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1. ### I Express the limit as a definite integral

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...
2. ### I Express the limit as a definite integral

Hi, PF, wrong again :smile:

6. ### I Express the limit as a definite integral

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!
7. ### I Express the limit as a definite integral

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...
8. ### I Express the limit as a definite integral

$$L(f,P)=\sum_{i=0}^{n-1}\inf_{t\in [x_{i},x_{i+1}]}=\sum_{i=0}^{n-1}\sup_{t\in [x_{i},x_{i+1}]}=U(f,P)$$ 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.
9. ### Solving a⁻¹ - b⁻¹ = b - a /(ab)

$$a^{-1}=a^{0-1}=\frac{a^0}{a^1}=\frac{1}{a}$$ 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...
10. ### I Express the limit as a definite integral

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...
11. ### How to solve this log equation?

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...
12. ### I The Basic Area Problem (introduction to the topic of integrals)

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!
13. ### I The Basic Area Problem (introduction to the topic of integrals)

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...
14. ### I The Basic Area Problem (introduction to the topic of integrals)

Hi, dear Forum, @Svein A quick search on the Internet has not given any clue. Could you give me some hint?
15. ### I The Basic Area Problem (introduction to the topic of integrals)

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...

46. ### Why does this function make it easy to prove continuity with sequences?

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...
47. ### Why does this function make it easy to prove continuity with sequences?

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...
48. ### Why does this function make it easy to prove continuity with sequences?

Fine, got the path
49. ### Why does this function make it easy to prove continuity with sequences?

##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...
50. ### Why does this function make it easy to prove continuity with sequences?

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...