The student changed something which was indeterminate ($\infty-\infty$) into something which was not ($\infty\cdot \infty$). How does that not merit a perfect score? Changing indeterminate expressions ...

https://www.theguardian.com/news/datablog/2013/may/31/times-tables-hardest-easiest-children There are links to a dataset in the article. As far as I can tell, this isn't a formal study: But some new ...

Coming from the perspective of someone who reteaches this material at the college level, neither the graph perspective nor the list of properties perspective really translate into a deep understanding ...

As an answer so that I can paste in a picture: I think that the problem with the one on the left is that it is possibly "good enough" that someone might think that it is to scale and that ...

What's wrong with this?: $$\frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial t} \frac{dt}{dt}$$ with $\frac{dt}{dt}... View answer Accepted answer 13 votes How about$\frac{1}{3}x^3-\frac{1}{4}x+\frac{1}{12}+\epsilon$? When$\epsilon=0$, this cubic has two distinct roots: a single root at$-1$and a double root at$\frac{1}{2}$. If we let$\epsilon>0$... View answer 11 votes Take the function$f(x)$and write the functions$\frac{1}{2}(f(x)-f(-x))$and$\frac{1}{2}(f(x)+f(-x))$. They are odd and even, respectively, and their sum is$f(x)$. Nothing fancy required so I ... View answer 10 votes Presumably, you are trying to reduce the chance that someone will get a correct answer by guessing. In that case, I think that 10 options are more than you need. Suppose that we have an$N$question ... View answer 10 votes You can make this$\epsilon/\delta$proof easy enough that an interested student should find the argument believable without experience with$\epsilon/\delta$proofs. Divide both the numerator and ... View answer 9 votes Questions about infinity are one way to go. e.g. 'Are there more natural numbers or even natural numbers?' Intuition says there are more natural numbers ($\mathbb{N}$) than even natural numbers ($2\...

Combining your last two methods: $3.9*7.5 = 4*8 - 0.1*8 - 4 * 0.5 + 0.1*0.5$, which can be thought of as computing the area of the big rectangle below, cutting off the two extra strips along the edge, ...

You should expect numerical integration to be "easier"  than symbolic integration because it is answering a fundamentally weaker question. That is, symbolic integration, if you can do it, gives you ...

I almost certainly would not introduce that integral to freshmen. I probably wouldn't introduce it to sophomores or juniors either; not without some particular application in mind. (I don't know of ...

Many linkages have this sort of behavior and are, perhaps, relevant to a mechanical engineer. For example, you could analyse a bicycle suspension --- a type of four-bar linkage. https://en.wikipedia....

I think that you probably need to give them some hands-on experience before any of it will sink in. I assume that they are CS students? After you give them the definition and a few examples, you could ...

How about $\mathbb{Z}[i]$ together with conjugation, thought of as a ring element? Conjugation and multiplication by $i$ do not commute.

It is worth considering that, if the ages would have been recorded as integers, rather than intervals, the assumption would have still been wrong in a similar but less obvious way. That is, a 25 year ...

There are many techniques of integration. Some of them, like integration by parts, are important theoretically. Integration by parts shows up in the derivation of the Euler-Lagrange equations in ...

You are asking a lot of them, even though each little bit of the problem isn't especially difficult and was probably covered in a class that they have taken. You need to remember a bit of trig/...

May I suggest giving them a bit of header code? i.e. Instead of using from math import sin try import math def sin(theta): return math.sin(math.radians(theta))

Per request, I am expanding my comment into an answer. Given a quadratic equation, $ax^2+bx+c=0$, the quadratic formula $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ will only involve complex numbers ...

I suspect that if $(\sin x)^x$ shows up in any physical situation, it will be highly specific and not really a natural or worthwhile thing for a calculus student to spend their time on. Perhaps ...

If your students really think that the limit is something that is realized after infinitely many steps, I think that you are doing them a disservice trying to dissuade them. Taking the limit is ...

It is worth noting that your definition of directional derivative allows for non-continuous functions to be differentiable (in all directions). It isn't enough to look at just straight-line paths for ...

Since you are doing this geometrically, I would suggest a different tact on your treatment of normal subgroups: You can think of them as stabilizers of certain extra decorations. For example, in a ...

I think that you may be selling short the value of a counterexample! They are quite useful for making sure that you have not proven too much. i.e. When you have a plausible but only semi-formal ...

Sometimes Stewart picks exactly the example I would have picked, even if I hadn't been prepping from his book. It is usually when the natural example is relatively simple. I wouldn't worry too much if ...

Honestly, both seem pretty bad and don't prove the statement in an obvious way. For example, there is confusion betwen irreps and the representations of individual elements. You can make the $G$-...