For questions related to the teaching and learning of mathematics. Note that Mathematics Educators StackExchange may be a better home for narrowly scoped questions on specific issues in mathematics education.

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4
votes
2answers
137 views

Good example showing why limits must exist in limit product rule

I'm looking for a way to show my calc 1 students not to use the limit laws without knowing that the individual limits exists. I could use $$\lim_{x\to 0} x^{2} \sin(1/x),$$ but by doing it wrong, one ...
3
votes
1answer
72 views

Being ready to tackle the math courses in my CS program

Here's my (long) story cut short. I was awful at math in high school. I did 4 years in the service and now I'm going to start college in just a few weeks. I am really nervous because I will have 5 ...
2
votes
0answers
52 views

Circular Logic and Continuity

So, I was doing a Calculus problem a few minutes ago and just recalled something that my real analysis professor said during a lecture years ago... To provide context, take the function $f$ defined ...
4
votes
5answers
389 views

What mistakes, if any, were made in Numberphile's proof that $1+2+3+\cdots=-1/12$?

This is not a duplicate question because I am looking for an explanation directed to a general audience as to the mistakes (if any) in Numberphile's proof (reproduced below). (Numberphile is a YouTube ...
0
votes
1answer
24 views

Finding arithmetic mean, with hidden ratios, HOW?

I have twelve items each with with their own percentage of 100%. I'll label them.. English(A):67 -MATH(B):60.5 -Biology(B):67.5 -Chemistry(C):80 -Physics(E):67 ARABIC(F):63 -Economics(G):78.5 ...
4
votes
3answers
94 views

What is the equivalent of musical ear training with regards to studying mathematics

When one aspires to be a professional musician, it is made clear that ear training is a very valuable skill that must be cultivated on a daily basis. The student is advised to put in the time and ...
1
vote
2answers
67 views

What is the linear transformation $ x \mapsto Ax $?

I am told by my textbook that, $ \text{Nul }A = {0} \text{ if and only if the linear transformation } x \mapsto Ax \text{ is one-to-one.} $ $ \text{Col }A = \Bbb R^m \text{ if and only if the linear ...
0
votes
0answers
23 views

3 variables 'linked' to one outcome (for lack of better words)

Paper route salary formula (reverse engineer...again for lack of a better word;) Ok suppose I have these variables: Pieces of mail : 302 Number of stops: 177 TotalWeight* : 272 Hours allocated: ...
24
votes
12answers
4k views

How do I make a student understand contradiction?

We were trying to prove that if $3p^2=q^2$ for nonnegative integers $p$ and $q$, then $3$ divides both $p$ and $q$. I finished writing the solution (using Euclid's lemma) when a student asked me ...
-20
votes
1answer
213 views

Solve $|z|+z=1+i$ [closed]

Solve the equation $|Z|+ Z=1+i$ please help me describe me how to solve it.
4
votes
2answers
97 views

I need a good reading list of the Masterworks of mathematics. [closed]

I'm interested in a solid list of the greatest textbooks in the major* fields as written by the "Masters". The problem I have is that after many years of a high-level, rigorous education in ...
0
votes
0answers
36 views

Are some Types of Proportional Relationships more useful than others?

Responses from a previous math stack exchange question (see link below) have lead me to this related question. Link to previous question Based on these responses, it seems that in just about any ...
-1
votes
1answer
46 views

a local government is planning to build new roads that will divide a forest into several fragments [closed]

a local government is planning to build new roads that will divide a forest into several fragments the roads can follow several different designs each of which design will minimize the edge effect on ...
2
votes
0answers
54 views

Is the precise definition of a limit worth covering in Calc 1?

I'm teaching calc 1 this semester and am trying to decide if I should cover the precise definition of a limit. My first impression was that I should. It helped me to gain a deeper understanding of ...
4
votes
4answers
133 views

Is the anti-derivative another way of saying the derivative?

This question is being posted so that I can understand a definition and concept. The definition - F(x) is an anti-derivative of f(x) if F'(x) = f(x) Therefore, $$F(x) = x^2$$ is an ...
0
votes
0answers
26 views

One Million Proficiency Efficiency Calculations

I have been playing the "Million" game on my apple device after discovering it through the trending applications in the search function. After beating the game I began to wonder to myself if there was ...
12
votes
9answers
807 views

Proof that doctors could relate to [closed]

I am supposed to present a mathematical proof to a lecture hall full of doctors in order to show them how mathematicians think. I'm having trouble picking a proof that will be easily followed by ...
1
vote
3answers
118 views

How to explain Clairaut-Schwartz's Theorem, $f_{xy}=f_{yx}$?

I am looking for a non-technical explanation of Clairaut's theorem which states that the mixed derivative of smooth functions are equal. A geometrical, graphical, or demo that explains the theorem and ...
2
votes
1answer
50 views

A different method for solving 2nd order ODEs

For constants $a,b$ there are many ways to find the solutions to $$y^{\prime\prime} + (a+b)y^\prime + aby = \phi(x). $$ Perhaps the most popular is to first solve the homogeneous case when $\phi(x) ...
0
votes
2answers
113 views

How complex can a Differential Equation be? [closed]

When I was first being introduced to differential equations, my teacher began by asking the other student and me to write down the most complex differential equation we could think of and gave us 10 ...
0
votes
0answers
15 views

Convention as to differentiability defined in terms of local extension?

Let $X \subset \mathbb{R}^{n}$. In differential topology or allied areas, people may define that a function $f: X \to \mathbb{R}^{m}$ is of class $C^{r}$ if and only if $f$ can be locally extended to ...
0
votes
1answer
140 views

calculate minimum of max value of set of numbers

Suppose to have a set of numbers: $$S:=\{ 1,2,3,9,10,56,58,60\}$$ How can i group this number like this: group 1:$\{1,2,3,9,10\}$ group 2:$\{56,58,60\}$ and then take "56" as minimum value of ...
-4
votes
1answer
133 views

Why is Distribution Prioritized Over Combining?

In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [$a(b + ...
0
votes
0answers
23 views

Identity and equality [duplicate]

What is difference between identity and equality in math? when to use identity and when equality? Most of math identity also defined as equality how to distinguish between identity and equality. in ...
0
votes
2answers
133 views

How do mathematicians find the underlying idea?

While reading through the lecture notes here (http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week2.pdf , page 22, last paragraph), I came across the following " Thus there must be some ...
2
votes
2answers
62 views

Variable in Math

What is formal definition of "variable" ? I cannot able to understand variable because some times it is varying or some time used for approaching. Arbitrary constant also confusing me. In Multi ...
2
votes
3answers
70 views

Intuitive explanation of second derivative test for functions of two variables.

I will be teaching multivariable calculus again this semester, and I am not so happy with the explanation I have for the second derivatives test for functions of two variables. QUESTION: What is a ...
11
votes
9answers
609 views

Motivation for the Definition of Compact Space

A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} ...
0
votes
2answers
32 views

Define temperature by clustering with math operators

I can´t figure out how to cluster the temperature for the weather in 3 optimal cases: hot, mild, cold My data contains: air temperature(the average daily value), max air temperature(highest daily ...
2
votes
1answer
261 views

What are the (most) essential/interesting sections of The Princeton Companion to Mathematics a junior math student better have a read?

I am studying The Princeton Companion to Mathematics, but it is heavy, the sections have different difficulties, and time is limited for me. Assuming that I have only some knowledge of undergraduate ...
0
votes
0answers
45 views

Is there a big difference between runge kutta 4th for ODEs vs SDEs?

I was working on 2nd, 4th order runge kutta method for stochastic differential equations. I saw 2nd formula for ODEs and SDEs. There is some difference between their formulas . Unfortunately I can't ...
0
votes
0answers
68 views

Good math websites for elementary geometry and algebra?

I would appreciate it if someone could suggest good websites where we can find English written geometry and algebra exercises for beginners, say junior and senior high school students. I have ...
0
votes
1answer
41 views

Program languages recommended for complexity theory

I am an undergraduate studying mathematics and one of my interests include complexity and computability theory. I have no experience in programming. The computability theory books I looked into didn't ...
4
votes
1answer
96 views

“A real polynomial of degree $n$ cannot have more than $n-1$ local extrema”: a proof without derivatives?

I am looking for a proof that does not use derivatives of the elementary theorem given in the title: Theorem: A polynomial $p:\mathbb{R}\to\mathbb{R}$ of degree $n$ cannot have more than $n-1$ local ...
2
votes
4answers
99 views

For what values of $p$ does this series converge?

This is a question we asked on a second semester calculus test. For what values of $p$ does this series converge? $$\sum_{n=1}^{\infty}\frac{\sin(1/n)}{n^p}$$ I believe that it actually can be shown ...
6
votes
2answers
337 views

Why don't we start studying calculus via series instead of the calculus on finite expressions?

It seems that historically, there were two trends on the idea of integration: Newton's work which depended on infinite series. Leibniz work which depended on the dream of integration of elementary ...
2
votes
3answers
72 views

Are there any significant differences between studying functional analysis from a normed space perspective versus a metric space perspective?

Does it matter if functional analysis was introduced from a normed space versus a metric space formulation? Are all major theorems from functional analysis (such as Banach contraction mapping, Hahn ...
2
votes
1answer
72 views

How To Prove Irrational Square Roots and Inequalities In Courant's Calculus Book? [closed]

Here's the proofs questions in a screenshot The first questions ask about proving the irrationality of non perfect squares. Numbers 3,5, and 6 ask for inequality proofs. I find it daunting that the ...
1
vote
0answers
99 views

Who knows Krotov's Method in Optimal Control Theory

I'm finishing my PhD thesis about applications of optimal control theory in the field of energy harvesting. In the course of my PhD I dealt with different ways to compute optimal controls, and I found ...
4
votes
7answers
163 views

How do i convince students in high school for which this equation: $2^x=4x$ have only one solution in integers that is $x=4$?

I would like to convince my student in high school level using a simple mathematical way to solve this equation: $$2^x=4x$$ in $\mathbb{z}$ which have only one integer solution that is $x=4$ . ...
0
votes
0answers
30 views

Hoeffding's inequality, number of samples required

I was deriving the number of samples required to qualify certain confidence bounds, at the end I am getting slightly different results from what is stated in my lecture notes. can anyone explain why ...
0
votes
1answer
68 views

Projecting functions onto planes

I understand the concept of projecting vectors onto the span of a vector but I'm having trouble projecting functions i.e How would I project the function cos(x) onto the vector that spans (1,1)?
2
votes
1answer
82 views

Alternative proof that base angles of an isosceles triangle are equal

The "classic textbook proof" of equality of base angles of an isosceles triangles which I studied in my school days is as follows: Let $\Delta ABC$ be a triangle with $AB = AC$ and let $D$ be the mid ...
5
votes
8answers
1k views

Is it too much rigor to turn a set into a vector space?

I was reading some online notes on vector spaces and one authors insisted on turning a set $\mathbb{X}$ into a vector space. I thought it was quite insane but maybe I am not seeing the point. The ...
1
vote
1answer
104 views

New idea to solve $\int x^n e^x dx$

I have this problem $$\int x^n e^x dx= x^ne^x -nx^{n-1}e^x +n(n-1)x^{n-2}e^x- \cdots+(-1)^nn!e^x $$ my try was to use integration by part . $$I_{n}=\int x^n e^x \, dx=e^x x^n -\int (nx^{n-1})e^x \, ...
4
votes
1answer
278 views

Books on complex analysis (Ahlfors, Conway and Lang)

To make my question slightly different from others, I would like to know how would you rate on the complex analysis books by Ahlfors, Conway and Lang? I had a basic course on complex analysis during ...
3
votes
7answers
303 views

Alternate ways to prove that $4$ divides $5^n-1$

I was working for various method to solve this: For all $n\in \mathbb N$: $4\;\mid\;(5^{n}-1)$. My try was: 1st: $$n=1 \to 4|5^1-1\\n \geq 2 \to 5^n=25,125,625,3125,...\\ n\geq 2 \to ...
2
votes
1answer
103 views

Difficult to read about different subjects simultaneously, should I leave one for now? [closed]

I learn math by reading books. Usually I read 3 books (about 3 different subjects) simultaneously and switch focus every couple of days. The books i'm studying right now are Rudin's functional ...
3
votes
1answer
68 views

Factoring bivariate quadratics with real coefficients (for high school students).

I was tutoring a Year 10 student last night (he's learning about quadratics). Unfortunately, we ran into a class of problems that I couldn't explain how to solve (beyond simply guessing and checking), ...
7
votes
5answers
442 views

New idea to solve this equation

I was teaching $\left \lfloor x \right \rfloor$ function properties and equation . I solved this equation in my class . My works are show below. Some students ask me for new Idea...,and now I am ...