# Tag Info

Accepted

### Whether to tell students how difficult (you think) a problem is

In a setting where students aren't working from a book with these labels on questions, is it worthwhile for the instructor to indicate to students where the work they are asking them to do falls on a ...
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### 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

### 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 ...
• 3,516

### 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 ...
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### 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 ...

### 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 ...
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### 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 ...
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### 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 ...
• 3,516

### Whether to tell students how difficult (you think) a problem is

A Frame Challenge I think that the premise of the question is slightly flawed: I do not think that the distinction between "exercises" and "problems" is one of difficultyâ€”an ...
• 8,225

### 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 ...
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### Whether to tell students how difficult (you think) a problem is

Donald Knuth certainly thought so: he graded the exercises in his monumental work The Art Of Computer Programming from 00 â€˜An extremely easy exercise that can be answered immediately if the material ...
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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 ...
• 10.8k

### 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 ...
• 20.8k

### 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 ...
• 5,630

### 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 ...
• 25.6k

### 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 ...
• 3,516

### 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: $$dN/dt = rN$$ ...
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### What to do with "wild goose chase" or "quantum leap"-types of incorrect solutions when you ask students to prove/show something?

It's quite likely to be a consequence of the belief that they have to answer the question. When they can't work it out, when they've gone around in circles and got lost, but still they have to give an ...

### 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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### 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 ...
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### 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. ...
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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+...