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It's a removable discontinuity. That expression is 0/0 at x=2, which you cannot make sense of (so 2 is not even in the domain of the function defined by that expression). However defining f(2)=-1/3 gives you a smooth function on R \ {5}.

Being orthogonal to the plane is the same as being parallel to the normal of the plane (our ambient space is three dimensional, so normals to a plane are uniquely defined up to scalar multiplication), so just apply a) using the normal to the plane.

Re: University Statistics Discussion Marathon Think about rolling a dice and doubling the outcome vs rolling two dice and summing them. The first RV only takes even values and takes each value in {2,4,...10,12} with equal likelihood. The second RV can take any value in {2,3,...,11,12} and...

Re: University Statistics Discussion Marathon As I just said, 4X and 3Y are irrelevant to the problem. We have \textrm{Var}(\textrm{Total Mass})=\textrm{Var}\left(\sum_{j=1}^4 X_j+ \sum_{j=1}^3 Y_j\right)= 4\cdot\textrm{Var}(X)+3\cdot\textrm{Var}(Y)\\ \\= 4\cdot 144+3\cdot 64 = 768.\\ \\...

Re: University Statistics Discussion Marathon Re: my comment on independence. We can see the extreme case of how this method breaks down if there are exactly 4 men and 3 women in the whole office. Then there is only one way to choose the occupants of the elevator, and so the total mass will...

Re: University Statistics Discussion Marathon Where are you getting 4X and 3Y from? You are not multiplying single R.V's by a constant, you are summing independent identically distributed R.V's. (These things aren't actually independent, but this is a good approximation if the number of...

Who knows his motivations? But it is clear from the outset that frequently reprinting a book that is used by many university courses will earn you a lot of money. The actual changes from version to version are very slight from what I have seen. There is even a separately printed version with...

Yes, reprinting a large and expensive book so frequently with trivial additions is one way to get rich.

That's pretty subjective. I wouldn't recommend Stewart for anyone planning on majoring in mathematics (or certain areas of physics). The emphasis is on using calculus as a tool to solve problems. (Generally fairly routine applications of the included theorems, but there are some ones that can...

Spivak is comfortably the best book I have seen to learn single-variable calculus properly for the first time, but it's probably not that suitable a thing to work on in HS. (His "Calculus on Manifolds" is also a gem imo, and his differential geometry series is classic although I have not read...

Re: First Year Uni Calculus Marathon That's more like it :).

Re: First Year Uni Calculus Marathon I wouldn't say they are trivial. (But they are easy consequences of specific theorems. A correct proof should be very short.) You can't rely on intuition too much here as funny things can happen in analysis. I can tell you that just changing your wording...

Re: First Year Uni Calculus Marathon What do you mean by saying f must be constant for every c? What does it mean to be constant at a point?

Re: First Year Uni Calculus Marathon And some much easier ones for students who don't want to do the above questions: E1. Prove that if a function f: (a,b) -> R is differentiable and f'(x) is non-negative in this interval, then f is non-decreasing in this interval. E2. Prove that if a...

Re: First Year Uni Calculus Marathon It's just providing the function for my question B2 (2nd question in second post) which is not too much of a spoiler, but now someone should definitely be able to prove that. B1 is a bit trickier, but for any student who wants to assume Rolle's and doesn't...

Re: First Year Uni Calculus Marathon Note: To be clear on nomenclature, when I say a function is differentiable on [a,b], I am being lazy. I actually mean that f is a continuous function on [a,b] that is differentiable on (a,b). No assumptions are made about the existence of one-sided...

Re: First Year Uni Calculus Marathon It is slightly tricky to spot, but the core idea is the same as in the proof of the mean value theorem. So perhaps I should make this a leadup exercise. The mean value theorem asserts we can find a c in (a,b) with f'(c)=(f(b)-f(a))/(b-a) if f is a...

Re: First Year Uni Calculus Marathon $A1. Suppose $f$ and $g$ are differentiable functions on the interval $[a,b]$ such that $g(a)\neq g(b)$. Prove that there exists $c\in (a,b)$ such that\\ \\ $\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}.$\\ \\ Hint: The main ingredient is applying the...

Re: First Year Uni Calculus Marathon Anything smaller than the first positive solution to \sec\left(\frac{\pi\epsilon}{2(1+\epsilon)}\right)=1+\epsilon^2. It's about 0.117041.

Re: First Year Uni Calculus Marathon This is only true for sufficiently small epsilon.