Calculus Exercises for Japanese High School Students

[YAMATO2201]

Please look at EXERCISE #3 in the following exercise sheet.

Calculus Exercises for Japanese High School Students (SAMPLE)
(This file is intended for returnees from English-speaking countries.)

Have I made any mistakes?

 

[tedmc]

I think "sketch the region" is vague.
Do you mean "sketch/draw the graph"?

 

[YAMATO2201]

I think "sketch the region" is vague.
Do you mean "sketch/draw the graph"?

Does the following version work for you?

Sketch the region in the plane consisting of all points (x,y) such that ....

 

[tedmc]

Do the points form a line or an area?

 

[YAMATO2201]

Do the points form a line or an area?

They form an area.

 

[tedmc]

Call it "an area" then - sketch/draw the area covered by the points..
I think it is not necessary to state "the plane" if you are only taking about two-dimension, i e. the x and y axes.

 

[YAMATO2201]

Call it "an area" then - sketch/draw the area covered by the points..

I don't have the faintest idea why the "region" does not work for you.

I think it is not necessary to state "the plane" if you are only taking about two-dimension, i e. the x and y axes.

I think "in the plane" is optional.

 

[tedmc]

I don't have the faintest idea why the "region" does not work for you.

The term "region" sounds alien to me as far as maths is concerned. It would be appropriate when we talk about geography.
Why do you prefer the word?

 

[YAMATO2201]

The term "region" sounds alien to me as far as maths is concerned. It would be appropriate when we talk about geography.
Why do you prefer the word?

I just found this:

Find the area, S, of the shaded region enclosed by the given curve, the given line and the x-axis.
http://mathcentral.uregina.ca/QQ/database/QQ.09.08/h/rogerson2.html

 

[bubbha]

The word "region" is commonly used in mathematics. It has a rigorous definition in topology, but is used more generally in calculus, geometry, or other branches where line segments, areas, volumes etc. are covered.

 

[YAMATO2201]

Do the points form a line or an area?

The region described in the exercise is {(x,y) | 0<x≤1 ∧ 0<y≤1 ∧ x≠y}.

This set is not a 'region' under the following definition because it has two connected components.

Definition 4.45. A set in R[SUP]n[/SUP] is called a region if it is the union of an open connected set with some, none, or all its boundary points.
T.M. Apostol: Mathematical Analysis, 2nd edition

(Please note that under this definition, every region is connected.)

However, the exercise is within the context of non-rigorous calculus, so it would be legitimate to use the "region".

 

[YAMATO2201]

Is it common in high school mathematics to use "verify" in place of "prove"?
(Please look at EXERCISE #5, (1).)

 

[tedmc]

The region described in the exercise is {(x,y) | 0<x≤1 ∧ 0<y≤1 ∧ x≠y}.

This set is not a 'region' under the following definition because it has two connected components.

Definition 4.45. A set in R[SUP]n[/SUP] is called a region if it is the union of an open connected set with some, none, or all its boundary points.
T.M. Apostol: Mathematical Analysis, 2nd edition

(Please note that under this definition, every region is connected.)

However, the exercise is within the context of non-rigorous calculus, so it would be legitimate to use the "region".

So there is the technical definition of "region" which has to do with subsets, and the general meaning which is the area of a surface bounded by lines. I think you mean the latter but you have used "region" so as not to be confused with "area" as in a measurement of the surface.

I have seen "prove that..." more often in maths than "verify..", though they mean the same thing. The latter is more common in science e.g. in carrying out experiments.

 

[YAMATO2201]

there is the technical definition of "region" which has to do with subsets,

There's more than one definition of "region".

A connected open set in ℂ will be called a region.
Stein and Shakarchi: Complex Analysis, Princeton University Press

I think you mean the latter but you have used "region" so as not to be confused with "area" as in a measurement of the surface.

I used the word "region" simply because I thought the context would make its meaning clear. However, it is true that "area" can mean "the quantitative measure of a plane or curved surface". (area) "Determine the area consisting of all points (x,y) such that ..." might cause some confusion. ;-)

 

[YAMATO2201]

I have seen "prove that..." more often in maths than "verify..",

I heard through the grapevine that in some Asian countries, the majority of education is conducted in English. What about in your country?

 

[tedmc]

I heard through the grapevine that in some Asian countries, [STRIKE]the majority[/STRIKE] most of the education is conducted in English. What about in your country?

Malay (similar to Indonesian) is the national language of Malaysia, being the language of the majority race, the Malays, (the population comprises 50% Malays, 23% Chinese, 7% Indians, 12% indigenous people and 8% other) but English is widely used in private sector. The medium of education is Malay, with English as the second language. English, being a legacy of the British colonial days, used to be the medium of education until the 70s, . The public universities now use Malay as the medium of instruction while the private universities use English. There are also tri-lingual schools at primary level to cater to students whose mother tongues are Chinese and Indian. Due to the diverse population, the spoken languages are even more diverse, the main ones being Malay, English, Chinese (with the various dialects) and Indian (Tamil). The switch of the language of education from English to Malay has resulted in the decline in standard of the English language in Malaysia, unlike in Singapore where it is still the main language.

The other Asian countries where English is widely used are those that are the former British colonies - Singapore, Borneo (part of Malaysia), Burma, India, Pakistan, Sri Lanka and Hong Kong. Engish is also the main language in the Philippines from US influence. Thus, they are in a better position than countries where there is hardly any exposure to English in everyday life - Thailand, Vietnam, Indonesia, China, Korea, Taiwan and Japan.

 

[Tdol]

English is also the main language in the Philippines from US influence.

I lived in the Philippines for three years. Most professional people can speak it fairly well, but the genuine native speakers are the elite. I would actually put English behind local languages like Tagalog, though if you go into a shop, you can't get served in English easily enough.

 

[tedmc]

By main language, I mean it is the medium of instruction in schools, the public and private sectors. The native Tagalog is most widely spoken of course.

 

[Tdol]

It is used in schools, though I think this may be a tool for exclusion for some.

 

[YAMATO2201]

Thank you very much, tedmic. :-D

I seem to have made a terrible mistake. I should have written "... most classes are conducted in English".