@AliCaglayan $SX$ and $\Sigma X$ are homotopy equivalent if $X$ is a CW complex. You should read Ch 0 of Hatcher, really, before jumping ahead to Ch 4.
There's no way to make sense of $n^{\bf v}$ where $\bf v$ is a vector in $\Bbb R^3$. Things that work because of complex analysis are very special to $\Bbb C$.
Once it knows you know how to do a type of problem correctly, say, 3 times, it should move on. Kids who mess it up should get to repeat it another 12 times, but, really ...
Oh, pipes. It's some convention that those engineers made up. It has nothing to do with usual uses of the word. In- stands for inside. Ob- usually suggests outside or beyond.
So they're thinking about starting at the bottom of the (inside of the) pipe and measuring from in to ob.
High school (and many college) teachers seem to mess up the actual definition of continuity. They think the function $f(x)=1/x$ is NOT continuous. It's perfectly continuous.
@MeowMix It was alright, I did not learn integrals, but a typical "lesson" looked like a few example questions which you had to infer their meaning from. Here's an example of something I'd have to look at to learn projections: imgur.com/a/nFXsR
@MeowMix So it was mere computation and rarely ever proofs. I did poorly in Advanced Functions (93% avg.) because they never explained or even showed a picture of the unit circle. In a sense I am self taught, never having heard a lecture.
Here's another common misconception that's slightly easier. If I have a graph of a function $f : [-1, 1] \to [-1, 1]$, and I see a "kink" at the graph (like a bend, as in $|x|$) I can immediately say $f$ is not differentiable there. But if the graph has no "kinks", is it necessarily differentiable? (ping @Meow).
@MeowMix I bought a college calc book to learn them, I'm not sure why my HS calc course didn't teach them- I think they should be taught with derivatives.
@TedShifrin I was thinking about the derivative of a function which is the instantaneous rate of change, and the integral of a function which is the area under a curve.. I have never even heard of this antiderivative!
^^ @Ted If there's nothing else I fault the IB with, it's that they literally just define the integral as the primitive, and say a definite integral is evaluating it at the endpoints
Demonark: On the other hand, AP teachers are coached on exactly what to tell their students to write in the free response questions. In particular, on the Mean Value Theorem problem, don't bother wasting your time explaining that the function satisfies the hypotheses of the theorem. Just write the result.
I would take off serious points in college for that.
@Dodsy: I would want to see some steps, probably. Like what form of 1 did you multiply by? I don't want you to write out $(a+b)(a-b) = a^2-b^2$, I guess, but ...
