0
votes
1answer
68 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
42 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
47 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
180 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
0answers
37 views

Mathematical Probability and Statistics( all the math need)

I would like some suggestions about mathematical techniques and knowledge are required to understand and master 2nd year undergraduate probability and statistics. I am mature student with some ...
0
votes
1answer
12 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
333 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
60 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 $n\rightarrow\infty$, ...
3
votes
0answers
34 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
72 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 ...
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
67 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
60 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
37 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
46 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
20 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
31 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
32 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
25 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
127 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
115 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
70 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
85 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
53 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
97 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
36 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
1answer
245 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
35 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
68 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
53 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 ...
1
vote
0answers
26 views

Battery between liftimes

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with $\theta$ $= 2$ (measured in years). Find the probability that a battery of this type will have ...
1
vote
2answers
74 views

Why does $\lim_{n \to\infty}a_{n+1} = \lim_{n\to\infty}a_n$?

Assume that $\{a_n\}$ is a convergent sequence. How to use the definition of a limit of a sequence to prove that
0
votes
1answer
67 views

Differential map on the vector space of polynomials: Kernel and Image

Given the $V_n$ is the vector space of polynomials of degree $\leq$ n over $\Bbb R$ So $M_D = \begin{bmatrix} 0 & 1 & ... & 0\\ 0 & 0 & 2 &... 0 \\ 0 & 0 &0 &.\\ . ...
1
vote
2answers
111 views

Help with the proof of the Witch of Agnesi curve

$a=1$ (The radius is 1). How do I prove that if we talking about $P=(x,y)$, then: $$y=\frac{8}{x^2+4}$$ I'd like to get any help! Thank you!
1
vote
2answers
59 views

Prove $\lim\limits_{x\to a} \frac{1}{g(x)} = \frac{1}{m}$

Suppose $\lim_{x\to a} g(x) = m$ and $m \neq 0$ then, $$\lim_{x\to a} \frac{1}{g(x)} = \frac{1}{m}$$ My attempt to prove this is using the epsilon-delta definition of a limit and here's what I have ...
0
votes
3answers
167 views

How do i solve delta epsilon proofs for quadratic equations?

For the $\lim\limits_{x\rightarrow 2} (x^2 + 5x - 2) = 12$ i need to show how to find a $\delta$ such that $|f(x) - L| < \varepsilon$ for all $x$ satisfying $0 < |x - a| < \delta$ Help is ...
0
votes
3answers
97 views

$\epsilon$-$\delta$ proof of $\lim f(x)g(x) = \lim f(x)\lim g(x)$ for $x\to a$

Can someone please hold my hand and guide me through this proof. I saw it in spivaks examples and yet it does not make sense! Your help is very appreciated
3
votes
2answers
107 views

If $f$ is differentiable and $\lim_{x→0} f'(x) = L$, then $f'(0) = L$.

True/False. (c) If $f$ is differentiable on an interval containing zero and if $\lim_{x→0} f'(x) = L$, then $f'(0) = L$. 1. How to presage proof by contradiction? Proof by contradiction. ...
2
votes
1answer
42 views

Does all non-monotonic continues functions have $x_0 \in \mathbb{R}$ such that $f'(x_0)=0$?

Given $f\colon\mathbb{R} \to \mathbb{R}$, $f$ is differentiable on $\mathbb{R}$ and the $\lim_{x \to \infty}f(x)$ does not exists . show/prove formally that there exists $x_0 \in \mathbb{R}$ such ...