calculus w1 t

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Page 1: Calculus w1 t

week 1

created by Mira Musrini B

week1Mira Musrini MT 1

Page 2: Calculus w1 t

Visions : to become the best technology information and comunications polytechnic in south east Asia. Misions : 1. To create and develop profesional workers who

have high integrities and ready to work in national and international industry.

2. To do some reasearch and innovations for industry and society

3. To do some society services 4. To do system development that can leverage

added value for institutions and stakeholder.

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Page 3: Calculus w1 t

As a student completes this course, he or she should be able to :1. understand the concept of vectors and solve

any problems related to vectors. 2. understand the concept of matrix and solve any

problems related to matrix and solution of linear equation.

3. understand the concept of functions, draw sketchs of these functions and solve any problems related to the topic.

4. understand the concept of the limit and solve any problems related to the limit.

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Page 4: Calculus w1 t

5. understand the concept of continuations and solve any problems related to the continuations.

6. understand the concept of diferentiations and solve any problems related to the differentiations.

7. understand the concept of integrations and solve any problems related to the integrations

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Page 5: Calculus w1 t

Mid Test = 30% Final Test = 30% Pre Test = 10% Post Test = 15% Assignment = 15% 5x + 3x

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Page 6: Calculus w1 t

Student must :a. wear proper clothing b. come to the class no later than 15 minutes. c. bring the courseware d. pay attention and focus in the class. e. wear proper shoes

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Page 7: Calculus w1 t

Students are not allowed to :a. smokingb. eat food c. chatting with others while faculty

speaking at the front of the class.

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Page 8: Calculus w1 t

If some students don’t follow the rules, then

Faculty must give the student additional assignment for the penalty.or Faculty can use his/her own right to send the student out of the class.

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Page 9: Calculus w1 t

w1 : vectorsw2 : matrix and the operations of matrix w3 : -matrix scalar multiplications and

matrix multiplications - introductions to systems of linear

equationsw4 : - elementary row operations - determine the inverse matrix with elementary row operations. - use inverse matrix to find solutions of systems of linear equationsweek1Mira Musrini MT 9

Page 10: Calculus w1 t

w5: Number systems w6 : Functions dan graphics of functions

such as linear ,quadratic, circle, ellips etc. Absolute functions.w7 : Domain and range of functions Composite functions. Domain and range of composite

functionsw8 & w9 : midtest.

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Page 11: Calculus w1 t

w10 : limit of functions properties of limit , two sides limit the existency of limit w11 : limit of trigonometri functions L’Hospital’s Rule. limit at infinity w12 : basic concept of continuity continuity and the existency of limit basic concept of differentiable one side differentiable differentiable and continuity .

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Page 12: Calculus w1 t

w13: the techniques in differentiations. Derivatives on trigonometry functions Chain rule Higher derivatives w14: Applications differentiations sketch graph of

some functions by determine : monotony functions, extreme values , concativity, point of inflections, Asymtote of Functions.

W15 : indefinite integrations properties of indefinite integration sigma notations.

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Page 13: Calculus w1 t

w16: definite integrations properties of definite inetgrations. w17 : final test

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Page 14: Calculus w1 t

Introduction: Vectors can be represented geometrically as directed line segments or arrows in 2-space or 3-space; the directions of arrows determine the directions of vectors, and the length of the arrow describes its magnitude.

ABv

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Page 15: Calculus w1 t

The tail of the arrow is called the initial point of vector , the

tip of arrow is the terminal point .

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A is the initial point , B is the terminal point.

Page 16: Calculus w1 t

Equivalent vectors :

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The vectors with same length and same directions, such as those in the figure are called equivalent.

If V and W are equivalent we write:

V = W

Page 17: Calculus w1 t

The negative of has the same lenght as ,but is oppositely

directed

v v

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Page 18: Calculus w1 t

Definition: if v and w are two vectors , then the sum v+w is the vector determined as follows: Position the vector w so that its initial point coincides with the terminal point of v.

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Page 19: Calculus w1 t

Definitions: if v and w are any two vectors, then the difference of v and w is defined by :

)( wvwv

week1Mira Musrini MT 19

wv

Page 20: Calculus w1 t

component vector in 2 space are (u1.u2) u

week1Mira Musrini MT 20

Page 21: Calculus w1 t

Component vector is (u1,u2,u3) P

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Page 22: Calculus w1 t

2 –spaceconsiderthen ),( 2211 wvwvwv

week1Mira Musrini MT 22

),(),( 2121 wwwvvv

Page 23: Calculus w1 t

consider and k is a real number

then

),( 21 vvv

),( 21 kvkvvk

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Page 24: Calculus w1 t

1 h.

)( g.

)( f.

)()( e.

0)( d.

00 c.

)()( b.

.a

uu

ulukulk

vkukvuk

uklulk

uu

uu

wvuwvu

uvvu

week1Mira Musrini MT 24

Page 25: Calculus w1 t

consider

norm of vector is22

21 vvv

week1Mira Musrini MT 25

),( 21 vvv

v

Page 26: Calculus w1 t

consider

distance between P1 and P2 is the norm of is

),(),( 212211 yyPxxP

21PP

week1Mira Musrini MT 26

212

212

21221 )()()( zzyyxxPP

Page 27: Calculus w1 t

Example :

the norm of vector is

Example :

the distance d between points P1(2,-1,-5) and P2(4,-3,1)

)1,2,3(u

14)1()2()3( 222 u

week1Mira Musrini MT 27

11244)51()13()24( 222 d

Page 28: Calculus w1 t

if u and v are vectors in 2-space and 3-space and θ is the angle between u and v ,then dot product or Euclidean inner product u.v is defined by :

║u ║║v ║ cos θ if u≠0 and v≠0

u.v =

0 if u=0 and v=0

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Page 29: Calculus w1 t

Example 1 : the angle between the vectors u=(0,0,1) and v =(0,2,2) is 450 . Find u.v .

Solution:

22

1)220)(100(cosuu.v 222222

v

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if u=(u1, u2) and v=(v1, v2) are two vectors in 2-space , then coresponding formula is

u.v = u1v1+u2v2

if u=(u1, u2,u3) and v=(v1, v2,v3) are two vectors in 3-space , then coresponding formula is u.v = u1v1 + u2v2 + u3v3

Page 30: Calculus w1 t

2

1

66

3

vu

u.vcos

6211v

61)1(2u

222

222

week1Mira Musrini MT 30

Example 2: Consider the vectors u=(2,-1,1) and v =(1,1,2) . Find u.v and determine the angle θ between u and v

Solutions :

u.v = u1v1 + u2v2 + u3v3 =(2)(1) +(-1)(1) +(1)(2)=3

Orthogonals vectors :

consider u and v are two nonzero vectors that prependiculas if only of u.v=0

Page 31: Calculus w1 t

0. ifonly and if 2

0. ifonly and if obtuse is

0. ifonly and if acute is

thenembetween th angle theis and zeronon are and vectors theif b.

).( , is that . a. 2/12

vu

vu

vu

vu

vvvvvv

0v if0. and,0 if 0. d.

).().().( c.

..).(b.

.. a.

vvvvv

vkuvukvuk

wuvuwvu

uvvu

week1Mira Musrini MT 31

Page 32: Calculus w1 t

Definition

) , - , (

notationt determinanin or

),,(

by defined vector theisproduct cross

thenspace,-3in vectorsare ),,( and ),,( if

21

21

31

31

32

32

122131132332

321321

vv

uu

vv

uu

vv

uuvu

vuvuvuvuvuvuvu

vu

vvvvuuuu

week1Mira Musrini MT 32

Page 33: Calculus w1 t

consider the vector

these vectors each have lenght 1 and lie along the coordinate axes the call standard unit vectors in 3 –space.Every vector in 3-space can be represented in terms of

)1,0,0()0,1,0()0,0,1( kji

kji

,,

week1Mira Musrini MT 33

kvjvivvvvvvvv

321321321 )1,0,0()0,1,0()0,0,1(),,(

Page 34: Calculus w1 t

jkiijkkij

jikikjkji

kkjjii

0

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Page 35: Calculus w1 t

vu

vu

week1Mira Musrini MT 35

is orthogonal to both u and v , if u and v is non zero vectors. The direction of can be determined using following “right hand rules”

Page 36: Calculus w1 t

For example: if u=(1,2,-2) and v=(3,0,1) , then

kji

kji

u x v21

21

31

31

32

32

321

321 vv

uu

vv

uu

vv

uu

vvv

uuu

kji

kji

u x v21

21

31

31

32

32

321

321 vv

uu

vv

uu

vv

uu

vvv

uuu

week1Mira Musrini MT 36