Math 152-02 Lab 4/15/09 Michael Pelsmajer and Allen FlavellWriting ProjectAs the writing project, the idea is for you to give careful explanations, using your own words.You might think of this as if you have been asked to help write a few pages that will become part of a calculus book. This gives you an idea of the level of detail to include, the level of care you have to put into it, and your audience (similar to your peers in this class). You won't use Maple code that much, except to perform calculations and to create illustrations. Having a few well-chosen pictures and calculations is immensely helpful to making a clear explanation.Some general guidelines:1. Explain what you are doing, where you are going with it, why each step is correct. Err on the side of being specific.2. Say what you mean, and mean what you say. Use correct terminology. 3. Don't pretend to understand if you don't. I will insist that you explain things in enough detail so that you won't be able to fake it. 4. Use correct notation. For example, "Pi/6" is not as good as "\317\200/6". Neither is this (Maple input format) except for Maple input. Fancy text like this is is okay, when there's a reason for it.5. Be neat and well-organized. Delete any extraneous text or calculations, including these guidelines!Grading Policy1. This project is worth 5% of the overall grade in this course. 2. There are only two grades possible: an "A" and an "E". That is, if you do not produce an "A" level project, then you will receive at most 50/100 for this assignment. 3. I (Pelsmajer) will grade the asssignments, giving you suggestions as to how to improve your work. You will have a chance to revise your work - more than one chance if necessary - so everyone should be able to get and "A" if they work at it. However, if you don't put enough effort into doing a revision, I will return it to you ungraded. (Minimum: You must address all of my comments; if you don't know how, see me.) Note that there's only a few weeks of the semester left, so you can't wait too long between revisions.Appoximating a series (infinite sum) by a partial sumBefore beginning, run this:restart:
with(plots):with(plottools):1. We can specify a sequence just like a function.a := n -> 7*n/(n^3+1);
a(1),a(2),a(3),a(4),a(5);
It's convenient to define partial sums as follows:A := n -> sum(a(i), i=1..n);A(1),A(2),A(3),A(4),A(5);1.a. Calculate these three values again, without "A". (You should use "a".)
1.b. The following calculations give numbers that you see above as well. Explain.A(4)-A(3);2. Here's how you can graph the sequence and series, where the rectangle's areas are the terms:display(seq(polygonplot([[j-1,a(j)],[j,a(j)],[j,0],[j-1,0]],color="SkyBlue",style=patch,view=[0..5,0..8]),j=1..5));
display(seq(polygonplot([[j-1,A(j)],[j,A(j)],[j,0],[j-1,0]],color="SkyBlue",style=patch,view=[0..5,0..8]),j=1..5));2.a. Improve part 1.b so that it uses the above plot to illustrate the explanation. In particular, whenever you mention a number or an expression in your explanation, you should describe what this corresponds to in the plot (e.g., "which is the area of the first and last rectangles in the lower plot", or whatever). You can combine plots using "display", as long as the "view" is the same on each plot.For example, look at this:display(
seq(polygonplot([[j-1,a(j)],[j,a(j)],[j,0],[j-1,0]],color="SkyBlue",style=patch,view=[0..15,0..8]),j=1..4),
seq(polygonplot([[j-1,a(j)],[j,a(j)],[j,0],[j-1,0]],color="LightGreen",style=patch,view=[0..15,0..8]),j=5..15),
plot(a,0..15,view=[0..15,0..8])
);
The infinite sum is the area of all the boxes, assuming that the green boxes continue to infinity.
2.b. In the above plot, what is the total area of the blue boxes? Express the answer as a sum, and also evaluate the sum to get a number.2.c. In the above plot, what is the total area of the green boxes (assuming that they continue to infinity)? Express the answer as a sum, only.
If we use the partial sum A(4) to approximate the infinite sum, this is the error of the approximation.While it may not be clear how to find the total area of the green boxes, the above plot suggests a way to find a number that bounds the error from above. (This is the same as the technique described in class a couple days ago.)2.d. Find a number X such that the error is less than X. Explain, referring to the above plot.3. Consider the infinite sum of this sequence, from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk= to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSgmaW5maW47RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlNGOQ==.b := n -> (-1)^n/n;3.a. Copy the code from the first two plots in part 2, and modify it to show the first eight terms of the sequence, and the first eight partial sums.3.b. The fourth partial sum approximates the infinite sum.
How close is it to the truth?
(Find a number that is greater to or equal to the error.)3.c. Explain part 3.b, referring to the plots.3.d. Calculate a partial sum 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 that is within 0.001 of the infinite sum.
3.e. Explain your work, referring to the plots.
(Tip: Your first job is to choose N properly.)