as a position vector with tail at the origin, the head will be at the point
(1,2). In this interpretation, the point (1,2) and the vector <1,2> are
very closely related. If we are interpreting the vector <1,2> as some
other physical quantity (for example, a force), there is no direct
connection to the point (1,2). Another difference between coordinate pairs
(a,b) and vector components
When we are dealing with vector-ouput functions, we often interpret the vector-output as a position vector. We then sometimes go back and forth between thinking geometrically about the arrows and the points. Each view (arrow and point) is useful for different purposes. Only from experience will you pick up on which view to work in for a given purpose.
Each of the movies below shows a curve begin traced out by a vector-output function. The vector view is shown in green and the point view is shown in red. You can use the controls to play the movies and to step through each frame one at a time.
I have posted a first draft of a trigonometry review. If you are feeling rusty on trigonometry, you might try working through this to remind yourself of some important details. I haven't tested this out before so I don't know how effective it is. If you try it, I welcome your feedback with suggestions for improvement. Check back later for updated versions.
Below is another movie showing a line being traced out by a vector-output function. Think of the function as R(t)=R0+t d. The vector R0 is shown in blue, the vector t d is shown in green, and the sum R0+t d is shown in red. I have displaced the green vector slightly so that you can see both this vector and the line being traced out.
Below are movies for the vector-output functions from Problems 1 and 2 of the worksheet we were doing in class today.
Kaylie found a web site showing a nice application of the vector-output functions we played with in class last Friday (and are shown in the movies above).
We started looking at the calculus of vector-output functions today. It is easy to compute the derivative of a vector-output function if we are given the components: just differentiate each component as a "regular" function. The new thing here is to understand how to think geometrically about the derivative vector. For Problems 9 and 10 in Section 2.5, it is worthwhile to make careful plots using graph paper.
We will have an exam on Friday that covers material through Section 2.5. There will be some emphasis on vector-output functions since this material is new as opposed to review.
I will be available much of the afternoon tomorrow (Tuesday). If you have questions, come find me or contact me to set up a specific meeting time.
Each of the movies below shows the output curve of a vector-output function r(t) along with the derivative vector r'(t). The derivative vector is drawn with tail at the point corresponding to t on the curve. The movie shows the derivative vector moving as t increases.
The first movie is for the helix r(t)=(cos(t), sin(t), t).
The next three movies are for the vector-output functions on the handout we looked at in class.
Some interesting issues have come up in class that we might not have time to fully address. If you want to explore these further, try your hand at these challenge problems. I am happy to talk with anyone about these. Come find me in my office.
We are now looking at definite integrals. Next we will study antiderivatives. These are related by the Fundamental Theorems of Calculus. Many of you are familar with these ideas from your high school calculus course. If so, this is an opportunity to take a deeper look at things. In particular, you can focus on the issue of how we prove the theorems that we use.
I've assigned some additional problems on definite integrals that require use of the definite integral properties we discussed in class.
Drawing slope fields by hand can be tedious. You need to draw a few for yourself to be sure you understand the idea. After that, you can use technology to draw slope fields. There are many Java applets available on the web. One I have tried and liked is Slope Field Calculator by Marek Rychlik at the University of Arizona. The Java code takes some time (perhaps 10-20 seconds) to load so be patient. For the problems in Section 3.2, you should both sketch a slope field by hand and use this Java applet to produce a slope field.
We are now turning our attention to differential equations. You can view this as a generalization of the antiderivative problem. We use the same tools: knowledge of derivatives, slope fields, and Euler's method. It is useful to use technology in drawing slope fields and computing with Euler's method.
You may have noticed that we have skipped some of the material in Chapter 4, specifically Sections 4.2 and 4.3. We may come back to this material at the end of the semester if there is time.
We will have an exam on Friday covering definite integrals, antiderivatives, the Fundamental Theorems of Calculus, and differential equations. The relevant sections from the text are 3.1-3.3 and 6.1-6.3. The kinematics problem of going from acceleration to velocity to position is also relevant.
As homework, you should submit Problem 2 from the Population models handout we worked on in class. Last week, I e-mailed a solution to the first problem. One person commented that the attachment did not open so I've included it here for you to download if you need.
I've written up a handout on line integrals giving the example we did in class on Wednesday. This includes various methods for evaluating a line integral. The handout also has two problems. The first is the one we worked on in class on Friday. The second involves forces and displacements in space as opposed to the plane. You should work on these as homework. I've also listed some problems from Section 7.3 on the usual homework list.
In class last week, I mentioned the program Mathematica. Wolfram, the company that produces Mathematica, has a web site called The Integrator that allows you to use the integration features of Mathematica online.
We have a campus license for Mathematica so the full program is available for you to use in many of the computer labs. The program has many features including symbolic calculations and mathematical graphics. I used Mathematica to produce the animations for vector-output functions given in previous notes.