# Questions for first year students at the University. [closed]

I will help teach in a introductory class in mathematics for engineers in applied math at the University. Anyone have any good and cool favorite questions or know where I can find some? Anything is welcome

For the moment there is some questions like

• write: $1 + 2 + 3 + ... + 100$ as a sum

• show that $XY-YX = I$ has no solution for square matrices.

• $p | (n^2 - n)$ ($p$ is prime).

Thanks

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## closed as too broad by amWhy, Hans Lundmark, clark, Jack M, studiosusJun 4 '14 at 17:28

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The second one there is, imo, unsuitable for an introductory maths class...and even for engineers! Most probably they don't have the faintest idea what a matrix is (as most first yearers), why there is no commutativity and etc. – DonAntonio Jun 4 '14 at 16:08
Some pedantry - I would argue that $1+2+3+\dotsb+100$ already is written as a sum. Maybe you mean "using $\Sigma$-notation" or something like that? – Matthew Pressland Jun 4 '14 at 16:09
For lesson openers (what I assume you're looking for), I think asking why and in what ways $1 - \epsilon$ is a good approximation to $\frac{1}{1 + \epsilon}$ when $\epsilon$ is near $0$ makes more sense than what you have suggested. – Dave L. Renfro Jun 4 '14 at 16:21

Ask them to prove that the product of any $k$ consecutive integers is a multiple of $k!$. And then just show them how powerful the formula for $\binom{n}{k}$ they learned in high school really is.

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1. Ask them to prove that there are infinitely many primes. I always found that to be a beautiful result.

2. Prove that $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$. I guess engineers will be interested in methods to linearize equations.

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Try asking students to solve for $x$ in the equation $x+\sin(x)=0$.

The solution $x=0$ seems trivial, but it can only be solved by numerical methods, a course of which engineering majors will benefit greatly from.

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You don't need numerical methods to solve it. Guess-and-check combined with some analysis will get you there. – Antonio Vargas Jun 4 '14 at 16:45
What kind of analysis do you mean? I was told by my engineering numerical methods professor that this can only be solved with a numerical methods approach. – user140170 Jun 4 '14 at 16:49
Well knowledge that $\sin 0 = 0$ would lead me to notice that $x=0$ solves the equation. Now $\left.\frac{d}{dx}\right|_{x=0} (x+\sin x) > 0$, so for $x > 0$ small we know that $x + \sin x > 0$, and for $x < 0$ small we know that $x + \sin x < 0$. For all $x$ we have $\frac{d}{dx} (x+\sin x) \geq 0$, so we may conclude that $x + \sin x > 0$ for all $x > 0$ and $x + \sin x < 0$ for all $x < 0$. Hence $x = 0$ is the only real solution. QED. – Antonio Vargas Jun 4 '14 at 16:56