# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $n\rightarrow\infty$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\epsilon$-$\delta$ 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
### 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. ...
### 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 ...