Math 291 Lab 7 Due Monday March 12, 2018

Instructions: Turn in your lab by emailing it to fagerstrom@mnstate.edu. You should email both your raw TEX (.tex) file and your compiled document (in .dvi, .ps, or .pdf form).

You will be graded on both your raw TEX code and the accuracy of your compiled document. Name your file in the format of JaneDoeLab2.tex Use a textwidth of 7.5 inches and a textheight of 9.5 inches. Set your top margin and oddside margin appropriately so that it fits nicely on the page when built.

1. Create a command in the preamble called \di that is a shortcut for \displaystyle. Then use that command to typeset the following sentence:

The Rieman integral is often defined in Calculus books as lim_{n→∞} ∑_{i=1}^{n} f (x_{i}) (Δx_{i}) = ∫_{a}^{b} f(x) dx.

2. Create a command that, when given a number and a function of x, will typeset the limit of the function as x approaches the number. Then use your command to typset the following. Make sure that your command can be called either in math mode or out of math mode, and acts the same regardless. When using the command to typeset the limits below, call your command within math mode at least once and outside of math mode at least once.

(a) lim_{x→3} (5x^{2} − 7x + 2)
(b) lim_{x→5} (\frac{x^{2} − 25}{x^{2} − 6x + 5})
(c) lim_{x→∞} (tan^{−1}(x))
(d) lim_{x→ \frac{π}{2}} (tan(x))

3. Create another command similar to that in #2 above, that, when given a number and a function of x, will typeset the limit of the function as x approaches the number, but this time make the number an optional argument with ∞ as the default value. Use it to typeset the following.

(a) lim_{x→∞} (\frac{x+3}{x+2})
(b) lim_{x→−2^{+}} (\frac{x + 3}{x + 2})

4. Define another command with at least three inputs and demonstrate its use at least two times. What it does and what its required inputs are are up to you.

5. Load the amsthm package and create an enumerated definition environment, an enumerated axiom environment, an enumerated note environment, and an enumerated theorem environment. Then use them to typeset each of the following. (Note that you may need to reset the counter for some of these.) [Source: Euclid’s Elements, by Thomas L. Heath]

(a)

Definition 1. A point is that which has no part.

Definition 2. A line is breadthless length. Definition 3. The extremities of a line are points. Definition 4. A straight line is a line which lies evenly with the points on itself. Definition 5. A surface is that which has length and breadth only.

(b) Axiom 1. Things which are equal to the same thing are also equal to one another. Axiom 2. If equals be added to equals, the wholes are equal. Axiom 3. If equals be subtracted from equals, the remainders are equal. Axiom 4. Things which coincide with one another are equal to one another. Axiom 5. The whole is greater than the part.

(c) Note 8.1. In Euclid’s Elements, a distinction is made between postulates and axioms. Note 8.2. In Heath’s translation, axioms are called common notions. Note 8.3 (My opinion). Heath’s translation also has frequent and long footnotes that refer back to the original Greek, and are often more of interest to someone who is interested in the nuances of a language and its translation than they are in the math being discussed.

(d) Theorem 1. There are infinitely many prime numbers.

Proof. Assume that there are finitely many prime numbers, given as p_{1}, p_{2}, p_{3}, …, p_{n} for some natural number n. Note that since 1 is not prime, 1 ≠ p_{n_{k}} for any k. Consider the number a = (p_{1} · p_{2} · p_{3} ⋯ p_{n}) + 1. This is one more than a multiple of each of our prime numbers, and since none of them are equal to 1, is therefore not a multiple of any of our prime numbers. So a is a number that is not divisible by any prime less than it, so it would be prime. But that contradicts our assumption that there are a finite number of primes. Therefore, there are an infinite number of primes.