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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, studiosus Jun 4 at 17:28

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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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 at 16:08
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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? –  Matt Pressland Jun 4 at 16:09
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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 at 16:21

3 Answers 3

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 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 at 16:49
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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 at 16:56

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