# All Questions

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### Mean value theorem for integrals: how does the sign matter?

The mean value theorem is that $\int_a^b f(x)g(x) dx = f(\xi) \int_a^b g(x) dx$ for some $\xi \in [a,b]$ if g(x) does not change sign on $[a,b]$. However, I can't see why the sign matters. There is a ...
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### Integrals in Reverse

I'm asked what solid this integral represents(the integral is used to obtain the volume of a solid, we are given this). I see that since we have the 2pi, this is probably a volume obtained by using ...
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### Which functions could be discussed in a calculus course?

In a typical calculus course, students are exposed to many functions. Some have their origin in a field of study, while others are simply combinations of polynomials, trig functions, exponentials, and ...
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### How to solve $\lim_{n \to \infty} \frac{x^n}{1+x^n}$

How do you find the limit of $$\lim_{n \to \infty} \frac{x^n}{1+x^n}$$when n goes to infinity? I've no idea how to solve this type of questions. I think when $x=1$ it equals to a half and when $x$ ...
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### Biking uphill and downhill

During an interview, I was asked "If you can bike 20 mph uphill and 30mph downhill, and you have 1 hour to bike, how far or how long should you ride uphill before turning back." While a very ...
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### Is the number of subsequential limits of a sequence always countable

I know that a sequence can have many different subsequential limits but is the number of subsequential limits always countable? How do we know?
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### Tough trigonometric identity

Prove that $$\cot 13^o\cot 23^o \tan 31^o\tan35^o\tan41^o = \tan 75^o$$ I managed to rearrange it to the form $$\tan 31^o\tan 35^o\cot 49^o = \cot 15^o\tan 23^o\cot 77^o$$ and in this form we have ...