Mathematics-I course file

7/30/2019 Mathematics-I course file

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SCIENCE & HUMANITIES

Course File

.

Department: SCIENCE AND HUMANITIES

Name of the Subject: MATHEMATICS - I

Subject Code: 51002

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JOGINPALLY B.R. ENGINEERING COLLEGE

YENKAPALLY(V),MOINABAD(M),R.R.DIST,HYDERABAD

Department of

SCIENCE &

HUMANITIES

Course File Year:2009-2010

2. Department: S&H

3. Name of the Subject: Mathematics-1

4. Subject Code: 51002

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JOGINPALLY B.R. ENGINEERING COLLEGE

YENKAPALLY(V),MOINABAD(M),R.R.DIST,HYDERABAD

Department of

SCIENCE &

HUMANITIES

Course Status Paper(Target, Course Plan,

objectives, Guidelines

etc.)

Year: 2009-2010

Target:

1. Percentage Pass: __________90%_

2. Percentage above 70% of marks: 60%____________

Course Plan:

(Please write how you intend to cover the contents: that is, coverage of units

by lectures, guest lectures, design exercises, solving numerical problems,

demonstration of models, model preparation, or by assignments etc.)

a. Design exercises,b. Solving numerical problems,c. Model preparation by assignments etc

On completion of the course the student shall be able to:

To utilize the mathematical concepts in other subjects To apply the mathematical knowledge and logical thinking in

other subjects

4. Method of Evaluation:

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3.1. Continuous Assessment Examination: Yes / No

3.2. Assignments: Yes / No

3.3. Questions in class room: Yes / No

3.4. Quiz as per University Norms: Yes / No

3.5. Others: Make the students to solve the problems on the board(Please Specify)

5. List out any new topic(s) or any innovation you would like to

introduce in teachingthe subject in this semester:

6. Guidelines to study the subject:

1. Mathematics has played a fundamental role in the formulation ofmodern

Science since the very beginning; a scientific theory is a theory that has an

adequate mathematical model.

2. The Mathematics that can be applied today covers all the fields of the

mathematical science and not only some special topics; it concerns

Mathematics of all levels of difficulty and not only simple results and

arguments.

3. The sciences continue to require today new results from ongoing research

and

present multiple new directions of inquiry to the researchers, but the rhythm

of the

contemporary society makes the time lapse substantially shorter and the

request more urgent.

4. The capabilities of scientific computation have made numerical

simulation, an indispensable tool in the design and control of industrialprocesses.

5.To develop an understanding of the basic principles governing the

conditions of rest and motion of particles and rigid bodies subjected to the

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action of forces; to develop the ability to analyze and solve problems in a

simple and logical manner

Expected date of completion of the course and remarks, if any:

Unit Number: 1 30-9-09






Unit Number:7 17-4-10


Remarks (if any):

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Schedule of instruction

Unit No: 1

S.No

Date Numberof Hours

Subject Topics Reference

1 1 Sequences,Convergent,Divergent,Oscillatorysequences

S.Chand &Dr.C.Sankara

2 1 Bunded and Unbounded sequences,limit of asequence


3 2 Infinite series:Properties of theseries,Geometric Series,Auxiliary Series(p-

Series),Series of positive terms


4 3 Comparison test,DAlemberts Ratio test S.Chand &

Dr.C.Sankara5 2 Raabes Test S.Chand &

Dr.C.Sankara

6 1 Cauchys Root Test,Cauchys Integral Test S.Chand & Dr.C.Sankara

7 1 Alternating Series, Lebnitzs test S.Chand & Dr.C.Sankara

8 2 Absolute and Conditional Convergence S.Chand & Dr.C.Sankara

Unit No: 2S.

No

Date Number

ofHours


1 1 Rolles Theorem S.Chand

2 1 Lagranges Mean Value Theorem

Altrnate form of Lagranges Mean Value

Theorem

S.Chand

3 1 Cauchys Mean Value Theorem S.Chand

4 1 Taylors Series,Taylors Theorem with

Lagranges form ofRemainder,Alternative Form

S.Chand

5 1 Maclaurins Series,Maclaurins Theorem

with Lagranges form of Remainder

S.Chand

6 1 Functions of Several variables S.Chand

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7 1 Jacobians,Properties S.Chand

8 1 Functional Dependence S.Chand

9 1 Taylors Theorem for Two Variables S.Chand

10 2 Maxima and Minima of a function of two

or more variables

S.Chand

11 1 Constrained Maxima and Minima:

Lagranges method of undeterminedmultipliers

S.Chand

Unit No: 3

S.No

Date Numberof

Hours


1 1 Curvature S.Chand & Dr.C.Sankaraiah

2 2 Formula for Radius of

Curvature,Circle ofcurvature

S.Chand &

Dr.C.Sankaraiah

3 2 Radius of curvature at theorigin

S.Chand &Dr.C.Sankaraiah

4 1 Pedal Eqution, Formula for

Radius of curvature for the

pedal eqn

S.Chand &

Dr.C.Sankaraiah

5 1 Centre of Curvature,Co-ordinates of centre of

curvature


6 2 Evolute,Properties of

evolutes

S.Chand &

Dr.C.Sankaraiah

7 2 Envelopes,Method of

finding envelope

S.Chand &

Dr.C.Sankaraiah

8 1 Curve Tracing S.Chand &

Dr.C.Sankaraiah

9 1 Curve tracing in Cartesian

form

S.Chand &

Dr.C.Sankaraiah

10 2 Curve Tracing in ParametricForm


11 2 Curve Tracing in Polar form S.Chand & Dr.C.Sankaraiah

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Unit No: 4

S.

No

Date Number

ofHours


1 1 To find the length of the arcof a curve in Cartesian co-

ordinates


2 1 Polar Co-ordinatesS S.Chand &

Dr.C.Sankaraiah

3 2 Volume of Revolution

4 3 Volume Formulae forParametric Eqn,Volume

Formulae in Polar Co-ordinates,Volume between

two solids

5 1 Surface Area of Revolution

6 1 Multiple IntegralsDouble Integral

7 1 Triple Integral

8 1 Double Integral in Polar

Form

9 1 Region of Integration

10 2 Change of order of

integration

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Unit No: 5

S.

No

Date Number

of

Hours


1 3 Differential EqnsIFirst

Order&First Degree)

Introduction:OrdinaryDifferential Eqns

Solution of adifferential Eqn

S.Chand &

Dr.C.Sankaraiah

2 1 Exact Differential Eqns

3 1 Integrating Factors

4 2 Eqns Reducible to exacteqns(methods to find

integrating factors)

5 1 Linear Eqns

6 2 Bernoullis Eqns

7 2 Applications-Orthogonal

Trajectories(CartesianForm,Polar Form)

8 2 Law of Natural Growthor Decay,Newtons Law

of Cooling

Unit No: 6

S.

No

Date Number

of

Hours


1 2 Linear differential eqnsComplementary fn


2 1 Particular IntegralMethods of finding P.I

-do-

3 2 Cauchys Homogenious LinearEqn

-do-

4 1 Legendres Linear Eqn -do-

5 2 Linear Differential Eqns of

Second Order-Method of

-do-

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Variation of Parameters

6 2 Applications

Bending of Beams

-do-

7 1 Boundary Conditins -do-

8 1 Electrical Circuits -do-

9 1 Simple Harmonic Motion -do-

Unit No: 7

S.No

Date Numberof

Hours


1 1 Introduction of Laplace

Transforms

S.Chand &

Dr.C.Sankaraiah2 2 Linearity Property,First

shifting,Second

shifing,Change of Scale

Properties

-do-

3 2 Laplace Transform of

standard fns:Multiplication by t

Division by t

-do-

4 2 Laplace Transform of

Derivatives

-do-

5 2 Laplace Transform of Integrals

-do-

6 1 Laplace Transform of Periodic fns,L-T of unit step

fn

-do-

7 2 Inverse Laplace

Transforms:First shiftingtheorem,Second Shifting

theorem,Second shifting

theorem

-do-

8 2 Change of scaleproperty,Inverse Laplace

Transform of Derivatives

-do-

9 1 Inverse Laplace Transform

of Integrals

-do-

10 1 Multiplication by powers of

s

-do-

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11 1 Division by powers of s -do-

12 1 Convolution Theorem -do-

13 1 Application of L-Ts -do-

Unit No: 8

S.

No

Date Number

ofHours


1 2 Vector Differentiation S.Chand & Dr.C.Sankaraiah

2 2 Gradient of scalar Function -do-

3 1 The Divergence of a Vector

Function

-do-

4 2 Curl of a Vectorfunction,Laplacian Operator

-do-

5 2 Vector Integration -do-

6 2 Surface Integrals -do-

7 2 Volume Integrals -do-

8 2 Greens Theorem in the

Plane,Application of Greenstheorem

-do-

9 1 Gauss Divergence Theorem -do-

10 2 Stokes Theorem -do-

This Assignment/Tutorial is concerned to Unit Number: 1

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(Please write the questions/problems/exercises. Which you would like to giveto the students)

Q 1: Test for convergence the series n

xn(x>0)

Q 2: Find the interval of convergence of the series

7

x.

6

5.

4

3.

2

1

5

x.

4

3.

2

1

3

x.

2

1x

753

+++

Q 3: Test for the convergence of the series, )0x......(10

x

5

x

2

x1

32

>++++

Q 4:Test whether the series

= +1n 2 1nncos

converges absolutely

Q 5:Find the interval of convergence of the series .............)x1(2

1x1

12+

+



Q 1:Show that8

1

335

1

3>

using Legranges mean value theorem.

Q 2:Find c of Cuachuys mean value theorem for f(x)= ,x g(x)=x

1in

[a,b] where 0++=


(Please write the questions/problems/exercises. Which you would like to give

to the students)

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Q 1:Find the curvature at the point

2

a3,

2

a3of the curve axy3yx

33 =+ .

Q 2:If 2,1 are the radius of curvature of the curve at the extremities of

any chord of the cardioids r= )cos1(a which passes through the pole.

Show that9

a16,

22

2

2

1=

Q 3:Considering evolutes as the envelope of the normal find the evolutes of

the parabola. Is ax4y2=

Q 4:Find the envelope to the family of circles thoroughly the origin and

whose centres lies on the ellipse 1b

y

a

x

2

2

2

2

=+

Q 5:Trace the curve22 )a5x)(a2x(ay9 =



Q 1:Find the length of the arc of the parabola ax4y2= cut off by its latus

rectum.

Q 2:Find the perimeter of the cardioid )cos1(ar += . Also show that the

upper half of the cardioid is bisected by the line x = /3.

Q 3: Find the volume of the solid generated by revolving the ellipse x2/a2 +

y2/b2 = 1 about the major axis.

Q 4: Evaluate the dzdydxzxy2

taken through the positive octant of the

sphere x2 + y2 + z2 = a2

Q 5: Evaluate by transforming into polar co-ordinates dydxyxy 22a

0

xa

0

22

+


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Q 1: Solve: (y2 2xy) dy = (x2 2xy)dy

Q 2: Solve: (x3y2 + xy) dx = dy

Q 3: Solve xdx

dy+ y = x3.y

Q 4: If the air is maintained at 300 C and the temperature of the body cools

down from 800 C to 600 C in 12 min. Find the temperature of body after 24mi

Q 5: Find the orthogonal trajectories of the family of the parabolas y 2 = 4ax



Q 1: Solve: (D2 + 4D + 4 ) y = 18 cos hx

Q 2: Solve: ym + 2.yn yp 2y = 1 4y3

Q 3: Solve by the method of variation of parameters yn + 4y = tan 2x

Q 4: Solve: x2sin.e8y13dxdy6

dxyd x32

2=+

Q 5: Solve xsinxydx

yd2

2

=+ by the method of variation of parameters.



Q 1: Find the laplace transforms of the following:

(i) tttet

3cos33sin2432

++(ii)sin(wt+ )

Q 2: If L[f(t)] = 3

2

)1(

15129

+

s

ss

, find L[f(3t)] using change of scale property

Q 3: Find the inverse Laplace transform of)2)(1(

4

++ ss

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Q 4:Using convolution theorem evaluate L-1

)4)(1( 22 ++ ss

s

Q 5:Using Laplace transform solve (D

2

+2D-3)y=sinx,y(0)=y(0)=0



Q 1: Find the directional derivative of (xyz2+xz) at (1,1,1) in a direction of

the normal to the surface 3xy2+y=z at (0,1,1)

Q 2: Using Divergence theorem, evaluate ++s

zdxdyydzdxxdydz where

s:x2+y2+z2=a2

Q 3: Verify divergence theorem for 2x2yi-y2j+4xz2k taken over the region of

the first octant of the cylinder y2+z2=9 and x=0,x=2

Q 4:Verify Greens theorem for ++c

dyxdxyxy22

)( , where c is bounded by

y=x and y=x2

Q 5Apply Stokes theorem to evaluate ++

c

xdzzdyydx

. Where c is the curveof intersection of x2+y2+z2=a2 and x+z=a

Internal Quiz Marks

S.No.

Quiz Test Maximum

Marks

BestMarks

WorstMarks

Remarks

1Quiz Test

1

10

2Quiz Test

210

3Quiz Test

310

4Quiz Test4

10

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5Quiz Test

510

* indicate if any remedial tests were conducted, if any.

Please note:

1. The question papers in respect of quiz test 1, 2, 3, 4 and 5 of this

subject should be included in the course file.

2. Model question paper which you have distributed to the students in the

beginning of the semester for this subject should be included in thecourse file.

3. The list of seminar topics you have assigned, if any may also be

included here.

4. The J. N. T. University end examination question paper for this subject

must be included in the course file.

5. A record of the best and worst marks achieved by the students inevery quiz tests must be maintained properly.

6. A detailed / brief course material / lecture notes if prepared may besubmitted in the HODs office.

7. Xerox copies of at least 5 answer sheets, after duly signed by thestudent on verification of the evaluated answer script.

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Mathematics-I course file

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