Math's Problem and Solution

Exercises
1.Evaluate :
Int dx/[(x+2)(3-x) from 1 to -1
2.Find int 3^[(2x+1)^1/2]dx
Show that the surface x^2-2yz+y^3= 4 is perpendicular to any member of the family of surfaces x^2+1=(2-4a)y^2+az^2 at the point of intersection (1,-1,2)!
Evaluate
Int dx/ (5+3cos x) by using the substitution tan x/2= u
Prove that 1/1.3 +1/3.5 + 1/5.7+ … =sigma nequals 1 to infinite 1/(2n-1)(2n+1) converge and find its sum.


Answered By Heni Ayu Pertiwi:
1.Method 1
Int dx/[(x+2)(3-x)]^1/2 = int dx/ (6+x-x^2)^1/2 = int dx /[ 6- (x^2-x)]^1/2 = int dx / [ 25/4 –(x-1/2)^2]^1/2
Letting x-1/2= u, this becomes
Int du/ (25/4 – u^2)^1/2 = sin ^-1 u/ (5/2) + c = sin^-1 (2x-1/5) + c
Then
Int dx/ [(x+2)(3-x)]^1/2 from -1 to 1 = sin^-1 (2x-1/5) from -1 to 1 = sin^-1(1/5) – sin^-1(3/5) = sin^-1 . 2 + sin^-1. 6
Method 2
Let x-1/2=u, as in method 1, now whwn x= -1, u=-3/2, and whwn x=1, u=1/2. Thus,
Int dx/[(x+2)(3-x)]^1/2 from -1 to 1 = int dx / [ 25/4 –(x-1/2)^2]^1/2 from -1 to 1= int du/[25/4 – u^2] from -3/2 to ½ = sin^-1 u/(5/2) from -3/2 to 1/2 = sin^-1 . 2 + sin^-1. 6

2.Let (2x+1)^1/2 = y , 2x+1= y^2 then dx = y. dy and the integral becomes
Int 3^y. ydy integrate by parts, letting u= y, dv= 3^y
Then du=dy, v= 3^y/ (ln 3) , and we have
Int 3^y.y dy = int u dv – int v du
=y.3^y/(ln 3) – int 3^y/ (ln 3) dy
=y.3^y/(ln 3) – 3^y/(ln 3)^2 + c
3.Let the equations of the two surfaces be written in the form
H= x^2 – 2.y.z + y^3 -4=0 and K=x^2+1-(2-4a)y^2-az^2=0
Then,
Vf= 2xi+(3y^2-2z)i-2yk, Vk=2xi-2(2-4a)yj-2azk
Thus, the normals to the two surfaces at (1,-1,2) are given by
N1= 2i-j+2k
N2= 2i +2(2-4a)j-4ak
Since, N1.N2=2.2 -2(2-4a)-2.4a=4-4+8a-8a=0
It follows that N1 and N2 are perpendicular for all a and so the required result follows.
4.Sin x/2 = 4/(1+u^2)^1/2, cos x/2 =1/(1+u^2)^1/2
Then, cos x= cos^2 x/2 – sin^2 x/2= 1-u^2/ 1+u^2
Also du= ½ sec^2 x/2 dx
dx=2cos^2 x/2 du= 2du/1+u^2
thus the integral becomes int du/u^2+u =i/2 tan^-1 u/2 +c +1/2 tan^-1 (1/2 tanx/2) +c
5.Un= 1/(2n-1)(2n+1)= ½[1/(2n-1) – 1/(2n+1)], then
Sn= U1+U2+U3+u4+….+Un=1/2(1- 1/3)+1/2(1/3 – 1/5)+…+1/2(1/2n-1 – 1/2n+1) = ½ (1-1/3+1/3-1/5+1/5-…+1/(2n-1)-1/(2n+1))= ½(1 – 1/2n+1)
Since limit n approache to infinite Sn = limit n approache to infinite ½(1- 1/2n+1)=1/2, the series converges and its sum is 1/2

Now, I am Aware that English is very Important for Communicating Mathematics at International Level

English is considered to be the most important and common language of the world today. A great number of people understand and use English in every part of the world. Because of its importance I have chosen English as a second language after Indonesian.

English is the most useful language. Being good at English, we can travel to any place or any country we like. We shall not find it hard to make others understand what we wish to say.

In Fact, English has become the international language. Everywhere people talk English. Because of his popularity, innumerable books have been written in English. The English help to spread ideas and knowledge to all the corners of the world. There is no subject that cannot be learned in the English language. So one of my ways to improve my English is read books on variety of subjects: economics, philosophy, physiology, mostly books which linking to Mathematics such geometry, calculus,algebra, history of mathematics,etc.

To acquire correct and fluent English I should learn basic rules of English Grammar and practice speaking English. Like idiom:"Practice makes perfect". I should frequently listen to native speakers through videos, television, and radio. Taking action is better than waiting, even if it's just a small step.

Now I am aware that English is important. Being aware of the preeminence of the English language in every aspect of man's activity, I have been learning it great zeal and avidity. I remember the phrases: If you dream it, you can achieve it.

Now I am study in Mathematics study program. That's mean I should have communicated English for Mathematics. As we know, English has been used in all important meeting and conferences at International level. As a person who know English, easily get more knowledge from many parts of the world. It is for all these reasons that I want to improve my English. So...What will I do? The most frequently heard question in my life maybe "What will I do?". My answer to this question can start changes in my life. Then I am describing the how and why of what I will doing.

I will study more diligent to improve my English for Mathematics. That's the way to enter International level. Positive attitude and action will lead me to better life through Mathematics because Mathematics not isolated from part of our life.

Dictionary of Mathematics A-G

A
Abad century
Absis abscissa
Akar root, radix
Akar Kuadrat square root
Akibat implication
Aksioma axiom
Alat instrument
Alat pembagi dividers
Algoritma algorithm
Aljabar algebra
Analisa analysis
Anggota member
Arah direction
Aturan,hukum rule
Awal initial
B
Bagian Berganda compound proportion
Bagian dalam interior
Bagian Luar exterior
Balok cuboid
Bandul pendulum
Bangun,bentuk shape
Bangun,struktur structure
Banyaknya(jumlah) quantity
Baris row
Barisan sequence
Barisan Fibonacci Fibonacci sequence
Batas boundary
Bayangan image
Bebas Independent
Belahan section
Belah ketupat rhombus
Belasan teens
Benda-benda plato platonic solids
Benda-benda putaran torus
Benda ruang solid
Bentuk baji cuneiform
Bentuk nyata significant figure
Bentuk telur ovoid
Bentuk trapezium trapezoid
Berayun oscillate
Berat weight
Berat jenis specific gravity
Berat keseluruhan gross weight
Berbanding langsung direct proportion
Berbentuk bilangan numerical
Berdimensi tiga three-dimensional
Berhadapan subtend
Berikutnya succession
Berkelompok associative
Berpasangan corresponding
Berpencar distribute
Bertemu,berpotongan meet
Bertentangan paradox
Berurutan consecutive
Besaran berganda compound quantity
Besarnya magnitude
Bidang plane
Bidang banyak polyhedron
Bidang empat tetrahedron
Bilangan number
Bilangan berarah directed number
Bilangan bertanda signed number
Bilangan-bilangan asli natural number
Bilangan bersahabat amicable number
Bilangan-bilangan nyata riil number
Bilangan-bilangan prima prime number
Bilangan bulat integer
Bilangan campur mixed number
Bilangan ganjil odd number
Bilangan genap even number
Bilangan irasional irrational number
Bilangan kardinal cardinal number
Bilangan khayal imaginary number
Bilangan kompleks complex number
Bilangan kuadrat square number
Bilangan majemuk composite number
Bilangan sempurna perfect number
Bilangan yang dibulatkan rounded number
Bintik,noktah dot
Bola sphere
Bukti proof
Bulat telur oval
Bunga interest
Bunga berganda compound number
Bunga tunggal simple interest
Busur arc
C
Cacah digit
Cacah dua bit
Cairan dan gas fluid
Cakram disc
Cara kesatuan unitary method
Cara penguraian decomposition method
Catatan notation
Cekung concave
Cembung convex
Cepat fast
Cincin ring
D
Dadu die (jamak: dice)
Daerah domain
Dasar,unsur elementary
Dasar level
Dasawarsa(10 tahun) decade
Data data
Datar flat
Daya,pangkat power
Derajat degree
Deret series
Deret hitung arithmetic progression
Desimal decimal
Detik second
Diagram diagram
Diagram Venn Venn diagram
Dimensi dimension
Ditabelkan(tabulasi) tabulate
Dua dimensi two dimensions
Dua kali,rangkap double
Dua suku binomial
E
Elips ellipse
F
Faktor factor
Faktorial factorial
Faktor persekutuan common factor
Faktor prima prime factor
Fungsi function
G
Gabungan union
Gambar figure
Gambar-gambar bidang plane figure
Gambar-gambar sebangun similar figure
Gambar sama dan sebangun congruent figure
Garis line
Garis bandul plumb line
Garis bangun contour
Garis berarah directed line
Garis bilangan number line
Garis bujur meridian
Garis-garis mencong skew lines
Garis keliling perimeter
Garis lintang latitude
Garis lurus straight line
Garis miring solidus
Garis panah arrow line
Garis patah broken line
Garis pokok base line
Garis potong secant
Garis sudut-menyudut diagonal
Garis tengah diameter
Garis tinggi altitude
Gaya force
Gesekan friction
Gigi(mesin) gear
Goresan locus
Grafik graph
Grafik balok block graph
Grafik gambar pictogram
Grafik garis line graph
Grafik kolom column graph
Grafik panah arrow graph
Grafik ruji bar graph
Gram gram/gramme
Gravitasi gravity
Gros gross
Grup group

What I've Done and What I'll Do about Improving My Compentence of English for Mathematics?

When Mr. Marsigit ask that question to me, I immediately to think hard. After that, I get numerous answers to the first question.What I've done about improving my competence of English for Mathematics?
It's hard question for me. When I was in senior high school, I tried to improve my English skills (reading,writing,speaking,listening,etc) used some ways such as:

• learn a new word every day
• watch an English Film at least once a month
• change my computer and mobile phone language settings from Indonesian to English
• Watching TV in English helps a lot
• Join an English course
• Read novels, books, newspapers, magazines which can enrich my vocabulary and help me with structure, I don't try to understand every word but I try to understand the overall meaning of a sentence of passage
• If I'm to tired to activity practice just relax and listen to some English for songs or radio stations

Now, I'm not student of Senior High School. I'm student of university study program Mathematics. That means, I should can communicate mathematics in English. I've more responsibility to study hard. I'll study more hard to improve my competence of English for Mathematics. Reading mathematics book is the first step that I do, then I write down the words I don't know. After that, translate them. It will be more difficult but never impossible.

I would like to read mathematics book because it will enrich my vocabulary. Make a habit of reading book regulary mathematics book as many as I can find. Again that should be fun to make sure the books I choose are not too difficult for me. It's great idea. To reach the goal, I must have motivate .

Book Review

PREFACE

First of all, we be grateful to Allah because we still given ability to review Mathematics for Junior High School year VIII. We would like to thank to Mr Marsigit who give us opportunity to review Mathematics for Junior High School year VIII.

There are some benefit that we get while review Mathematics for Junior High School such as be remember with grammar and tenses, know more about the different between curriculum mathematics now and 6 year ago, etc.

We hope this review book useful. It is intended to students and teachers of Junior High School year VIII as an opinion before buy mathematics book.

To release of this review book has been made possible due to the assistance and contributions of various people who cannot mention one by one. To all who involved in this preparation of this review book, I would like to express my high appreciation and gratitude.

Comments and suggestions to improve the contents of this review book are always welcome.

Yogyakarta, April 2009
Reviewers

written by: Enti Dwiningsih

I.CONTENT

There are 7 chapters in this book. All of them are used to learn mathematics for junior high school year VIII. The material divided into 2 units; that is:
UNIT I : Algebra
Chapter:
1. Algebra and its Applications
2. The Relation and Functions
3. The Equations of a Straight Line
4. The System of Linear Equations in two variables
UNIT II : Geometry and measurement
Chapter:
5. The Pythagorean Theorem
6. A Circle
7. Polyhedral
I think, the content of this book is complete enough. There is a previous in every chapter that make the readers know what will they learn in that chapter. All of the chapters are arranged by some subtitle. It is complete with the definitions, examples (and its problem solving), and more exercise. To remember the material or some formula there is an “be remember” in some page in every chapter. It’s content some formula or material briefly. So, it can help the reader (especially the students) to remember the materials of every study easily.

Beside the exercise of every subtitle, there is an exercise in the end of every chapter that content of some mathematics problems suitable to the material of every chapter. There are 20 multiple choices problems and 5 essays. At the end of every unit, there is an evaluation. This book also be completed with “final evaluation” at the last part. Student must solve 30 multiple choices problems and 10 essays.

Generally, this book is very good and interest for student year VIII. Its content is suitable to the curriculum of study in junior high school in Indonesia at this moment (suitable to KTSP) . The problem in the exercise and evaluations are realistic problems. We can find the problems in our daily activities. So, it makes the student can imagine and solve the problem easier.

The excellence of this books is because it is a bilingual books. So, there are two languages (Indonesian and English language) in one book. Every Indonesian page translate to English directly. It is very useful, because with reading this books, students not only can improve their mathematics skill and knowledge but also develop their English. I think this book is suitable to some school to be world class school in Indonesia (school with international school standardization).

Over all, the content of this book is complete enough and simple (not too difficult to understand all of material mathematics on it). The point plus of this book is about it bilingual. So, I recommend this book to all of student year VIII in Indonesia. I hope it can help you to learn mathematics easier.

written by: Artika Kristianingrum

II. PROBLEM

In this book, there are seven chapters. There are exercises in each chapter. In this book also existed three evaluations : evaluation 1 available after 5 chapter, evaluation 2 available after last two-chapter and evaluation final available in the end. Evaluation was made to examine how far the students understand the substances which are in this book.

In this book there are the problem examples which are clarified in each chapter. The problem examples which are given consist of substances which are stated. In examples is given many examples problem solving which is enable the students to choose the easiest way. there are problem solving. After given examples, there are exercises for students. The form of exercises is varied and still consist all of the substances. Method of problem solving is stated and sequential.

written by: Soffia Anisa H.A.C

III. INFORMATION

Book mathematics for junior high school year VIII by Marsigit give complete information about its chapter, so that easy to understand. Reasons information of this book easy to understand are:
1. Each chapter explained in detail that is by sub chapter
2. Presented with bilingual that is in Indonesia and in English, so English people also can study it.
3. Each sub chapter explained theoretically, given example exercise, and exercise. So, after understand explained fill chapter, then can understand example exercise, so can finish exercise. In this book there are seven chapters, each chapter explained in detail became sub chapter.
4. explained fill chapter not out from studied problem.
5. There are chapter explained with picture ( more use picture . For example, in chapter six is circle, this book more use picture.

To limiting reader to understand information of this book, depended from reader. But, in general this book present information in detail.

written by: Dian Tri Handayani

IV. INTEREST

Mathematics for Junior High School year VIII which is written by Mr. Marsigit is different from the other mathematics book commonly. This book is published in bilingual edition (Indonesian-English). We can get many advantages by learn this book such as get mathematics knowledge. The students of junior high school year VIII also can increase vocabulary owing to mathematics.
This Book is suitable used to the students or teachers of international school standardization and students which want to continue their studies abroad. On the other hand, this book will inspire them to study better.

Not all people like reading book with black-white color. Some prefer like reading full color book. Mathematics junior high school year VIII as a full color book that will stimulate students more interest to study. Talk about the substance , this book explain the substances clearly. In the end of chapter, there is exercise which consist of 30 multiple choices and 10 essay. The purpose is students can apply what has been him learn. This book has some of ancient mathematician story such Al-Khwarizmi.

Written by: Enti Dwiningsih


Group III:

1. Enti Dwiningsih (chairman)
2. Soffia Anisa H.A.C (member)
   
3. Dian Tri Handayani (member)
   
4. Artika Kristianingrum (member)
   

The Sum of The Angles of a Triangle

In this explanation will be discussed how to prove the theorem of the sum of the angles of a triangle. Pay attention to the explanation below:

Theorem:

"The sum of the angles of a triangle is 180"

Let the straight line BC

Let C be a point between B and D

Let E be a point such that A and E are on the same side of line BD

Suppose CE to be the line through C parallel to line BA

Playfair's Axiom:

"Through a given point there can be only one straight line parallel to a given straight line"

Considering to the playfair's axiom, line CE is the only one straight line that parallel to line BA

Theorem 1

"If two parallel line are cut by a tranversal, the alternate interior angles are equal"

Since AB and CE are parallel and AC as tranversal, by theorem 1 we obtain:

the measure of ACE angle equal to the measure of BAC angle

Theorem 2

"If two parallel line are cut by a tranversal, the corresponding angles are equal"

Since AB and CE are parallel and BD as tranversal, by theorem 2 we obtain:

the measure of ECD angle equal to the measure of ABC angle

Corollary:

"For any triangle, the measure of an exterior angle is the sum of the measures of its two remote interior angles."

The measure of ACD angle is the sum of the measure of CAB angle and the measure of ABC angle.

As we know, the sum all of angles in straight line is 180

So we obtain,

The sum of the measure of BCA angle and the measure of ABC angle equal to 180

Since the sum of the measure of CAB angle and the measure of ABC angle equal to the measure of ACD angle

Thus,

The sum of the measure of BCA, the measure of CAB angle and the measure of ABC angle equal to 180

which was to be proved

Math Words and Phrases

Chord is the segment of a secant joining the two points of intersection with the circle. Diameter can mean any chord passing throught the center.

Plane Section is the intersection of a plane and a surface or a solid.

Arithmetic Sequence/ Arithmetic Progression:
A sequence, each term of which is equal to the sum of the preceding term and a constant, written
a, a+d , a+2d, ..., a+ (2n-1)d
where a is the first term, d is the common difference or simply the difference, and a+(2n-1)d=l is the last, or nth ,term.

Edge is a line or a line segment which is the intersection of two plane faces of a geometric figure, or which is in boundary of a plane figure. Examples are the edges of a polyhedron and of polyhedral angle, and the lateral edge of a prism.

Altitude is a line segment indicating the height of a figure in some sense (or the lenght of such a line segment).

SET
A set is a collection of elements defined in such a manner that it is possible to determine what elements are in the set and what elements are not in the set. It's assumed also that the elements of a set are distinct. In he material that follows, sets are denoted by name or by capital letters, as the set of integers, or the set A.
Just what items are to be included as the members of a particular set can be indicated in several ways, as listed below.

• A={a,b,16,Jhon}.The members of set A are a,b,16 and Jhon, and this is indicated by listing the elements are enclosing them with a pair of braces {}.
• B={0,1,2,3,4,5,...,98}.The members of set B are the "counting numbers" from 0 to 98 inclusive. The three dots,...,indicate that the pattern established by listing the first several elements is to be used in obtaining the remaining members of the set. The 98 following the three dots indicates that the last elements of the set is 98. This method should not be used unless the pattern is rather obvious.
• C={0,1,2,3,4,5,...}. Set C consists of all the "counting numbers". The three dots indicate that the established pattern is to be used to obtain the remaining members of the set, and the absence of a numeric symbol following the three dots indicates that the set goes on "forever".
• D={x/x has been a president of the United States}. Set D is defined with what is reffed to as the set-builder notation. This notation is read as : the set of elements x such that x has been a president of the United States . The braces indicate that it is to be a set; the x portrays a representative element of the set; the vertical line is read "such that"; and the remainder of the notation spells out the conditions necessary for an item to be a member of the set.

THE EMPTY SET OR NULL SET
Just as its name implies, the empty set is the set that contains no elements. In set algebra the empty set plays a role similar to the role played by zero in arithmetic. The empty set is a subset of every set, including itself.
Some examples of the empty set:
A={b/b is a human being and b is 20 feet tall}
B={}

PARALLEL
Two straight lines on a two-dimensional surface are parallel if they do not intersect(cross) no matter how far they are extended. In figures, pairs of arrow like >,>>,and>>> will mean that the lines are parallel.

whole number

A natural number;0,1,2,3,4,...

PROBABILITY
The probability of an event is the ratio of the number of time it occurs to the large number of trials that take place; the mathematical model of probability is a positive measure which gives the measure of the space the value 1.

INTEGRAL FUNCTION
Integral Function is a function taking on integer values
Integral consist of two kind:

• definite integral / definite Riemann Integral
• indefinite integral is an integral at least one of two whose limits of integration is finite.

ALTERNATE INTERIOR ANGLES
are the angles between a pair of line that switch sides of a third line.

VERTICAL ANGLES
are the two opposite (that is non-adjacent) angles formed by two intersecting lines. Vertical is a relationship between pairs of angles, so you cannot call one angle a vertical angle.

A SEGMENT
a segment is a region bounded by a chord and an arc lying between the chord's endpoints.

Taken from various resources such:
http://www.mathpropress.com/glossary/glossary.html
Mathematics Dictionary Fourth Edition by James
College Preparatory Mathematics 2 (geometry) by Salle, and friends
Fundamentals of College Mathematics by Donald Herrick
Dictionary of Mathematics second edition by McGraw-Hill

http://en.wikipedia.org/wiki/Circle

INTRODUCTION TO ENGLISH 1

What is the first thing that crosses your mind when you hear English?

When I ask that question to people, I usually get numerous responses. Some directly think about international language. Some refer to tenses, gerund, modals, conjunction, etc.

Well, it's nice responses. English is useful and so is mathematics. Therefore, I still going study English in University. I'm student of pure mathematics and I want be able to communicate Mathematics in English.

It was February 17, Mr. Marsigit, M.A had opened his lesson "ENGLISH 1" in my class. First of all, he introduce himself to us. Dr. Marsigit, M.A came from Central Java province. Now, he is 51 years old. I guess that his educational background had helped him to became a successful man. He also said about his journey experiences round the world.

Then he tought us how to make a blog and became follower of his blog. Besides, he gave us advise about commpetence.
How to became commpetence:

• Motivation
• Make sure your Behaviour or attitude support your purpose
  
• Knowledge
• Skills
• Experience

Finally, he closed our first lesson . I would like to thank to Mr. Marsigit because of him point of view, I get many advantages.