CBR Calculus Diferential

 

CRI TI CAL BO BOO OK RE PO PORT RT “CALCULUS DIFERENTIAL: LIMIT”  

L ect ctur ure er : H ana D H ut uta abar at, M,S M,Si  i  

CREATED BY: PALAGUNA SIAHAAN 4193131017

STUDY PROGRAM S1 BILINGUAL CHEMISTRY EDUCATION FACULTY OF MATHEMATICS AND NATURAL SCIENCE STATE UNIVERSITY OF MEDAN 2019

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PREFACE Praise the authors say the presence of God Almighty, for His blessings so that the author can complete the Critical Book Report (CBR) to fulfill the assignments of Differential Calculus course. In the preparation of this task or material, not a few obstacles faced by the author. Therefore the author would like to thank the lecturers who have helped in the smooth writing of this CBR. In writing this CRITICAL BOOK REPORT, the author has tried to present the best. However, there may still be errors in the writing. The writer hopes to get criticism and input from readers. The author also hopes that this t his CBR can provide information and have benefits for all parties.

Medan, 23rdSeptember 2019

Author

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TABLE OF CONTENTS PREFAC PRE FACE E .......................................................................................................... 2 TABLE TAB LE OF CON CONTEN TENTS TS ................................................................................... 3

CHAPTER I : I NTRO NTRODUC DUCTI TI ON  ........................................................................ 4 1.1Background 1.1Backgr ound..................................................................................................... 4 1.2Objective......................................................................................................... 4 1.2Objective 1.3Benefits 1.3Bene fits ........................................................................................................... 4

CHAPTER CHAPT ER I I : DI SC SCUS USSIO SION  N  ............................................................................. 5 2.1 Identity Identity of Book .............................................................................................. 5 2.2 Summary Summary of Book ........................................................................................... 3 2.2.1 Book 1 .........................................................................................................6 2.2.2 Book 2 .........................................................................................................8 2.2.3 Book 3 ....................................................................................................... 12

CHAPTER CHAPT ER I I I : ASS ASSESSM ESSMENT  ENT  ........................................................................ 13 3.1.1 Advantages of Book 1 ................................................................................ 13 3.1.2 Advantages of Book 2 ................................................................................ 13 3.1.3 Advantages of Book 3 ................................................................................ 13 3.2.1Disadvant 3.2.1Dis advantages ages of Book 1 ............................................................................ 13 3.2.2Disadvant 3.2.2Dis advantages ages of Book 2 ............................................................................ 13 3.2.3Disadvant 3.2.3Dis advantages ages of Book 3 ............................................................................ 13 CHAPT CHA PTER ER IV: CLO CLOSIN SING G .............................................................................. 14

4.1.Conclusion 4.1.Conclusio n ................................................................................................... 14 4.2. Suggestion Suggestion ................................................................................................... 14 REFERE REF ERENCE NCES S ................................................................................................. 15

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CHAPTER I : I NTRO NTRODUC DUCTION TION 1.1 I ssu ssue e B ackg ckgrr ound The background of writing this paper is to fulfill the tasks given by calculus lecturers.Limit are one of the foundations or foundations in analysis so that mastery of various concepts and  principles of limit limit functions can help in solving a problem in everyday life. A function can be analyzed based on the idea of going up or down, optimizing, and turning points using the limit concept. In the following sections, we will try to observe various real problems and study several cases and examples to find derivative concepts. In everyday life, we often encounter the rate rat e of change. The rate o off change is closely related to speed. In the following following discussion, it will focus on comparisons co mparisons between two different books on limit material.

1.2 Objective 1. Describe the concept of o f algebraic function limits limits using real context and apply them appropriately, systematically, and creatively. 2. Understanding the characteristics of the limit function and solving problems with these attributes appropriately and responsibly. 3. Choose an effective strategy and present a mathematical model in solving real problems about the limits of algebraic functions precisely, systematically, systematically, and creat ively.

1.3 Benefits Based on the background above, the objectives to be achieved in this critical book report are: 1. To deepen knowledge about the Limit 2. To add insight into how to criticize books. 3. To find out which book is good and quality

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CHAPTER II : DISCUSSION

 2.1

I denti nti ty of Book Book A.Book 1

Title of Book

: KalkulusUntukPerguruanTinggi

The Author

: Mhd Daud Pinem, S. T ., M. T .

The Publisher

: RekayasaSains, Bandung

Published Year

: 2015

B. Book 2 (Main Book)

Title of Book

: Calculus Early Trascendental, 10th Edition

The Author

: Anton Bivens Davis

The Publisher

: Laurie Rosatone

Published Year

: 2012

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C. Book 3

Title of Book

: Kalkulus Integral danAplikasinya

The Author

: Didi Diditt Budi Nugroho

The Publisher

: GrahaIlmu, Yogyakarta

Published Year

: 2012

 2.2 Sum Summ mary of Boo Bookk  A. Bo B ook 1 The limit of a function f (x) for x approaches a certain value such as c, written lim f (x) = L, which means that if x is close but different from c then f (x) is close to L. The limit is very closely related to derivative because the limit limit result of a function f (x) is also a result of the derivative of the function and can be written as follows. dx

 f  ( x   x)    f  ( x )

0 f’(x)= dy = lim  x

 

 x

T he heo or em o off L i mi t  

lim

 

lim

 

lim

 

lim

 

lim











k=k

 x  a

k.f(x) = k. lim f(x) = k.F

 x  a

 x  a

   f  ( x  )    g ( x)= lim

 x  a

lim

g(x) F  G

 x  a

[ f  ( x  ). g ( x )]  = [lim  f  ( x)].[   lim  g ( x)] = F.G  x  a

 x  a

 x  a

f(x) 

 x  a

  f  ( x) 

lim  f  ( x)

 x a

 g ( x)  =  xlima  g ( x) =

 x  a

 

 F  G

 



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n

 =  F      lim  f   ( x)     li lim m ( )    f   x  xa   x a n

n



 x

 

lim



 x



  1  1     x 

=

e

 

1

 

lim



 x 0

1   x

 

x

=

e

 

 Determine the Limit Value with Direct Substitution If the number is substituted, the limit value is obtained, then this method is called substitution. substitution. Howeve, if the limit results are not the others methods must be used. For Fo r example: 0     ,    , dan  . The other way is by factoring and multiplying with pear numbers 0   

 Determine the Limit Value with Factorisasi Factorisasi The aim of this way is to miss the number that make the limit value is undefined.  Determine the Limit Value with Cross to the friend’s friend’s Number   In general, we use this t his way to solve the question in roots form  Apllication of the Limit Theorem in Undefine It says the limit theorem at infinity is when x goes to infinity



 x  

 . Usually the form of a

given problem is deep polynomial rank form (multi-rank) fractions functios. The following are m

common form of undefined limit: limit:

m1

lim ax bx  px qx

 x 





n 1

n





m 2

cx rx

n 2

 

...

 

...

T r i gono gonom metr y L Lii mi t As is well known that trigonometry is contained in it sine, cosine, tangent, cotangent, secan, and a nd cosecan forms in the limit trigonometry, the following limit values have been bee n determined as equal : lim  x 0

sin sin  x

 x

1

;

lim  x 0

tan  x

1

 

 x

So, to determine the trigonometric value of limit if the result of o f the limit limit is not defined by changing the shape of the limit into a form which can be resolved.

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As already mentioned above, that the t he limit limit is very closely c losely related with a derivative that is a derivative of a function can be obtained with method limit. If a function of  y  f  ( x) , so in   

general its derivative form can be written as follows follows : '

dy

'

 

 f   x   x    f   x 

 y   f   x   dx  lim

 x

  0  x 

 

The atmosphere of a fuction A function  f      x   is called continuous at x = a if :

  lim  f  ( x)    L( defined )  



 x  a

   f  (a)



 



define [for x=a function of f(x) has the value

  L= f(a) (value of condition 1 and condition to must be same)   Left value of limit = right value of limit





Note :If

one of the conditi co ndition on above is not met, then the function f(x) said to be not continuos (discontinuous).

B . B ook 2 The concept of a “limit” is the fundamental building block on which all a ll calculus concepts are based. In this section we will study limits informally, with the goal of developing an intuitive feel for the basic ideas. In the next three t hree sections we will focus on computational methods and  p  precise recise definitions.  definitions.    Now that we have seen how limits arise in various various ways, let us focus on the limit concept itself.. The most basic use of limits is to describe how a function behaves as the itself t he independent variableapproachesagivenvalue. Forexample,letusexaminethebehaviorofthefunction:



 

2

lim  x   x

a



  x  1   

3 

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ONE-SIDED LIMITS

The limit in (6) is called a two-sided limit because it requires requ ires the values of f(x) to get closer and closer to L as values of x are taken from either side of x = a. However, some functions exhibit different behaviors on the two sides of an x-value x -value a, in which case it is necessary to distinguish whether values of x near a are on o n the left side or on the right side of a for purposes of investigating limiting limiting behavior. be havior. For example, consider co nsider the function

Withthehelpofacalculatingutilitysetinradianmode,weobtainTable1.1.2. The data in the table Withthehelpofacalculatingutilitysetinradianmode,weobtainTable1.1.2. suggest that: lim  x 0

si sin n

 x

1

 

 x

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THE RELATIONSHIP BETWEEN ONE-SIDED LIMITS AND TWO-SIDED LIMITS In general, there is no guarantee that a function f will have a ttwo-sided wo-sided limit at a given point a; that is, the values of f(x)may not get closer and closer to any single real number L as x→a. In this case we say that lim

f(x) does not exist

 x  a

Similarly, the values of f(x) may not get closer and closer to a single real number L as x→a+ or as x→a-. In these cases we say that lim  x a

f(x) does not exist



or that lim  x 

f(x) does not exist 

a

In order for the two-sided limit of a function f(x)to exist at a point a, the values of f(x) must approach some real number L as x approaches a, and tthis his number must be the same regardlessofwhether x approaches a fromtheleftortheright. fromtheleftortheright. Thissuggeststhefollowing result, which we state without formal proof.

INFINITE LIMITS

Sometimes one-sided or two-sided limits fail to exist because the values of the function increaseordecreasewithoutbound. Forexample, considerthebehaviorof f(x)=

1 x  for

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near 0. It is evident from the table and graph in Figure 1.1.15 that as x-values are taken closer and closer to 0 from the right, the t he values of f(x)=

1  x

 are positive and increase without w ithout bound;

and as x-values are taken tak en closer and closer to 0 from the left, the valuesof f(x)=

1  x

 

arenegativeanddecreasewithoutbound.

VERTICAL ASYMTOT

C. B ook 3 Limit Rules

If L, M, c, and k are real numbers lim  f  ( x)   L and lim  g ( x)   M  , then  x c

 x c

1.  Addition Rule: lim  f  ( x)   g ( x)    L   M     x c

The limit of the sum of the two functions is the sum of each limit. 2.  Deduction Rule: lim  f  ( x )   g ( x )   L   M     x c

The limit of the difference between the two functions is the difference from each limit.

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3.  Multiplication Rules: lim  f  ( x)  g ( x)   L    M     x c

The limit of multiplication of two functions is the multiplication of the respective limit. 4.  Rule of Multiplicati Mu ltiplication on Constants: Co nstants: lim k . f  ( x)    k   L    x  c

The limit of a constant multiplied by a function is a constant multiplied by the limit o off a function. 5.  Divisi Division on Rules: lim

 x c

 f  ( x)

 L



 g ( x)

 M 

 

The limit of the division of two functions is the division of each limits. 6.  Rank Rules: If r and s are non-zero null numbers which have no common factor, then lim

 x c

r 

r 

 s   

s

 f  ( x)   L

r 

 , where  L s is a real number. (If s is is even, then it is assumed L> 0.)

The limit of the rational rank of a function is the rank of the function limit, where the result is real. 7.  If  P ( x)  an x

n 1

 

a 1 x  

 

n

    P c   a  c

lim  P   x  x c

n

n

n



...  a0  is a polynomial , so

 



a

n 1

c

n 1

 

 ... 

a0  

8.  If P(x) and Q(x) are polynomials and a nd Q(c) ≠ 0, so

lim

 x c

 P c  Qc 

 

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CHAPTER III: ASSESSMENT

 3.1.1  3.1 .1 Adv Adva anta ntages ges of Boo Bookk 1 After reading the first book, I find some advantages adva ntages of this book. This boo book k contains many examples include the ways to solve. The author explain the ways clearly and step by step. There are many formulas that we often use.

 3.1.2  3.1 .2 Adv Adva anta ntages ges of Boo Bookk 2 After reading the second book, I find some advantages of this boo book. k. This book not o only nly contains examples include the ways to solve, many exercises, but also the basic concept of the formula with graphics.

 3.1.3  3.1 .3 Ad Adv vanta ntages ges of Bo Boo ok 3 After reading the third book, I find the formula which often used. The author writes the simple formula so we can use the formula to solve the problem easily.

 3.2.1Di  3.2 .1Disa sad dvanta ntages ges of Boo Bookk 1 After reading the first book, I am not too interest to read the book because of the display, the paper and we don’t find any graphic. graphic.  

 3.2.2Di  3.2 .2Disa sad dvanta ntages ges of Boo Bookk 2 After reading the second book, I just understand the topic from the examples and graphics because the theories of this book boo k is less than the first book.

 3.2.3  3.2 .3 D isa isad dvanta ntages ges of Bo Boo ok 3 After reading the third book, I don’t find basic concepts, examples, exercises, and graphics.

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CHAPTER IV: CLOSING

4.1.Conclusion So, from three books that I have read, I decide that the second is the best book.So, in this CBR we use the second book as a main main book. The second book has the most advantages. They are, examples include the ways to solve, exercises, but also the basic concept of o f the formula with graphics. The first book is better than the third book because the first first book contains more examples include the ways to solve, and exercises while the third book just contains the simple formula without examples and exercises. The first book has most examples with the ways to solved than another book. But, the second book has more exercises than another book.

4.2. Sugg S ugge esti sti on After we read three books thatI have chosen, I have suggestions. They are : t he paper should be changed tto o be better paper. And also   For the first book, I suggest that the



the first book, actually must be included the graphic graph ic with the its explanation. expla nation.

  The second book, I suggest that the book should have more theories.   For the third book, I suggest that the book not only included tthe he simple formula but also





included some some example with the ways to solve and some exercises to train the readers understanding about limit concept.

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R E F E R E NCE S Pinem, Mhd. Daud. 2015. Kalkulus Untuk Perguruan Tinggi. Tinggi. Bandung: Rekayasa Sains Davis, Anton Bivens. 2012. Calculus Calculus Early Trascendental . Laurie Rosatone  Nugroho, Didit Budi. 2012. Kalkulus 2012.  Kalkulus Integral dan Aplikasinya. Aplikasinya. Yogyakarta: Graha Ilmu.

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