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For each interval ##[x_i,x_{i+1}]## of any partition: ##L(f,P)## approximates the area in that section under the function by means of a little rectangle of height the lowest value ##m_i## that the function takes in the interval. ##R(f,P,c)## approximates the area in that section under the...

Hi, PF, wrong again :smile:

Hi, PF, let's give another try $$\inf_{t\in{[x_i,x_{i+1}]}}f(t)\le f(t_i)\le \sup_{t\in{[x_i,x_{i+1}]}}f(t)$$ $$\sum_{t=0}^n \inf_{t\in{[x_i,x_{i+1}]}}f(t)(x_{i+1}-x_i)\le \sum_{t=0}^n f(t)(x_{i+1}-x_i)\le \sum_{t=0}^n \sup_{t\in{[x_i,x_{i+1}]}}f(t)(x_{i+1}-x_i)$$ $$\lim_{||N||\to...

Not right 😒

Hi, PF $$\underset{t\in[x_i,x_{i+1}]}{\inf{f(t)}}\leq f(t_i)\leq\underset{t\in[x_i,x_{i+1}]}{\sup{f(t)}}$$ $$\displaystyle\sum_{t=0}^n{\inf f(t)}\leq\displaystyle\sum_{i=0}^n{f(t_i)}\leq\displaystyle\sum_{t=0}^n{\sup f(t)}$$ $$L(f)=\displaystyle\lim_{t_i...

Definetely true Yeah, runnig in circles, it's the raw evidence Is there an equivalence relationship between partition tags and bounded Riemann sums, as claimed in the original post? Quoted above is the way to prove it, if so? If that is the case, could I be given the next step? Regards!

Hi, PF I will quote the textbook to see if I have solved the question, that is, how can I state first ##x_{i-1}=\frac{(2i-1)}{n}<c_i<\frac{2i}{n}=x_i## for each ##i##, and observe that the sum is indeed a Riemann sum for ##f(x)## over ##[0,2]##: "Note in Figure 5.13 that ##R(f,P,c)## is a sum of...

$$L(f,P)=\sum_{i=0}^{n-1}\inf_{t\in [x_{i},x_{i+1}]}=\sum_{i=0}^{n-1}\sup_{t\in [x_{i},x_{i+1}]}=U(f,P)$$ This is my first step, but shouldn't be the last. Working on it. Brilliant remark. Unfortunately, the link to Wikipedia is only a hint to me.

$$a^{-1}=a^{0-1}=\frac{a^0}{a^1}=\frac{1}{a}$$ When it comes to functions that are not constants, the rule changes: Remark at "Calculus - A Complete Course 7th ed - Robert A. Adams and Christopher Essex": "Do not confuse the ##-1## in ##f^{-1}## with an exponent. The inverse ##f^{-1}## is not...

Hi, PF, there goes the definition of General Riemann Sum, and later the exercise. Finally one doubt and my attempt: (i) General Riemann Sums Let ##P=\{x_0,x_1,x_2,\cdots,x_n\}##, where ##a=x_0<x_1<x_2<\cdots<x_n=b##, be a partition of ##[a,b]##, having norm ##||P||=\mbox{max}_{1<i<n}\Delta...

Laws of Logarithms If ##x>0##, ##y>0##, ##a>0##, ##b>0##, ##a\neq 1##, and ##b\neq 1##, then (i) ##\log_a 1=0## (iii)##\log_a {(xy)}=\log_a x+\log_a y## (iii)##\log_a {\left(\dfrac{1}{x}\right)}=-\log_a x## (iv)##\log_a {\left(\dfrac{x}{y}\right)}=\log_a x-\log_a y## (v)##\log_a {(x^y)}=y\log_a...

Nice. Quite sure. Not familiar to improper integrals. However, I understand what you explain. Fine. Nice again. Your post is exciting, comprehensible, and it suits my background perfectly. In essence, perfect for me, helpful. Thanks!

The last thing I just read in the texbook is similar to your quote: "If ##f## is continous on ##[a,b]##, then ##f## is integrable on ##[a,b]##". Two sentences later remarks: "We cannot, however, prove this theorem yet". Conclusions: First, thank you; Second, I will continue reading the textbook...

Hi, dear Forum, @Svein A quick search on the Internet has not given any clue. Could you give me some hint?

Hi, dear PF I need some advice, or better said, opinions. One of the functions suggested on post #10 was $$f(x)=\begin{cases}{1}&\text{if}& x\in [0,\pi]-\{\pi/2\}\\0 & \text{if}& x=\pi/2\end{cases}$$ Now comes the controversy: if it is an avoidable discontinuity, it becomes a rectangle; And I...

Hi,PF Upper and lower Riemann sums The lower (Rieman) sum, L(f, P), and the upper (Riemann) sum, U(f, P), for the function ##f## and the partition ##P## are defined by: $$L(f,P)=f(l_1)\Delta x_1+f(l_2)\Delta x_2+\cdots+f(l_n)\Delta x_n=\sum_{i=1}^n f(l_i)\Delta x_i$$ $$U(f,P)=f(u_1)\Delta...

Hi, @Svein, were you intruducing Riemann Sums and Partitions? I've turned the page of the textbook, and I have stumbled upon something very similar to your quote, and I sense that more powerful for the calculation of areas (see attached figure): If ##A_i## is that part of the area under...

Hello, dear PF, I lack the knowledge to face this exciting question for now. I have made the decision to focus on the textbook. Thanks! P&L

Hi, something like this? It's cosine function, ##x\in{[0,\frac{\pi}{2}})\cup{(\frac{\pi}{2},\pi]}##

Good question. Continous in ##[0,\frac{\pi}{2})\cup(\frac{\pi}{2},\pi]##. Is this the suggested new condition? Intriguing indeed, in 48 hours I'll give a try. Regards!

Just the same, I would say. Now I realize. Is...It's the same thing. But I didn't see it. Best whises, @Mark44 !

Hi PF, I will try to give an example: ##y=2+\cos x##. The plot is a decreasing function in the interval ##[a,b]=[0,\pi]##. Let's make a partition of this interval. The points would have a shape like ##a=x_0=0<x_1<x_2<x_3\cdots<x_{n-1}<\pi=x_n=b##. Let's see what happens: no matter how big is...

Hi PF There goes the quote: The Basic Area Problem In this section we are going to consider how to find the area of the region ##R## lying under the graph ##y=f(x)## of a nonnegative-valued, continous function ##f##, above the ##x##-axis and between the vertical lines ##x=a## and ##x=b##, where...

Hi @jtbell, look for the best value for money first. The wheels are, in my personal opion, as important as the engine itself. It's pure safety on the move. Best wishes!

Wonderful post, @jtbell !

Corollary: "In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorfism or in some contexts linear function is a mapping ##V\rightarrow{W}## between two vector spaces that preserves the operations of vector...

Hi, PF, to better understand everything posted, I've looked for an example to be a little specific about what we discuss. It is better to separate the thread in two steps, and then join together (i) Given two vector spaces ##V##, ##W##, a linear mapping ##S:\;V\to W## is a map preserving the...

Hi, PF, there goes the quote from Calculus 7th ed. by Robert A. Adams and Christopher Essex: "When adding finitely many numbers, the order in which they are added is unimportant; any order will give the same sum. If all the numbers have a common factor, then that factor can be removed from each...

The living oldest person nowadays is this lady: https://en.wikipedia.org/wiki/List_of_the_verified_oldest_people Greetings, PF!

Well, in international maritime law, when two ships collide, the responsibility is, a priori, of both. I haven't seen the video; I sense it's biased. Personally think it's not worth it. ¿keys to failure in academia? My Jesuit education would direct me to look for the part of me that failed...

Thanks. I've always thought that this cat is been amplified, decontextualized, trivialized, all the words ending in a disturbing "ed" :frown:. Actually, the cat is just a fingered metaphor

Hi, @Seyara, welcome. No way: the coin has two sides. Here, everywhere. No comparison with the cat. Probability has nothing in common with quantum mechanics. Nevertheless, regard previous post. Mine is just an intuitive, naive, etc, opinion.

Hi, @BadgerBadger92. Newton: he put the first brick, provided all the background, to which we are all devoted. HNY

In Basque, yes, "a thousand thanks, from the heart". PD @FactChecker, I feel I should have written a few more about #30, but I've plunged headlong into the next chapter, integration. Love, greetings, PF.

Furthermore, it's the real thing:wink: Definitively, compañero. Yes. This... Was as simple as @PeroK knew; as hard as I turned it. A thousand thanks, milesker, bihotzez. HNY, PF!

Hi, isn't this the function? $$f(x)=\begin{cases}{\left|{x}\right|}&\text{if}&x\in\mathbb{Q}\\0&\text{if}&x\not\in\mathbb{Q}\end{cases}$$ FCh, which is the function?

Hi, PF, @Mark44, I wanted to explore every step carefully. The goal was to ensure that we have a composition of functions of the real numbers in the real numbers and eventually ensure the prevalence of step D over E.

Hi, PF, d) step is what we want to prove; it is a consequence of e) step; ##\xymatrix{ A=\mathbb{N}\ar[r]^{x_n} \ar[dr]_{f(x_n)} & x_{n\in\mathbb{N}}\subset \mathbb{R}\ar[d]^f \\ & \mathbb{R} }## I've been given the proof for an arbitrary sequence: Let ##\{x_n\}_{n=1}^{+\infty}## be an...

You don't need to choose. ¡Qué pasada! What a blast! Certainly there are many kinds of intelligence. They are genious, I mean the American football players...

Attempt (i) I'm dealing with a piecewise function. (ii) A piecewise function can be continuos if each function that makes it up is continous. (iii) (iv) ##f(x)=\begin{cases}{\left |{x}\right |}&\text{if}& x \in \mathbb{Q}\\ 0 & \text{if}& x \not \in \mathbb{Q}\end{cases}## (v) -...

All right, ##\Big\{x_k\Big\}_{k=1}^{\infty}## with ##x_{k}\in{\mathbb{R}}## for all ##k##, such that ##x_{k}\rightarrow{0}## might be, if (I wonder), ##k\in{\mathbb{R}}## (so some terms might be rationals, others not. If I would be right in my previous sentence...Am I? PD: PF, I've found where...

Hi, @Spathi, the study focused on quantitative or qualitative relationships ?.

Attempt to understand this approach The distance between ##f(x)## minus ##\lim_{x\rightarrow{\infty}}{f(x)}=f(0)## is less or equal than that of the argument: hence, it shows a significant trend line of ##x##

Very enriching words. But I was asking the Forum to check my point of view about these two limits to obtain ##e##, which I consider as a ##\lim_{x\rightarrow{\infty}}##, for ##x\in{\mathbb{R}}##, on one side, and the equivalent way to reach ##e##, through the limit of a sequence...

Hi, @FactChecker, the given function is continous only at one point, at zero: $$f(x)=\begin{cases}{\left |{x}\right |}&\text{si}& x \in \mathbb{Q}\\ 0 & \text{si}& x \not \in \mathbb{Q}\end{cases}$$ It is just to provide an example where to prove continuity with sequences. It is not...

Hi, take this as something speculative, for I'm just guessing. Let ##\epsilon>0##. We choose ##N\in\mathbb{N}## such that ##N>\dfrac{1}{\epsilon}##. Such a choice is always possible by the Archimedean property. To verify that this choice of ##N## is appropiate, let ##n\in{\mathbb{N}}## satisfy...

Good remark. My intention was to began proving that, as ##n\rightarrow{\infty}##, some ##\{a_n\}\rightarrow{0}##, and go pulling the thread. I suppose this is not the way. You mention to give the definition of continuity. At a point, or at an interval?. At an interior point ##c## of its domain...

Fine, got the path

##f(x)=\begin{cases}{\left |{x}\right |}&\text{if}& x \in \mathbb{Q}\\ 0 & \text{if}& x \not \in \mathbb{Q}\end{cases}## To evaluate the limit of ##f## when ##x## tends to ##0## just notice that if the sequence ##\{x_n\}_{n=1}^\infty## tends to ##0##, then ##f(x_n)=\begin{cases}{\left...

I've been given the proof, but don't understand; to calculate the limit of ##f## when ##x## tends to zero it's enough to see that if ##\{x_n\}_{n=1}^\infty## is a sequence that tends to ##0##, then...