calculus w1 t

week 1

created by Mira Musrini B

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

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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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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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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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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

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

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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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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.

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

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

directed

v v

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

)( wvwv

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wv

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

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Component vector is (u1,u2,u3) P

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2 –spaceconsiderthen ),( 2211 wvwvwv

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),(),( 2121 wwwvvv

consider and k is a real number

then

),( 21 vvv

),( 21 kvkvvk

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1 h.

)( g.

)( f.

)()( e.

0)( d.

00 c.

)()( b.

.a

uu

ulukulk

vkukvuk

uklulk

uu

uu

wvuwvu

uvvu

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consider

norm of vector is22

21 vvv

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),( 21 vvv

v

consider

distance between P1 and P2 is the norm of is

),(),( 212211 yyPxxP

21PP

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212

212

21221 )()()( zzyyxxPP

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

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

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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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

2

1

66

3

vu

u.vcos

6211v

61)1(2u

222

222

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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

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

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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

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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

,,

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kvjvivvvvvvvv

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

jkiijkkij

jikikjkji

kkjjii

0

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vu

vu

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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”

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

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