17 votes

Good examples of functions defined as definite integrals of elementary functions?

The gamma function is very useful in counting problems (among others) and is seen as an extension of the factorial function into the reals. It is defined as: $$ \Gamma(z) = \int_0^\infty t^{z-1}e^{-...
  • 171
17 votes
Accepted

Good examples of functions defined as definite integrals of elementary functions?

It seems that the key term here may be the somewhat non-specific-sounding special functions. By googling for a few examples (Erf, Si, Li) I came across a Table of Special Functions and, on the Lists ...
17 votes

How to use false theorems or proofs?

I would not recommend putting false proofs onto the board unless you immediately (within the same class period) point out their falsity, and make an assignment to find it. For smaller mistakes that I ...
  • 18.3k
14 votes

Good examples of Lagrange multiplier problems

A place that Lagrange Multipliers comes up is in the proof of the real spectral theorem. Namely, let $A$ be a symmetric matrix. Define $f: \mathbb{R}^n \to \mathbb{R}$ by $f(v) = v^\top A v$. If ...
12 votes

Good examples of functions defined as definite integrals of elementary functions?

My first take is $$ \ln(x) = \int_1^x\frac1t dt. $$ Granted, some texts introduce the natural log of the inverse of $\exp$ but other texts define $\ln$ as above and the $\exp$ as the inverse. If I ...
  • 2,962
11 votes

Challenge questions for extremely bright kids

Here are some suggestions for problem sources in English. Some of them are appropriate for very bright students studying geometry or Algebra II, but might nonetheless prove too difficult for students ...
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11 votes

Name the heuristic: exploiting the legitimacy of the questioner

The heuristic described here is one manifestation of what Polya (1945) and others thereafter refer to as trying a special case. I do not know of a more specific term for the context that you have put ...
11 votes

Good examples of functions defined as definite integrals of elementary functions?

The function $\displaystyle \text{Li}(x) = \int_2^x \frac{1}{\log t} \,\, dt$ comes up in the study of the distribution of primes. Specifically, the number of prime numbers less than $x$ is ...
10 votes

How to use false theorems or proofs?

Wondering Why are induction proofs so challenging for students?, I thought of trying this as a possible assignment after introducing induction: Find the flaw in this induction proof. Claim: $3n = 0$ ...
9 votes

Are there any benefits to having an entire course's homework problems available from day one?

I did this for an introductory calculus course at a US state university. My reason was that I wanted to assign more homework than my colleagues typically do. On the first day of class I warned ...
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9 votes

How can you elicit the $\log x = {\log} \cdot x$ error?

Writing from a software engineer's point of view, it's a fact that mathematics uses a notation that's highly ambiguous. If you don't know that $log$ is used to denote some logarithm function, then ...
8 votes

Is there a framework to study the mathematical competence in problem-posing?

Nice question! Let me add one reference to your list: Silver, Edward A. "On mathematical problem posing." For the learning of mathematics (1994): 14(1) 19-28. (PDF download link.) Silver cites ...
8 votes

Good examples of functions defined as definite integrals of elementary functions?

No one has still mentioned Fresnel functions: $S(x)=\int_0^x \sin(t^2)dt$ and $C(x)= \int_0^x \cos(t^2)dt$ They are (of course) very relevant in signal analysis and in studying diffraction. What is ...
8 votes

Good Examples of Questions to have Students Ponder Over Without Paper

A possibility, requiring one definition: What is a tiling of the plane with an infinite supply of congruent copies of a single tile (technically, a monohedral tiling). This can go as deep as you'd ...
8 votes

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

Here are some example questions. The graph of the function $f$ is given above. Evaluate the following limits. If the limit is infinite, write $\infty$ or $-\infty$ as appropriate. If the limit does ...
8 votes

How can you elicit the $\log x = {\log} \cdot x$ error?

Ask the student to critique this work: Solve for $x$: $\sqrt{x} = 3$ Easy: $x = \frac{3}{\sqrt{\phantom{x}}}$. I have tried this a small number of times, and it has worked so far. The students ...
7 votes

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

I'll argue that this will be likely not feasible for a test question. There's several points in the calculus progression where there's a "hard bottleneck" of some sort, but once you get past ...
6 votes

How to use false theorems or proofs?

There is a common type of exercise in French's mathematical secondary educators curriculum: give the student a (true or alleged) student's answer to a typical problem in your class, and ask them to ...
6 votes

Good examples of Lagrange multiplier problems

A question with two constraints might make the method seem preferable to finding a parameterization (which I assume is the "easier" technique you refer too in the OP). For example, maximizing $f(x,y,...
6 votes

Challenge questions for extremely bright kids

I work with gifted elementary school students, but one of my favorite sites, nrich has challenging problems that you could use for older gifted students. Try looking at secondary problems for stages ...
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6 votes

Are there any benefits to having an entire course's homework problems available from day one?

I teach at community college. I often publish the homework problems at the beginning of the semester, listed by section. I have never had a student work ahead (that I know of). And I have had a few ...
  • 18.3k
6 votes

Good examples of functions defined as definite integrals of elementary functions?

A (cata)caustic is formed by the reflection of light, such as the cardioid in this coffee cup: G.B. Airy showed in [Airy, "On the intensity of light in the neighbourhood of a caustic," Transactions ...
  • 4,689
6 votes
Accepted

Interesting but very easy epsilon-delta problems?

I suggest using rational functions. Students are used to evaluating limits of rational functions because such examples are prevalent in most calculus courses. Moreover, I think the work required to ...
6 votes

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

Give a function together with some properties of the function, but do not give a formula, and ask for the derivative. An example: Let $f: \mathbb{R} \to \mathbb{R}$ be a function which enjoys the ...
5 votes

Is there a framework to study the mathematical competence in problem-posing?

The answer to your question is yes. Check out the recent textbook: Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing: From research to effective practice. ...
5 votes

Mathematical Task with Various Solutions

Here is one I enjoyed from middle school. This was a project: I think we had a whole week to experiment, and discuss, and come up with a solution. Consider a rectangle a 231 by 84 rectangle which ...
5 votes

How does Project Euler come up with such good problems so rapidly?

You can read on PE's website that questions are either Classic questions Questions that are derived from some theorem in some way Other questions are believed to be original There are now over 600 ...
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5 votes

Good Examples of Questions to have Students Ponder Over Without Paper

Perhaps logic puzzles would work in this case. Some classic examples are: You're traveling along a road and arrive at a fork. Two guides are posted, but one always lies and the other always tells ...
  • 6,579
5 votes

Question about the process of creation of problems and exercises in Mathematics

I will take a stab at an answer though clarifying what level of education we are talking about would help. I have never created problems for things like Qual exams (essentially masters exams) so I ...
  • 1,008
5 votes
Accepted

Can number theory help me create equations with nice solutions?

I think that the number theory in this case is pretty light. Let's suppose that we want some integer solution for $x$ and that $a,b,c$ are also integers. Then $x+b$ must be a perfect square, so let $N^...
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