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1

If the curve $x = \cos t + \cos 2t$, $y=\sin t + \sin 2t$ passes through the point $(-1,1)$, then there must be some value of $t$ such that $-1 = \cos t + \cos 2t$ and $1 = \sin t + \sin 2t$. Your first task is to find that value of $t$. Once you have that, you can evaluate the derivative (which you have correctly calculated) at that value of $t$, which ...


2

I like the pictures from chapter 0 of Hatcher for that purpose. And also Louis Kauffman made some nice pictures for the Fox calculus. And Thurston's 3-manifolds book has good pictures as well. "Normal maths works with squares, triangles, perfect circles; it counts beans. In topology we're freer to work with more interesting shapes." (cue picture from ...


1

This may be useful for a somewhat different perspective. My personal feeling is that it is (I would say) of foremost importance that you become comfortable and nurture a proficiency with proofs. The topics you mention, when studied in the depth I would think you want to explore all have a "proof component." While this point could be argued, typically in ...


2

Yes, there are all sorts of problems that mathematicians can and do solve for a living, either in an academic setting or in industry. For example, you might look at What Kind of Problems Might I Work On?.


0

You should take a look at the field called Operations Research. It combines many interesting elements of both mathematics and computer science. The problem like the one you linked to reminds me of some problems I solved working as a Research Scientist in industry. If the corporate environment does not suit you, then there are research labs and professorships ...


0

That octal is wrong. This should be obvious; $0.657_8$ is clearly somewhere between 0 and 1. In any case, the true method is this: $$0.657_8 = \frac{6}{8} + \frac{5}{64} + \frac{7}{512} = \frac{431}{512} = 0.841796875$$


0

$0.657$ octal should be $$\frac{1}{8^1}\cdot6+\frac{1}{8^2}\cdot5+\frac{1}{8^3}\cdot7=0.841796875$$


2

Two books; I guess yesterday, somebody asked again about the question of Regiomantus, Finding the widest angle to shoot a soccer ball from the sideline using optimization!! I knew of this from some book i had 40 years ago, but it is in two books that can be purchased (or borrowed) , Heinrich Dorrie (translated) 100 Great Problems of Elementary Mathematics ...


1

This is another rectangular problem, but I like it because the student can take one of at least 3 approaches. A spider in on the floor in the north-west corner of a room. He would like to crawl to the south-east corner on the ceiling. What is the path that minimizes the distance that he has to crawl? Possible approaches When the spider reaches the edge ...


1

The examples you gave my themselves are elementary but good examples already. However, if you want to expand even more, you have many options. I'll name the ones I can think of at the moment. Economics has a lot of great maximization problems at various levels, especially microeconomics. Physics, chemistry, and biology use optimization problems a lot. An ...


1

Answer: Update 1: I thought about it and created this EXCEL WORKSHEET which I thought might be helpful. This seems to be the right way to do it. Earlier method was wrong. You will see four images, 1) When transaction cost = 0%, Option not to trade the interest and Option to reinvest are the same. 2) When transaction cost = 3% 3)when transaction cost is ...


0

To address your question about cancelling the $x^2$ and the $x$: In a fraction you can't cancel 'terms' (things added or subtracted). This can be seen with a numerical example: Let's look at $\frac{3+4}{6+7}$. Certainly just by doing the addition, this equals $\frac7{13}$. On the other hand, if you tried to cancel the $3$ with the $6$, you get ...


0

One of my most memorable moments in mathematics was when I was attempting to prove the formula for the volume of a sphere on my own. I hadn't been taught calculus yet and had no idea about it, but I was convinced I could solve the problem. I used an infinite amount of small disks and added their volume ( essentially the limit of a riemann sum, an integral, ...


4

At one point, I got frustrated with always lacking the math to understand the things I read about and made a cheeky poster expressing my frustration. With that format, I tried my best to cluster more related topics together. (I could add that thinking about the possible hierarchy made me ask this question. Later I started an online notebook, working out ...


1

The raw fields in your number are: $s = 0$ $e = 101010_2 = 42_{10}$ $m = 111110000_2$ Without knowing the specific format used, we can't be certain what your bit string represents. But if the IEEE conventions are followed, the exponent bias is 31, and the mantissa has an implied leading 1. In this case, the adjusted values are: $e' = 42_{10}-31_{10} = ...


0

Beginner List item List item ... Intermediate List item List item ... Advanced List item List item ...


1

Tupper's self-referential formula is a self-referential formula defined by Jeff Tupper that, when graphed in two dimensions, can visually reproduce the formula itself. It is used in various math and computer science courses as an exercise in graphing formulae. The formula was first published in his 2001 SIGGRAPH paper that discusses methods related to the ...


0

The $\sqrt{-1}$, complex analysis, and how real world problems could be solved "by these objects which don't exist."


2

Mathematics courses geared towards engineers are (generally) more applied, focusing on applications rather than theory. For example, at UBC, the Linear Algebra course for engineers concerns mostly solving systems of equations, geometry, determinants, eigenvalues/eigenvectors, and applications of the aforementioned topics. Conversely, the course for ...


0

it all depends. At the start of your undergrad, most math majors and engineers will take calculus 1 and 2 (the same one). However sometimes math majors may take the honours versions of those. By multivariable calc it may be a bit different with engineers focusing on vectors. and ofcourse engineers and math majors alike will take courses on ODE's and lots of ...


1

You could explain it in terms of rotation. Suppose you have a point $(a,b)$ in the plane. If you multiply it by a real number $s$ then you scale the length but preserve the direction, $(sa,sb)$. What if we want to multiply two vectors together in a way that multiplies their lengths together and adds their polar angles? This would allow us to rotate points ...


-2

The complex number (a, b) should actually be a + bi; where $i^2 = -1$. Therefore, $(a,b) * (c,d) = … = (ac-bd) + (ad+bc)i = (ac-bd, ad+bc)$.


0

Definitely review! Think of this as landing a job. One you look forward to advancing your career and/or happiness, to boot. But the employer wants you to be good at something you have skill at but haven't done in a while. Do you just play up the past and wing it from there, or do you re-familiarize yourself with that skill set as much as you can? The latter, ...


1

I was undergrad math-physics and did compsci on the side. Afterwards i worked a CS job for 2 years and then applied to a math masters program with a similar goal. Having now completed it, I can say that it should be doable with little prior study (I never had analysis in undergrad) so I never even prepared for it in grad school. Though I prepared nothing ...


0

Thank you for taking the time to reply, Dan. I found a satisfactory solution along the lines of your second response. A "real world" application of rational functions is the "Thin Lens Equation" which relates focal length, object distance, and real image distance. It is as practical as it gets. Cameras, eyeballs, magnifying glasses, etc all operate using ...


3

The founder of the Institute for Advanced Study, A. Flexner, once published a paper called "The usefulness of useless knowledge" http://library.ias.edu/files/UsefulnessHarpers.pdf. This paper justifies all theoretical sciences. I get a kick out of reading the article. I think about another one, "The Spirit and the Uses of the Mathematical Sciences", ...


0

For a 2-dimensional curve, try Paul Ma's Heart Equation $$(y - a|x|^b)^2 + (cx)^2 = d$$ Where   a b c d   are parameters that you can set. In my light-hearted blog on heart curves ==> http://onemanadreaming.blogspot.com.au/2014/08/mathematical-equation-of-love-heart.html There are some sample values for  a b c d.  You can try out other ...


3

Stumbled across this just now. Not sure of the source to give the appropriate credit. If someone knows where it is from please add a comment.


5

Mathematics - Form and Function by Saunders Mac Lane is a great book which tries to give some kind of "big picture". As a bonus, some of the chapters also contain "map-like" sketches of various areas of mathematics (IIRC), but that's not why I'm recommending the book.


5

Have you ever seen a river delta? At the root there is a river (logic) which splits into several streams (analysis, topology (geometry), algebra etc.). But these streams merge and split (algebraic geometry, differential topology, etc.). There isn't a single sequence of reliances that you could follow. It isn't necessary to go into the depths of logic for ...


11

Here is one map of mathematics that's been getting some attention lately:


0

It has to do with the geometry of the hyperbola. Specifically, that the function $x \mapsto \frac{1}{x}$ maps geometric sequences to geometric sequences. To demonstrate, let A denote the region above the abscissa and below the hyperbola $yx = 1$ and between $x = 1$ and $x = 2$. And similarly let B denote the region above the abscissa and below the hyperbola ...


2

A very satisfying visualization of the area of a circle.


0

In mathematics, a statement is either an axiom provable by using existing axioms contradictory to existing axioms (commonly called wrong) undecidable by using existing axioms Terms like "obviously true" and "self-evident" have caused severe problems in the history of mathematics and philosophy as a whole. Thus, they lead on a dangerous path and should ...


1

The object of the sentence $3<x$ is 3 but really the object we are interested in is $x$ so $x>3$ is preferable. Also, graphing $3<x$ is conceptually more difficult at that level and so the preference is $x>3$. A frequent error is to graph the wrong solution, showing that the student doesn't understand the answer. His answer is correct, just ...


1

$3<x \iff x>3$. That is, the statements $3<x$ and $x>3$ are equivalent. Your son's teacher is wrong. I personally prefer to put the variable first, but, by no means is not doing so wrong. It's just a matter of convention. The reason for the teacher's wanting to put the variable first is that it's more natural to express the solution, $x$, ...


1

For every chapter after the lecture you should sit down and try to attempt the problem sets. Usually there will be an easier problem that the instructor has gone through in class that will lead you through most of the exercises. When you get stuck go back and reread. Find examples in the text chapters, which you also should read and work through. ...


3

Many schools offer at least some free academic support such as tutoring or walk-in help sessions for their classes. You might investigate that possibility at your school. (It sounds like you don't want to 'invest' in a tutor.) As for your bigger question: should I drop maths altogether...you might want to ask yourself questions like these: (1) 'Why am I ...


4

To all the people that tells me: "I was always bad in maths" I answer: "That's because you were not ready to invest enough time for it" Improving your efficiency while learning is matter of experience, and sometimes tutor can help (in order to discipline yourself for example).


5

Here's my take, in diagram form: The diagram is organized from basic (top) to advanced (bottom). A solid arrow indicates a more or less definitive prerequisite. For instance, I consider numbers and sets a prerequisite for both groups and discrete math because working with sets is essential in both subjects, and because numbers and sets are a good place ...


2

You pretty much need #3 and #7 before anything. Which order you do these in doesn't matter. A lot of stuff on your list will require multivariate calculus, which is why I say #7. And in #3 you will build up your mathematical maturity/proof skills. Then a whole world will open up to you, so you could do whatever you find interesting. I wouldn't recommend ...


2

I suggest the following sequence: 3-5-2-8-4-7-1-9-10-6-18-11-14-12-13-16-15-17. Exact contents and prerequisites of your material may call for some permutations, but this shouldn't be too bad. The list of topics seems good to me. As you proceed, you will develop a taste that will tell you what you want to focus on and what requires clarification. When ...


1

counterexample is the best thing ever to disprove something . i dont know whats wrong with modern teachers , and see they want it in different way is annoying me somehow . counterexample is neat way , there is not such a thing more neat to disprove things like this , if you wanna do a formal proof , then u have to deal with infinitely of contradictions that ...


0

Southern New Hampshire Unviersity offers a BA in Math both on campus and online. This is a traditional, not-for-profit university.


0

Let $\phi=\dfrac{\sqrt{5}+1}2$ and note that $\phi^{-1} =\dfrac 1\phi= \dfrac{\sqrt{5}-1}2$. Note also that $1+\dfrac 1\phi=\phi$ and $1-\phi=-\dfrac 1\phi$. From your formula, $$F_n = \frac 1{\sqrt{5}}\left[\phi^n-(-\frac 1\phi)^n \right]$$ For $n=k$ and $n=k-1$, $$\begin{align} F_k &= \frac 1{\sqrt{5}}\left[\phi^k-(-\frac 1\phi)^k \right]\\ F_{k-1} ...


0

A good book for a first look at analysis is Bartle's real analysis. A very good introductory linear algebra text is Linear Algebra Done Right, by Axler. Another good linear algebra text is finite dimensional vector spaces by Halmos. For basic topology, nothing beats Munkres. Another good abstract algebra text is Basic Algebra 1 by Jacobson; another is ...


4

Consider this integral and the substitution $tx=y$:$$\int_1^u \frac{1}{x}dx=\int_t^{tu}\frac{t}{y}\frac{dy}{t}=\int_t^{tu}\frac{dy}{y}$$ From this follows: $$\int_1^t \frac{dx}{x}+\int_1^u \frac{dx}{x}=\int_1^{tu} \frac{dx}{x}$$ The geometric way of describing this is stretching the function $f(x)=\frac{1}{x}$ with a horizontal factor $t$, so it becomes ...


1

For real $x$, the gamma function provides a continuous (indeed, even smooth) function that gives the factorials at the positive integers. However, you can choose points in between the positive integers and define your function to be whatever value you want there, and still find a smooth function that fits those values plus gives factorials at the positive ...



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