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
19 views

question about the Darboux integral theorem proof

well, the sentence goes like this: Consider $f$ bounded function in $[a,b]$. $f$ is integrable IF AND ONLY IF $\forall\epsilon >0$ $\exists$ a partition $P$ of $\left[a,b\right]$ such that ...
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 ...
1
vote
0answers
32 views

Proof Validation Function From Integers to Rationals is Continuous

I am teaching myself real analysis, so any help is greatly appreciated. Let the function be defined as $F : Z \rightarrow Q$ where $Z$ is the set of integers and $Q$ is the set of rational numbers, ...
2
votes
0answers
61 views

Proof regarding derivatives and Mean Value Theorem.

Original question: $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Show that for any $c \in (a,b)$ that is not a point of maximum or minimum for $f'$, there exist $x_1, x_2 \in (a,b)$ ...
3
votes
1answer
63 views

$f$ is differentiable. If $\lim_{x \to c}f'(x)$ exists, then this limit must be $f'(c)$.

Please prove: Let $f:(a,b) \to R$ be differentiable function, and let $c \in (a,b). $ If $\lim_{x \to c}f'(x)$ exists and is finite, then this limit must be $f'(c)$. I tried doing it directly but ...
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?
0
votes
1answer
43 views

Help with proof for solving an ODE using contraction mapping theorem

I'm trying to follow a proof for solving the ODE $$\frac{df}{dx} = (f(x)+x)x$$ For $0 \leq x \leq 1$ with the initial condition $ f(0)=0$. The proof I am following goes like this Define ...
3
votes
1answer
74 views

How to prove l'Hospital's rule for $\infty/\infty$

I'm having trouble with this l'Hospital's rule wiki page(the proof of l'Hospital's rule): http://en.wikipedia.org/wiki/LHospital%27s_rule Well, in the case where the limit looks like $0/0$, it's ...
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 ...
0
votes
1answer
60 views

Epsilon delta proof

I know that the point of the proof is to show that you can get within $\epsilon$ of the limit, by giving a value that is within $\delta$ of $x$. But when solving for $\delta$ in terms of $\epsilon$ ...
0
votes
1answer
28 views

Error replacing integral of f with its midpoint rule approximation

here is a question I've been banging my head against. If f is continuous on [a,b] and differentiable on (a,b), and if there is a positive real number M such that |f'(t)| is less than or equal to M ...
2
votes
2answers
57 views

How to complete this epsilon delta proof

Prove $\lim_{x\to 1} {2+4x \over 3} = 2$ using the epsilon delta definition of a limit. if $0 < \left|x-1\right| < \delta$ then $\left|{2+4x \over 3}-2\right| < \epsilon$ scratch work for ...
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 ...
3
votes
1answer
63 views

Spivak Ch 11 Theorem 7

Can someone please explain how to supply a rigorous $\epsilon,\delta$ argument for this theorem as Spivak says ? My argument is: $f'(a)=\lim_{h\to 0} f'(\alpha_h)$ equivalent to ...
0
votes
2answers
29 views

Show that $f$ is everywhere differentiable and the partials commute

Take the function $$ f(x,y) = \begin{cases}\frac{x^3y -xy^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}. $$ Show that it is everywhere differentiable and that $D_{1,2}f(0,0)$ ...
2
votes
1answer
43 views

Showing that if derivative is 0, function is constant ($f: U \rightarrow \mathbb{R}$ where $U \subset \mathbb{R}^n$)

Here's the question: Suppose that $f: U \rightarrow \mathbb{R}$ is differentiable on the open subset $U\subset \mathbb{R}^n$, and $Df(x) =0$ for all $x\in U$. Show that $f$ is constant on $U$. My ...
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 ...
1
vote
0answers
44 views

Show that this is not differentiable at any point in $\mathbb{R}$

Define $\phi: \ \mathbb{R} \rightarrow \mathbb{R}$ by $$ \phi(x) = \begin{cases} x\ :\ 0\le x\le \frac{1}{2}\\ 1-x :\ \frac{1}{2} \le x \le 1 \end{cases}$$ And then extend ...
3
votes
1answer
86 views

Best proof of some theorems in calculus

I would like to choose (among the miriads of proofs) a well-structured, elegant, neat, clear proof of the first fundamental theorem of calculus; the second fundamental theorem of calculus; the mean ...
0
votes
0answers
29 views

Need help on analytically proving the monotonicity of an inexplicit integral

I have the following function which I have numerically investigated to be monotonically increasing in $\nu$: $D(x;\mu,\nu) =\frac{1}{\sqrt{2\pi \cdot \nu}}\int_{-\infty}^{\infty}e^{-\theta \cdot x} ...
1
vote
1answer
41 views

does F(x) = -F(1/x) for all x in domain of F for this specific F(x)?

I apologize in advance for my ignorance on how to type mathematical symbols in this editor. Let F be the function defined for $x > 0$ by $F(x)= \int_1^x e^{((t^2)+1)/t}\frac{dt}{t}$ Show 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 &.\\ . ...
2
votes
2answers
78 views

Another Epsilon-N Limit Proof Question

How to prove the limit of the following sequence using epsilon-N argument. $$a_n=\frac{3n^2+2n+1}{2n^2+n}$$ I took the limit to be $\frac{3}{2}$ and proceeded with the argument, ...
0
votes
1answer
108 views

Comparison Theorem for Integrals

Problem: Let $a>0$ and $b>a+1$. Use the Comparison Theorem to show that the following integral is convergent: $$\int ^ \infty _0 \frac{x^a}{1+x^b} \ dx$$ My attempt at this was that since ...
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 ...
1
vote
1answer
49 views

Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$

$f(x)$ is continuous on $[a,b]$. Now we define a new function $M(t)$, for every $t\in[a,b]$ $$M(t) = \sup_{a \leq x \leq t} f(x).$$ Prove formally that $M(t)$ is continuous on $[a,b]$. (sup = ...
3
votes
2answers
70 views

Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function. Prove formally that $P$ is onto $\mathbb{R}$

Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function i need to formally prove that $P$ is onto $\mathbb{R}$ my strategy so far ....... polynomial function is ...
4
votes
2answers
112 views

formal proof from calulus

Given $f:R \to R$, $f$ is differentiable on $R$ and $\lim_{x \to \infty}(f(x)-f(-x))=0$. I need to show that there is $x_0 \in R$ such that $f'(x_0)=0$ I am trying to prove it by contradiction .... ...
0
votes
1answer
47 views

Strange derivative

In this proof: http://www.math.hmc.edu/calculus/tutorials/mean_value/proof_mean.html Why does $g'(x) = f'(x) - \frac{f(b)-f(a)}{b-a}$?
1
vote
1answer
65 views

proof of $\frac{\partial \frac{\partial f(x,y)}{\partial x}}{\partial y}=\frac{\partial \frac{\partial f(x,y)}{\partial y}}{\partial x}$

I was at my physics class(electrodynamics).I saw a relation which frequently uses in my course.Relation is that $$\frac{\partial \frac{\partial f(x,y)}{\partial x}}{\partial y}=\frac{\partial ...
1
vote
2answers
98 views

prove or give a counter-example

I think I have solved it (please check) but I would like to see and (re)-learn how one writes a proper proof (including the mathematical signs) and little things (I might have missed), maybe even more ...
0
votes
1answer
46 views

Critical Points and Gradients/Derivatives

Plot the function $f(x)= 3+\cos(3x)-0.5\sin(5x)+0.2\cos\left(10x-\left(\frac{\pi}{4}\right)\right)$. Estimate how many critical points are on the interval $[0,2\pi]$. Consider $\mathbb{R}^{20} \to ...
0
votes
1answer
104 views

Help to clarify proof of Euler's Theorem on homogenous equations

Why is the last step (setting $\lambda = 1$ allowed? I have trouble accepting this because if I set $\lambda =1 $ at the very start, then: $f(\lambda x , \lambda y)=\lambda^r f(x,y)$ becomes ...
3
votes
2answers
210 views

Stuck with a tricky existence proof

Show that there exists a continuous function $f: [-1, 1] \rightarrow \mathbb{R}$ such $f(0) = 1$ and $f(x) = \frac{2-x^2}{2} \cdot f(\frac{x^2}{2-x^2})$ $\forall x \in [-1, 1]$ I tried putting ...
0
votes
1answer
63 views

Proof adding layers of constant width to a shape tends to an $d$-sphere as the number of layers tends to $\infty$

Good night, I've recently seen one of Victoria Hart's videos on Youtube (it wasn't about this, it was about Fibonacci numbers, and I found it on a comment in this site), and in it she said that if ...
0
votes
1answer
97 views

Formal Definition of Limit and Proofs

I'm having trouble understanding the formal definition of a limit... Let $f(x)$ be defined on an open interval about $x_0$, except possibly at $x_0$ itself. We say that the limit of $f(x)$ as ...
3
votes
3answers
110 views

Proof for $\sinh(x-y)$

Basically I need to prove that $\sinh(x-y)=(\sinh x)(\cosh y)-(\cosh x)(\sinh y)$ I could use the fact that $\cosh$ is an even function and $\sinh$ is an odd. I can prove that: $$\sinh(x+y) = \sinh ...
2
votes
1answer
53 views

Prove c exists for f:[0,1] -> R

Suppose that a function $f:[0,1] \to \mathbb R$ is continuous in $[0,1]$. Prove that there is a point $c\in [0,1]$ such that $\int_0^1 x^2 f(x)\,dx = \frac{f(c)}{3}$. Do I somehow use the mean value ...
0
votes
1answer
146 views

Prove that $f$ is not integrable on $[0,1]$

Prove that $f(x)= f(n) = \begin{cases} x, & \text{if $n \in Q$ } \\ -x, & \text{if $n\notin Q$ } \\ \end{cases}$ is not integrable on $[0,1]$ Here is what I got but I'm not so sure Let ...
1
vote
1answer
71 views

prove that if $L(f,P)=U(f,P)$ then $f$ is constant on $[a,b]$

Suppose that $f$ is a bounded function on $[a,b]$ and there exists a partition $P $of $[a,b] $such that $L(f,P)=U(f,P)$. Prove that $f$ is constant on $[a,b]$ I know that $L(f,P)=U(f,P)$ meaning $f$ ...
1
vote
1answer
91 views

prove that $\int(f(x)+g(x))dx= \int f(x)dx+\int g(x)dx$

Let $f,g$ be two functions defined on $A$. Supposed that $F$ and $G$ are anti-derivative of $f$ and $ g$. Prove that $\int(f(x)+g(x))dx= \int f(x)dx + \int g(x)dx$ Here is what I got. Let $H(x)$ ...
0
votes
0answers
53 views

Help to understand manipulations on limits and integrals - $\int_a^b \! c \, \mathrm{d}x=c(b-a)$

I'm reading this proof from here: and I don't understand how to reach $$\lim_{n \to \infty} \left(\sum\limits_{i=1}^{n}c \right)\frac{b-a}{n}$$ Specifically, why are we allowed to take out ...
1
vote
3answers
58 views

When do two functions differ by a constant throughout an interval (Fundamental Theorem of Calculus)

I'm reading the proof of the Fundamental Theorem of Calculus here and I don't understand the following parts (at the bottom of page 2): I don't know how to conclude that $G(x)-F(x)=C$ for a $x \in ...
0
votes
1answer
33 views

let $L_f, L_g, L_{f+g}$ be the lower integral of $f, g, and f+g$. Prove that $L_{f+g}\ge L_f+L_g$

let $f,g$ be two bounded function on $[a,b]$, let $L_f, L_g, L_{f+g}$ be the lower integral of $f, g, and f+g$. Prove that $L_{f+g}\ge L_f+L_g$ I don't know how to start
2
votes
3answers
431 views

Prove that if $f$ is integrable on $[a,b]$ then so is $|f|$?

Prove that if $f$ is integrable on $[a,b]$ then so is $|f|$. I can prove the converse of this is false, I also try using the definition of integrable function $f$, but I don't know what to do after ...