18
votes
Calculus problems arising from real research problems
Modeling Basketball Free Throws
by Joerg M. Gablonsky and Andrew S. I. D. Lang, SIAM Review vol. 47, no. 4, pp. 775-798, 2005, https://doi.org/10.1137/S0036144598339555
Abstract
This paper presents a ...
- 10.8k
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:
$$
\Gamma(z) = \int_0^\infty t^{z-1}e^{-...
- 171
17
votes
Accepted
Good examples of functions defined as definite integrals of elementary functions?
It seems that the key term here may be the somewhat non-specific-sounding special functions.
By googling for a few examples (Erf, Si, Li) I came across a Table of Special Functions and, on the Lists ...
- 18.4k
16
votes
How to use false theorems or proofs?
I would not recommend putting false proofs onto the board unless you immediately (within the same class period) point out their falsity, and make an assignment to find it.
For smaller mistakes that I ...
- 20.1k
13
votes
Calculus problems arising from real research problems
Consider Betz's law, which was worked out around $100$ years ago by three different scientists independently (in Germany, the UK, and Russia). It determines the maximum power that can be extracted ...
- 3,258
12
votes
Good examples of functions defined as definite integrals of elementary functions?
My first take is
$$
\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 ...
- 2,982
11
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$ ...
- 29.5k
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 ...
- 2,601
10
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 ...
- 29.5k
10
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 ...
- 441
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 ...
- 1,715
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 ...
- 24.3k
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 ...
- 21.2k
8
votes
Calculus problems arising from real research problems
Consider trying to find a curve that will let you link up two straight tracks smoothly, such as two straight parts of a roller coaster track. By "smoothly" meeting, we want the derivatives ...
- 3,258
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 ...
- 24.3k
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 ...
- 5,345
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 ...
- 20.1k
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 ...
- 9,079
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 ...
- 10.7k
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 ...
- 24.3k
6
votes
Calculus problems arising from real research problems
Consider the reflective property of a parabola: rays coming into a parabola parallel to its axis of symmetric will all bounce off the parabola (angle of incidence equals angle of reflection) and meet ...
- 3,258
6
votes
Calculus problems arising from real research problems
There should be a lot of interesting examples in ecology. The first one that comes to mind for me is population dynamics: Wikipedia Link
The simplest model is exponential growth:
$$ dN/dt = rN $$
...
- 211
5
votes
Is there a framework to study the mathematical competence in problem-posing?
The answer to your question is yes. Check out the recent textbook:
Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing: From research to effective practice. ...
- 18.4k
5
votes
How does Project Euler come up with such good problems so rapidly?
You can read on PE's website that questions are either
Classic questions
Questions that are derived from some theorem in some way
Other questions are believed to be original
There are now over 600 ...
- 1,007
5
votes
Mathematical Task with Various Solutions
Here is one I enjoyed from middle school. This was a project: I think we had a whole week to experiment, and discuss, and come up with a solution.
Consider a rectangle a 231 by 84 rectangle which ...
- 24.3k
5
votes
Can number theory help me create equations with nice solutions?
Begin with factored quadratic $(u-r_1)(u-r_2)=0$ so $u^2-(r_1+r_2)u+r_1r_2=0$. Choose one root positive, say $r_1>0$ whereas $r_2<0$ then we solve for $u$ using positive root,
$$ u = \sqrt{(r_1+...
- 10.7k
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^...
- 5,182
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 ...
- 1,018
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 ...
- 7,500
5
votes
Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?
This might not be what you want, but sometimes a backwards problem appears more difficult than it seems and the difficulty is immediately removed when you know and apply the definition of the ...
- 10.7k
Only top scored, non community-wiki answers of a minimum length are eligible