0
votes
4answers
54 views

Prove that for an increasing and differentiable function $f'(x) \ge 0$ holds.

Prove: If $f$ is a differentiable and increasing function then $f'(x) \ge 0$ for all $x$. Proof from my class notes: $$ f'(x) = f'_+(x) = \lim\limits_{\Delta x \to 0} \frac{f(x+\Delta x) - ...
0
votes
1answer
49 views

Show that $f^{[n]}(0)=0$ for all $n=0,1,2…$

Let $$f(x)=\left\{ {\matrix{ {{e^{ - {1 \over {{x^2}}}}},x \ne 0} \cr {0,x = 0} \cr } } \right.$$ Show that $f^{[n]}(0)=0$ for all $n=0,1,2,\cdots$ The proof: First note that for ...
0
votes
2answers
115 views

Any proof that verify why the limit of the difference is the difference of the limits?

I did a research on internet and books about why the difference of the limits is the difference of the limits, but i didn't get any result of this proof. I would appreciate if somebody can help me. ...
0
votes
2answers
131 views

Why the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0 is 1? [duplicate]

I need a rigorous proof that verify why the limit of $\dfrac{\sin(x)}{x}$ as $x$ approaches $0$ is $1$. I tried before but i do not know how start this proof. I would appreciate if somebody help me. ...
2
votes
1answer
29 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
1
vote
1answer
115 views

Understanding last step of a proof that “two trajectories cannot cross at a finite value of t” (Phase trajectories/nodes)

Note: This proof prefaced critical points at the origin for coupled first order ODEs. It was done before showing the asymptotically stable and unstable critical points: Improper, Proper, Spiral, ...
0
votes
1answer
28 views

Prove that for any $\{x_{m_k} \}\in I_n$, where $I_n$ are dyadic intervals, $\lim_{n \to \infty} x_{m_k} =c$

Proving for any $\{x_{m_k} \}\in I_n$ that:$$\lim_{n \to \infty}\{x_{m_k} \}=c$$ I have been trying to solve this problem, but i dont know how to write it properly, so i need your help whit ...
3
votes
3answers
131 views

Convergent or divergent $\sum_{n=1}^{\infty} \frac{e^nn!}{n^n}$?

Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$ \sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n} $$ I'm currently stuck.
1
vote
2answers
59 views

How to prove that the sequence is decreasing $a_{n}=\frac{ln(n)}{n^2}$

Is my way/proof good and completely mathematically rigorous? $a_{n}=\frac{ln(n)}{n^2}$ --> $a_{n+1}=\frac{ln(n+1)}{(n+1)^2}$ $\frac{ln(n)}{n^2} > \frac{ln(n+1)}{(n+1)^2}$ ...
0
votes
1answer
71 views

Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4 $. Somewhere along the proof we would ...
0
votes
1answer
41 views

How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
1
vote
1answer
45 views

Why does secant method converge

Assume $f$ is continuous and twice differentiable on $[a,b]$ such that $f'(x)>0$ and $f''(x)>0$, $x \in [a,b]$. If $f(b)>0$ and $f(a)<0$ and I choose $x_0=a$,why are we gauraunteed ...
0
votes
2answers
61 views

Proof of application of Mean Value Theorem

Two bicyclists begin a race at 8:00AM. They both finish the race 2 hours and 15 minutes later. Prove/explain that at some point during the race, the bicyclists are traveling at the same velocity. So ...
3
votes
2answers
195 views

Why is $\cos(2x)=\cos^2(x)-\sin^2(x)$ and $\sin(2x)=2\sin(x)\cos(x)$?

I was studying math.. and I just realized that I only just memorized these trigonometric equations, but I don't really know the reason behind them. So um... Why is $\cos(2x)=\cos^2(x)-\sin^2(x)$ and ...
0
votes
1answer
13 views

A question about the proof of the limit comparison test for series

A question about the proof of the limit comparison test for series: http://en.wikipedia.org/wiki/Limit_comparison_test About the the last part: $b_n(c-\epsilon)<a_n<(c+\epsilon)b_n$, to ...
19
votes
3answers
347 views

How to prove $ \lim_{n \to \infty} e^n \cdot \left( \sum_{k=0}^{n-1} ({k-n \over e})^k/k! \right)- 2 \cdot n = \frac 23$?

I observed for the function $$ f(n)= e^n \sum_{k=0}^{n-1}\left(\dfrac{k - n}{e}\right)^k \cdot \dfrac{1}{k!} \tag 1$$ with small $n$ that ...
0
votes
3answers
64 views

Convergence of sequence method, Math behind intuition

Now I want to find convergence of a sequence: $$ \lim_{n \to \infty} \sqrt[n]{4^n + 5^n}$$ Now I am pretty sure I have solved this using logic on inspection: $4^n \ll 5^n$ as $n\rightarrow\infty$, ...
2
votes
0answers
36 views

Prove Differentiation Multivariable

Given $f(x,y) = \frac{ xy^2}{x^2 +y^2}$ From defintion we know it is differentiable if: $\lim_{h\to 0}\frac{F(X+h)-F(X)-c*h}{|h|}$ exists, where $c$ is the gradient of the function. I have ...
0
votes
2answers
62 views

How would this problem need the Mean Value Theorem?

I'm asked to square the inequality and use the Mean Value Theorem to prove that $$\sqrt{1+x} < 1 + \frac{x}{2}$$ for $x>0$. Unfortunately, I don't really understand why I would need such a ...
0
votes
0answers
26 views

U-substitution proof by partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
1
vote
2answers
41 views

Show a double-sided infinite integral of $\sin(x+b)$ exists iff $b=n\pi$

More formally: Show that $$\lim_{a\rightarrow \infty} \int_{-a}^a \sin(x+b)$$ exists if and only if $b=n\pi$ for some $n \in \mathbb{Z}$. I get the intuition fine. The function is just a horizontal ...
4
votes
0answers
76 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
23
votes
2answers
2k views

Single Variable Integral

Compute the following integral \begin{equation} \int_0^{\Large\frac{\pi}{4}}\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\, ...
0
votes
0answers
32 views

Show if this is integrable (defined 1 on rationals, 0 else)

Define $f: [0,1] \rightarrow \mathbb{R}$ as $f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & x \notin \mathbb{Q} \end{cases}$ Find $\underline{\int_0^1f}$ and $\overline{\int_0^1f}$. Is ...
2
votes
1answer
48 views

The Fundamental Theorem of Calculus and Derivatives

How do I show this in a convincing manner? I know I need to use the Fundamental Theorem of Calculus, but I find it difficult to show any steps in between, as it appears obvious?
1
vote
2answers
70 views

Can we prove the formula for surface of revolution?

This is math. We like to prove things. However, proofs are rigorous processes (for a good reason) and are more than just "that idea looks like it could make sense". I've seen proofs for many ...
1
vote
3answers
61 views

How to go about proving that $\cos^2 x$ is everywhere differentiable?

My first line of reasoning was to try directly evaluating $$\lim\limits_{h \to 0}\frac{\cos^2 (x+h) - \cos^2 (x)}{h}$$ and showing such a limit existed for any x, but when $\cos^2(x)$ evaluates to ...
0
votes
1answer
38 views

Question about the Least squares method

We have $n$ dots: $(x_1,y_1)\cdots (x_n,y_n)$. We know that if we use the Least squares method we will get a line $y=mx+b$ that giving the minimal value for the function $w=\sum_{i=1}^n ...
1
vote
2answers
24 views

Heuristics for Lipschitz equivalence

I'm studying for my finals in general topology and when I look at the definition of the Lipschitz equivalence of two metrics as: Let $d(x,y)$ and $d'(x,y)$ be metrics on a non-empty set $X$. We ...
2
votes
0answers
51 views

Continuity of the right-hand derivative of a Convex function (help with the proof)

Hi everyone I have some trouble with one point in the following proof. Let $f$ be a convex function (strict convex function) on a real interval. If $f'_-(a)=f'_+(a)$ where $f'_-$ and $f'_+$ are ...
0
votes
1answer
21 views

Trying to show that a vector value function has equal mixed partial derivatives

Let $f: R^n \to R^n$. $||x||$ is Euclidean norm. Define $f(x) = xg(||x||)$. where $g: [0, \infty) \to R^n$ is differentiable on $(0, \infty)$. $g$ is not constant. I want to show that every $i \neq ...
1
vote
1answer
35 views

First derivative test and uniqueness of local extrema

This is the context in which my question lies. See below for the actual question. Let $f(x)$ be differentiable everywhere and have a minimum at $x^*$. Then for every $x$ in a proper neighbourhood ...
0
votes
2answers
19 views

Proving divergence for piecewise defined sums

Show that the series $\sum(-1)^{n-1}b_n$, where $b_n=1/n$ if $n$ is odd and $b_n=1/n^2 $ if $n$ is even, divergent. I'm completely stuck on how to start the problem. I was told that I should use ...
1
vote
1answer
34 views

Show that we can reorder mixed partials, if every partial is continuous

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
0
votes
1answer
27 views

Showing the Clairaut theorem in higher dimensions — partials commute

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
1
vote
0answers
50 views

The Flat Function

I have to write an essay on the flat function $$\text{flat}(x) = \begin{cases} e^{-\frac{1}{x^2}} & \text{for } x \ne 0 \\ 0 & \text{for } x = 0 \end{cases}$$ and I want to prove ...
0
votes
1answer
44 views

Question regarding trigonometry

I've got this thing on my mind : we know that $cos(x)$ is a periodic function , hence integral from $2(k-1) \pi$ to $2k \pi$ will yield the same value for any $k \geq1$. My question is , why is ...
0
votes
0answers
10 views

prove where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable and is not differentiable [duplicate]

Classify where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable -- where it's not, prove it, and where it is, prove it. For $x\neq 0$ and $y\neq 0$, we can just treat it as $f(a,b) = \sqrt{a+b}$ (where ...
12
votes
1answer
134 views

Why is an equation necessarily dimensionally correct?

I have just read a fascinating proof of the value of the integral $$ \int_{-\infty}^\infty e^{-ax^2} dx, $$ which proceeds by dimensional analysis, as follows: we know that we can write $$ ...
4
votes
1answer
119 views

A tough one: show that this is not differentiable at any point in R

Here's the question: Define $\phi: \ \mathbb{R} \rightarrow \mathbb{R}$ by $$ \phi(x) = \begin{cases}x & 0\leq x\leq\frac{1}{2}\\ 1-x & \frac{1}{2}\leq x\leq 1\end{cases}. $$ And then ...
0
votes
1answer
75 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by finding a linear function T

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
0
votes
2answers
86 views

The concept of $\epsilon - N$ proofs

I just don't understand how to complete $\epsilon - N$ proofs. I don't know what my goal is or why they prove what they do. I have asked two questions on here in the past, but I simply don't 'get it'. ...
1
vote
2answers
55 views

If $f$ is increasing on an open interval and continuous at endpoints, it's increasing on the closed interval.

Prove that if $f$ is increasing on $(a,b)$ and continuous at $a$ and $b$, then $f$ is increasing on $[a,b]$. The question then clarifies: "In particular, if $f$ is continuous on $[a,b]$ and $f'>0$ ...
1
vote
2answers
98 views

Using L'hopital's rule to solve problem.

Show that $$\lim_{x \to 0} \frac{-3x }{e^{x/3}}=0 $$ by L'hopital's rule. I know how to solve this without using L'hopital's rule. I was just reading about this and was wondering can we solve it ...
0
votes
1answer
38 views

Borsuk–Ulam theorem for $n=2$

How one can intuitively prove the following statement: At any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures. What about a rigorous proof?
1
vote
2answers
328 views

why area under curve or riemann sum equals to definite integral

i do get that Riemann sums is sum of infinite triangles with with infinitely small length. But definite integral is completely different you are taking anti derivative of f(x) at b and subtract anti ...
1
vote
0answers
36 views

expected value with integration

For the exponential distribution, $f(x)=(1/\theta) e^{-x/\theta}$ for $x>0,$ and $f(x)=0$ for $x \leq0$ $(i)$ Determine the exact value for the probability $P(0<X<3\theta).$ I need help ...
0
votes
2answers
34 views

Proof that $t^ne^{-t}\leq Ce^{-t/2}$ for all $n\geq 1$ and $t\geq 0$

How do I prove that $t^ne^{-t}\leq Ce^{-t/2}$ for all $n\geq 1$ and $t\geq 0$. I am not sure which type of proof to use, for example induction with two variables. The graphs suggest C can always be ...
0
votes
3answers
72 views

epsilon delta to prove $\lim_{x \rightarrow a} \frac{1}{f(x)}$

i was solving problems on my textbook.... and i became stuck. The question is: Let $a\in (- \infty , \infty ).$ Suppose $\lim_{x \rightarrow a} f(x)=L \neq 0$. Use the $\epsilon - \delta$ arguement ...
1
vote
0answers
57 views

Proof about lognormal distribution

I'm trying to prove a result about the lognormal distribution that seems to me to be fairly intuitive, but I can't get the proof to work. Basically, I'd like to prove that as the mean increases, the ...