fy b. tech. semester ii complex numbers and calculus · sanjay ghodawat university, atigre. course...

Sanjay Ghodawat University, Atigre. Course plan for F.Y. B.Tech. course FYT114 (CNC)

FY B. Tech. Semester II

Complex Numbers and Calculus

Course Code FYT114 Course Complex numbers and

Calculus (CNC)

Prepared by S M Mali Date 6/11/2017

Prerequisites Basic knowledge of results from Algebra.

Knowledge of Derivatives.

Knowledge of Definite and Indefinite integration.

Basic knowledge Geometry and Trigonometry.

Course Outcomes

At the end of the course the students should be able to:

CO 1 Find3 roots of Complex numbers and relate circular functions and hyperbolic

functions

CO 2 Obtain3 real and imaginary parts of a complex number.

CO 3 Discuss3 convergence of a series.

CO 4 Solve3

improper integrals.

CO 5 Solve4 differential equation of first order and first degree.

CO 6 Expand3 the given function in powers of x and ( x – a) and evaluate limits.

Mapping of COs with POs

POs

COs

a b c D e f g h i j k l m n o

CO 1

CO 2

CO 3

CO 4

CO 5

CO 6

Course Contents

Unit No. Title No. of

Hours

I

Complex Numbers

1. Introduction, Modulus and argument of a Complex Number.

2. Types of Complex numbers.

3. Algebra of Complex numbers

4. De Moivre's Theorem (Without proof) . Roots of complex numbers

06


by using De Moivre's Theorem

5. Expansion of sin(nθ) and cos(nθ) in powers of sinθ and /or cosθ.

6. Expansion of Sinn and Cos

n in terms of sines and cosines of

multiples of .

II

Hyperbolic Functions

1. Hyperbolic Functions. Definitions. Introduction.

2. Relation between Circular & Hyperbolic functions.

3. Formulae of Hyperbolic Functions (without proof).

4. Inverse hyperbolic functions.

5. Separation of a complex number into real and imaginary parts.

6. Logarithmic function of a complex variable.

06

III.

Infinite series

1. Sequence, series and properties of series.

2. Series of positive terms, comparison test, integral test.

3. Ratio test, D’ Alembert’s ratio test.

4. Root test, Cauchy root test.

5. Alternating series.

6. Series of positive and negative terms.

06

IV.

Improper Integral and special functions

1. Introduction to improper Integrals.

2. Improper integral of first and second kind.

3. Gamma functions and its properties.

4. Beta functions and its properties.

5. Relation between Beta and Gamma functions.

06

V.

Ordinary Differential Equations of First Order and First Degree:

1. Definition, order and degree of a differential equation.

2. Solution of DE of first order and first degree : Linear DE.

3. Solution of DE reducible to Linear differential equations.

4. Exact DE and DE Reducible to Exact differential equations.

5. Applications of DE to orthogonal trajectories.

6. Applications of DE to simple electrical circuits.

06


VI.

Expansion of Functions:

1. Meaning of expansion of functions

2. Expansion of a function by Maclaurians series

3.Standard Expansions

4. Integration, derivative by substitution method

5.Taylor’s Series

6.Indeterminate forms

06

Reference Books:

Sr.

No.

Title of Book Author Publisher/Edition Topics

1 Higher Engineering

Mathematics. Dr. B. S. Grewal.

Khanna

Publications,

Delhi. 41st edition.

All

2 Advanced Modern

Engineering Mathematics Glyn James.

Pearson Education

(2012). 3rd

edition. All

3 A textbook of Engineering

Mathematics.

N. P. Bali, Manish

Goyal.

Laxmi Publications

(P) Ltd., New Delhi

(2011). 8th

edition.

All


Mathematics.

H. K. Dass, S. and

Er. Rajneesh Verma

Chand.

S. Chand &

Company Ltd.,

(2011) New Delhi.

All

5 Advanced Engineering

Mathematics.

Peter V. and

O’Neil.

Cengage learning,

(2012). 7th

edition. All


Mathematics. Ramana B. V.

Tata McGraw Hill

Publishing

Company, New

Delhi, (2008).

All

Evaluation scheme

Lectures Tutorials Practical Credits Evaluation Scheme

Component Exam WT Pass

3 2 -- 4 Theory

(100)

FET 20

Min 50 CAT-I 15

CAT-II 15

ESE 50


Scheme of Marks

Unit No. Title Marks

1 Complex Numbers. 16

2 Hyperbolic Functions. 18

3 Infinite Series. 16

4 Improper Integral and Special functions. 16

5 Differential Equations of first order and first degree. 18

6 Expansion of functions. 16

Course Unitization

Unit wise Lesson Plan

Unit No 1 Unit Title Complex Numbers Planned Hrs. 6

Lesson schedule

Class

No. Details to be covered

1 Introduction of complex number, modulus, Argument and Algebra of complex

numbers.

2 Statement of De Moivers theorem and examples.

3 Roots of comlex number by using De Moivers theorem.

4 Expansion of sinnѳ, cosnѳ and tannѳ in powers of sinѳ, cosѳ and tanѳ.

5 Definition of circular functions in complex variable.

6 Logarithm of complex number.

Review Questions

Q1 Simplify

1.

7 5

12 6

cos 2 sin 2 cos3 sin 3


i i

i i

2.

2 3

9 5

cos5 sin 5 cos7 sin 7

cos 4 sin 4 cos sin

i i

i i

Unit

No Title

Course

Outcomes

No. of Questions in

CAT-I CAT-II

1 Complex Numbers. CO 1 Q.1 10 Marks

2 Hyperbolic Functions. CO 2 Q.2 10 Marks

3 Infinite Series. CO 3 Q.3 10 Marks

4 Improper Integral and Special functions. CO 4 Q1. 10 Marks

5 Differential Equations of first order and

first degree. CO 5 Q.2. 10 Marks

6 Expansion of functions. CO 6 Q.3. 10 Marks


1 sin cos

1 sin cos

ni

i

Q2 Show that

1. 1 cos sin

cos sin1 cos sin

ni

n i ni

2. 8 8

81 3 1 3 2i i

3. 5 3sin 632cos 32cos 6cos

sin

3 5

2 4

5tan 10 tan tantan5

1 10 tan 5tan

Q3 Express cos7 and sin 6 in terms of powers of cos and sin

Q4 Solve

1. 4 3 2 1 0x x x x

2. 9 5 4 1 0x x x

3. 6 0x i

4. 7 4 3 1 0x x x

5.

61

11

x

x

5 51 32( 1)x x

Q5 Find the continued product of all the values of

3/4

1 3

2 2i

Q6 Find all the values of

1/5

1 2. 1/5

1 i

Q7 Find nth root of unity and show that

1. Roots are in geometric progression

2. Sum of the all roots is zero

Product of all roots is 1

1n

Q8 Find the common roots of 4 1 0x and

6 0x i

Q.9. Prove that 1 + cos 2z = 2 cos2z

Q.10. Prove that Sin-1

z = - i log(iz ± ( 1- x2) )

Unit No 2 Unit Title Hyperbolic Functions Planned Hrs. 6

Lesson schedule

Class

No.

Details to be covered

1 Introduction of Hyperbolic functions and its properties


2 Relation between circular and hyperbolic functions

3 Inverse Hyperbolic functions

4 Separation of a complex number into real and imaginary parts

Review Questions

Q.1. Prove that sinh (ix) = i⋅sin x and sin(ix) = i sinh x

Q.2. Prove that sinh (2x) = 2 sinh x cosh x and cosh2 x – sinh

2 x = 1

Q.3. Prove that | sinh z |

2 = sinh

2 x + sin

2 y

and that | cosh z |2 = sinh

2 x + cos

2 y where z = x + iy

Q.4. Prove that tanh 3x = 3 tanh x + tanh 3x

1+3 tanh 2x

Q.5. If sin (A + iB) = x + iy prove that x2

cosh 2B +

y2

sinh 2B = 1

Q.6. Prove that tanh x is a periodic function with the period = 2i

Q.7. If tanh x = ½ , find the value of x and sinh 2x

Q.8. Find the value of tanh log x if x = 3

Q.9 Solve the equation 7cosh 8sinh 1x x for real values of x

Q.10 Prove that

1.

31 tanh

cosh 6 sinh 61 tanh

xx x

x

2. 7 1sinh sinh 7 7sinh5 21sinh3 35sinh

64x x x x x

3. 1 2sinh log( 1)z z z

4. 1 2cosh log( 1)z z z

5. 1 1 1tanh log

2 1

zz

z

6. 1 1

2sinh tanh

1

xx

x

7. 1 1tanh (sin ) cosh (sec )

8. 1 1coth log

2

x x a

a x a

1sech sin log cot2

Q11 If

x xtan tanh

2 2

, prove that 1. sinh tanu x 2. cosh secu x

Q12 Separate into real and imaginary parts

ii 2.

i

i 3. tanh x iy 4. 1tan ie 5. 1 3cos

4

i


Q13 If cos( ) (cos sin )i r i then prove that

1 sin( )log

2 sin( )

Q14 If sin( ) tan seci i , prove that cos2 cosh 2 3

Q15 If tan( ) sin( )i x iy , prove that

tan sin 2

tanh sinh 2

x

y

Q16 If cos

4u iv ec ix

, prove that

2 2 2 2 2( ) 2( )u v u v

Q17 If tan

6x iy i

, prove that

2 2 21

3

xx y

Q18 If 1 1 1cosh cosh ( ) coshx iy x iy a then prove that

2 2 22( 1) 2( 1) 1a x a y a

Q19 If tan( )i i and x,y are real prove that x is indeterminate and y is infinite

Unit

No 3 Unit Title Infinite Series Planned Hrs. 06

Lesson schedule Class

No.


1 1. Sequence, series and properties of series.

2 2. Series of positive terms, comparison test, integral test.

3 3. Ratio test, D’ Alembert’s ratio test.

4 4. Root test, Cauchy root test.

5 5. Alternating series.

6 6. Series of positive and negative terms.

Review Questions

Q.01 Discuss the convergence of –

(1) 1

2 +

1

4 +

1

8 + ⋯ ⋯

(2) 12 + 22 + 32 + 42 ⋯ ⋯

(3) 1 + 2 – 3 + 1 + 2 – 3 + 1 + 2 – 3 + ⋯⋯

Q.02.

Use comparison test to show that the series

(1) 1+ 1

2 +

1

3 +

1

4 + ⋯ ⋯ 𝑖𝑠 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑡


(2) 1

1 ! +

1

2 ! +

1

3 ! + ⋯ ⋯ 𝑖𝑠 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑡

Q.03. Use D’ Alembert’s Ratio test to show that the series

(1) 2

1 +

22

2 +

23

3 + ⋯ ⋯ is divergent

(2) 14

𝑒12 +

14

𝑒22 +

14

𝑒32 + ⋯ ⋯ is convergent

Q.04. Use Cauchy root test to show that the series

(1) 1

4 +

3

4

2

+ 1

4

3

+ 3

4

4

+ 1

4

5

⋯ ⋯ is convergent

(2) 1 +𝑥

2 +

𝑥2

32 +

𝑥3

43 + ⋯ ⋯ is convergent

(3) n+1 x n

nn +1

∞

𝑛=1 is convergent if x < 1, 𝑑ivergent if x ≥ 1

Q.05. Use Leibnitz test to discuss the convergence of the alternating series –

(1) 1

2 ∙

1

13 −

2

3 ∙

1

23 +

3

4 ∙

1

33 −

4

5 ∙

1

43 + ⋯ ⋯

(2) 1

1 ∙

2

3 −

1

2 ∙

3

4 +

1

3 ∙

4

5 −

1

4 ∙

5

6 + ⋯ ⋯

(3) 2

1 −

3

2 +

4

3 −

5

4+ ⋯ + −1 𝑛−1 ∙

(𝑛+1)

𝑛+ ⋯

Q.06. Examine whether the following series are absolutely convergent, conditionally

convergent or divergent.

(1) 1

1 ! −

1

2 ! +

1

3 ! −

1

4 ! ⋯ ⋯

(2) 1 − 1

2 +

1

3 −

1

4 +

1

5 ⋯ ⋯

Unit

No

4 Unit Title Special Functions Planned Hrs. 07

Lesson schedule

Class Details to be covered


No.

1 Improper integral of first and second kind and examples on it.

2 Gamma function and its properties

3 Beta function and its properties

4 Examples on Gamma and Beta function

5 Relation between Gamma functions and Beta functions

6 Examples on Relation between Gamma functions and Beta functions

Review Questions

Q.1 Evaluate the following integral.

Q.2. Determine if the following integrals are convergent or divergent and if convergent find

their value.

① ➁

Q.3. Determine if the following integral are convergent or divergent. If convergent find

their values.

① ➁ ③

Q4 Prove that Γ(n+1) = nΓn

Q5

Evaluate the following integrals 4

04x

xdx

b)

1

0 1log

x

xdx

c) 0

n axx e dx

d) 24

0

xa dx

Q6 Evaluate the following integrals

a)

1

3 5

0

(1 )x x dx b) /2

20 11 sin

2

d

c)

1

1

(1 ) (1 )m nx x dx

d) 1/47

3

( 3)(7 )x x dx

Q7 Show that

8 6

24

0

(1 )0

(1 )

x xdx

x

Q8 Show that

/2 2 1 2 1

2 2

0

sin cos 1( , )

( sin cos ) 2

m n

m n m nd B m n

a b a b

Q9 Prove that

/2 /2

0 0sin

dsin d


Q10

Prove that

1

3 30

2

3 31

dx

x

Q11

Evaluate

0

1x

axee dx

x

Q12

Prove that

1

0

1log

log 1

a bx x adx

x b

; a > 0, b > 0

Unit

No

5 Unit Title Differential equations of first order and

first degree

Planned Hrs. 06

Lesson schedule

Class

No.


1 Differential equation, Degree, Order and types of solutions

2 Exact Differential equation, Reducible to exact ( Rule 1,2)

3 Reducible to exact ( Rule 3,4), Examples

4 Linear Differential Equation

5 Non-linear Differential Equation

6 Examples

Review Questions

Q1 Explain Degree, order of differential equation

Q2 Solve 2(sin .cos ) cos .sin tan 0xx y e dx x y y dy

Q3 Solve 2 1 2 1 0x y dx y x dy

Q4 Solve 2 2 2 2 2 2 0x y a xdx x y b ydy

Q5 Solve 22 22 0a xy y dx x y dy

Q6 Solve 2 sin 0yx e dy y x x dx

Q7 Solve 11 cos log sin 0y y dx x x x y dyx

Q8 Solve

2 log

dy y

dx y y y x

Q9 Solve

2 2( ) 2 1

ydx xdy dx

x y x

Q10 Solve 2 3 2 2( 2 ) 2 3 0yxe xy y dx a x y xy dy

Q11 Solve 22 0xy y dx xdy

Q12 Solve ( ) 0yx xxe dx dy e dx ye dy

Q13 Solve 2 21 ( 1)

dyx x y x x

dx


Unit No 6 Unit Title Expansion of functions Planned Hrs. 6

Lesson schedule

Class No. Details to be covered

1 Meaning of expansion of functions.

2 Expansion of a function by Maclaurians series.

3 Standard Expansions.

4 Integration, derivative by substitution method.

5 Taylor’s Series.

6 Indeterminate forms.

Review Questions

Q1 Expand in powers of x

1. tanx

2. log(1+ex)

3. exsecx

4. 2

1 1 1tan

x

x

5. log tan4

x

6. 3 4 517 6 2 3( 2) ( 2) ( 2)x x x x in powers of x by Taylor’s theorem

7. tan x in powers of 4

x

8. log(1 sin )x by Maclaurin’s series

9. 1x

x

e by using standard expansions

10. 5 4 3 2 1x x x x x in powers of 1x and hence find (11/10)f

3 22 7 1x x x in powers x-2

Q2 Prove that

1. 2 4 6

logsec .....2 12 45

x x xx

2. 4

2 2 2sec 1 .....

3

xx x

3. 2 3 4

cos 111 ....

2 3 24

x x x x xe x

4. 3 4

2 5(1 ) 1 .....

2 6

x x xx x

5. 2 3

1/ 5log[log(1 ) ] .....

2 24 8

x x x xx


6.

2 3

log(1 ) ....2 3

x x xe x x

Q3

If 3 2 32 1x xy y x then prove that

2

1 ....3

xy x

Q4 If

2 3 4

....2 3 4

y y yx y then prove that

2 3

.....2! 3!

x xy x

Q5 Using Taylor’s Theorem

25.15 2. 0tan(46 36') 3.

0sin(30 30') 4. 0cos41

Q6 Evaluate

1. limx o

3 3

5 3

sin 2 sinx x

x x

2. limx o

2

2

sin

1

xe x x x

x xlog x

3. log(2 )cot( 1)1

lim x xx

4. limx

1

1 1 12 3 5

3

xx x x

5. limx o

tan3

sin 2

x

x

6. lim

y x

x y

x y

x yx o

7. limx o

1

2

112

x xx e e

x

8. limx o

2

2

sin

1

xe x x x

x xlog x

9. limx a

7tan2

2ax

a

10. limx

1

1 1 12 3 5

3

xx x x

11. limx o

tan3

sin 2

x

x


12. 0

limx

2

1

tan xx

x

13. limx a

1

2 2

sin

sin

a x

a x

a x

limx a

cot( )

log 2

x ax

a

Q7 If lim

x o 3

sin 2 sinx p x

x

is finite; find the value of p and the limit

Model Question Paper

Course Title : FYT114 : Complex Numbers and Calculus Max.

Marks

Duration 3 Hours 100

Instructions:

All questions are compulsory

Figures to the right indicates full marks

Use of non-programmable calculator is allowed

Q.

No

Marks

1 a Find all the values of

3/4

1 3i and show that their product is 8 5

b Prove that 2 4 6sin 7

7 56sin 112sin 64sinsin

5

c

Attempt any one of the following 6

(i) Prove that if the sum and product of two complex numbers are real

then either they both must be real or they are complex conjugates.

(ii) Prove that the n nth

roots of unity are in geometric progression.

2 Attempt any three 18

a If sin( ) (cos sin )i r i then prove that 2 1

cosh 2 cos 22

r 6

b If 5sinh cosh 5x x find tanh x 6

c If sinh θ + i φ = eiα , prove that sinh4θ = cos2α = cos4φ 6

d

If u = log tan 𝜋

4+

𝜃

2 , prove that cosh u = sec θ and tanh

u

2= tan

θ

2

6

3 a

Examine the convergence of the following infinite series 2

1 +

22

2 +

23

3 + ⋯ ⋯

5


b

Examine the convergence of the following infinite series 14

𝑒12 +

14

𝑒22 +

14

𝑒32 + ⋯ ⋯

5

c


Examine the convergence of the following infinite series

2

1 −

3

2 +

4

3 −

5

4+ ⋯ + −1 𝑛−1 ∙

(𝑛 + 1)

𝑛+ ⋯

Examine whether the following series are absolutely convergent,

conditionally convergent or divergent.

1

1 ! −

1

2 ! +

1

3 ! −

1

4 ! ⋯ ⋯

4

a

Evaluate the following integral.

5

b Determine if the following integral is convergent or divergent and if

convergent find its value.

5

c Attempt any one of the following 6

Evaluate the following integrals

𝑖) 𝑑𝜃

1 − 12

𝑆𝑖𝑛2𝜃

𝜋2

0

ii) Prove that

/2 /2

0 0sin

dsin d

5 Attempt any three 18

a Solve 2 2 2 2 2 2 0x y a xdx x y b ydy

b Solve 22 22 0a xy y dx x y dy

c Solve 2 sin 0yx e dy y x x dx

d Solve 2 21 ( 1)

dyx x y x x

dx


6 a

Expand tan x in powers of 𝑥 − 𝜋

4

5

b If

2 3 4

....2 3 4

y y yx y then prove that

2 3

.....2! 3!

x xy x

5

c


Evaluate

lim

y x

x y

x y

x yx o

Evaluate

limx o

1

2

112

x xx e e

x

Assignments / Tutorials

List of tutorials /assignments to meet the requirements of the syllabus

Assignment No. 1

Assignment Title Complex Numbers CO1, CO2

Batch I i. 1. Simplify

7 5

12 6



i i

i i

ii. 2. Show that 1 cos sin

cos sin1 cos sin

ni

n i ni

iii. 3. Find all the values of

3/4

1 3

2 2i

show that their product is 1

iv. 4. Solve 4 3 2 1 0x x x x

v. 5. Solve 9 5 4 1 0x x x

vi. 6. Solve for x and note all five roots

51

321

x

x

vii. 7. Prove that 2 4 6sin 7

7 56sin 112sin 64sinsin

viii. 8. If cos cos cos 0, sin sin sin 0 then show that

cos2 cos2 cos2 0, sin 2 sin 2 sin 2 0


Batch II ix. 1. Simplify

2 3

9 5

cos5 sin 5 cos7 sin 7

cos 4 sin 4 cos sin

i i

i i

x. 2. Show that 8 8

81 3 1 3 2i i

xi. 3. Find the continued product of all the values of

3/4

1 3

2 2i

xii. 4. Solve

61

11

x

x

xiii. 5. Solve 5 51 32( 1)x x

xiv. 6. Solve for x and note all roots 9 5 4 1 0x x x

xv. 7. Prove that 3 5

2 4

5tan 10 tan tantan5

1 10 tan 5tan

xvi. 8. If cos cos cos 0, sin sin sin 0 then show that

cos2 cos2 cos2 0, sin 2 sin 2 sin 2 0

Batch III xvii. 1.Simplify

1 sin cos

1 sin cos

ni

i

xviii. 2. Show that 3 5

2 4

5tan 10 tan tantan5

1 10 tan 5tan

xix. 3. Find the continued product of all the values of

3/4

1 3

2 2i

xx. 4. Solve 6 0x i

xxi. 5. Solve 7 4 3 1 0x x x

xxii. 6. Solve for x and note all five roots

51

321

x

x

xxiii. 7. Prove that 100 100 511 1 2i i

xxiv. 8. Find nth root of unity and show that

1. Roots are in geometric progression

2. Sum of the all roots is zero

3. Product of all roots is

11

n


Assignment No. 2

Assignment Title Hyperbolic Functions CO1, CO2

Batch I 1) If 1 3z i and n is an integer, then show that 2 22 2 0n n n nz z if n is

not multiple of 3

2) Define cosh & sinhx x . Also prove that 2 2cosh sinh 1x x

3) Prove 1 2cosh log( 1)x x x

4) If cos (cos sin )i R i then prove that 1 sin( )

log2 sin( )

5) If tan6

i x iy

then prove that 2 2 2

13

xx y

6) If tan ii e then prove that 1

log tan2 4 2 4 2

nand

7) Separate into real and imaginary parts of i) 1tan ie

Prove that 1 1tanh (sin ) cosh (sec )

Batch II 1. Solve the equation 7cosh 8sinh 1x x for real values of x

2. Prove that 1 2cosh log( 1)z z z

3. Ifx x

tan tanh2 2

, prove that 1. sinh tanu x 2. cosh secu x

4. Separate into real and imaginary parts

i

i

5. If tan6

x iy i

, prove that 2 2 2

13

xx y

6. If sin( ) tan seci i , prove that cos2 cosh 2 3

Prove that 1sech sin log cot2

Batch III 1. Prove that

31 tanh

cosh 6 sinh 61 tanh

xx x

x

2. Prove that 1 1

2sinh tanh

1

xx

x

3. Prove that 1 1tanh (sin ) cosh (sec )

4. Separate into real and imaginary parts 1 3cos

4

i


5. f cos( ) (cos sin )i r i then prove that 1 sin( )

log2 sin( )

6. If cos4

u iv ec ix

, prove that 2 2 2 2 2( ) 2( )u v u v

7. If 1 1 1cosh cosh ( ) coshx iy x iy a

then prove that 2 2 22( 1) 2( 1) 1a x a y a

Assignment No 3

Assignment Title Infinite Series CO3

Batch I Q1. Discuss the convergence of –

A. 1

2 +

1

4 +

1

8 + ⋯ ⋯

B. 12 + 22 + 32 + 42 ⋯ ⋯

C. 1 + 2 – 3 + 1 + 2 – 3 + 1 + 2 – 3 + ⋯⋯ Q2. Use comparison test to show that the series

A. 1+ 1

2 +

1

3 +

1


B. 1

1 ! +

1

2 ! +

1

3 ! + ⋯ ⋯ 𝑖𝑠 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑡

Q.3 Use D’ Alembert’s Ratio test to show that the series

A. 2

1 +

22

2 +

23


B. 14

𝑒12 +

14

𝑒22 +

14

𝑒32 + ⋯ ⋯ is convergent

Q.4.Use Cauchy root test to show that the series

A. 1

4 +

3

4

2

+ 1

4

3

+ 3

4

4

+ 1

4

5


B. 1 +𝑥

2 +

𝑥2

32 +

𝑥3

43 + ⋯ ⋯ is convergent

C. n+1 x n

nn +1

∞

𝑛=1 is convergent if x < 1, 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑡 𝑖𝑓 𝑥 ≥ 1

Q5.Use Leibnitz test to discuss the convergence of the alternating series –

(1) 1

2 ∙

1

13 −

2

3 ∙

1

23 +

3

4 ∙

1

33 −

4

5 ∙

1

43 + ⋯ ⋯


(2) 1

1 ∙

2

3 −

1

2 ∙

3

4 +

1

3 ∙

4

5 −

1

4 ∙

5

6 + ⋯ ⋯

Q.6 Examine whether the following series are absolutely convergent,

conditionally convergent or divergent.

1) 1

1 ! −

1

2 ! +

1

3 ! −

1

4 ! ⋯ ⋯

2) 1 − 1

2 +

1

3 −

1

4 +

1

5 ⋯ ⋯

Batch II Q1. Discuss the convergence of –

A. 1 +1

2 +

1

3 +

1

4 + ⋯ ⋯

B. 1

1+2+

1

1+ 22+

1

1+ 23 ⋯ +

1

1+ 2𝑛+ ⋯ ⋯

C. 𝑆𝑖𝑛2𝑛

3𝑛∞1

Q2. Use comparison test to show that the series

A. 2

1 ∙ 2 ∙3 +

4

2 ∙ 3 ∙4 +

6

3 ∙ 4 ∙5 + ⋯ ⋯ 𝑖𝑠 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡

B. 1 + 1

22 +

22

33 +

33



A. 3

24 +

32

25 +

33


C. 1

2 1 +

𝑥2

3 2 +

𝑥4

4 3 + ⋯ ⋯ is convergent if 𝑥2 ≤ 1


A. 1

3 +

2

5

2

+ 3

7

3

+ 4

9

4

+ 5

11

5


B. 1

2+

2𝑥

3 +

9𝑥2

42 +

64𝑥3

53 + ⋯ ⋯ is convergent if x < 1

C. an − xn

1+n2

∞

𝑛=1 is convergent if x < 1 , a < 1


Batch III Q1. Discuss the convergence of –

A. 1

1+ 2−𝑛 ∞1

B. 𝑛2+2

2𝑛2+3 ∞1

C. 1

𝑛 !∞1

Q2. Use comparison test to show that the series

A. 1

𝑛∞1 𝑖𝑠 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑡

B. 1 ∙2

32∙ 42 +

3 ∙4

52 ∙ 62 +

5 ∙ 6

72∙ 82 + ⋯ ⋯ 𝑖𝑠 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡


A. 1

3 +

2!

32 +

3!


B. 𝑥

1 ∙2 +

𝑥2

3 ∙4 +

𝑥3

5 ∙6 + ⋯ ⋯𝑥 > 0 is convergent if x < 1

and the test fails when x = 1


A. 23n

32n

∞

𝑛=1 is convergent

B. an

1+n2

∞

𝑛=1 is convergent if a < 1

Q.5.Use Leibnitz test to show that the series

(−1)n

n2+ 2n+2

∞

𝑛=1 is convergent

Assignment No. 4 Assignment title Improper Integral and special functions CO4

Batch I 1) Prove that Γ(n+1) = nΓn

2) Evaluate the following integrals


i)

4

04x

xdx

ii)

1

0 1log

xdx

x

iii)

0

n axx e dx

iv) 24

0

xa dx


i)

1

3 5

0

(1 )x x dx ii) 1/47

3

( 3)(7 )x x dx

4) Prove that

/2 /2

0 0sin

dsin d

5) Prove that

1

3 30

2

3 31

dx

x

6) Prove that

1

0

1log

log 1

a bx x adx

x b

; a > 0, b > 0

7) Verify the rule of differentiation under integral sign for the integral 2

1

0

tan

ax

dxa

8) Define error function and state and prove any two properties of error function

Batch II 1) Evaluate the following integrals

a)

0

n axx e dx

d) 24

0

xa dx

e) 2 2h xe dx


1)

/2

20 11 sin

2

d

2)

1/47

3

( 3)(7 )x x dx

3) Show that

8 6

24

0

(1 )0

(1 )

x xdx

x

4) Show that

/2 2 1 2 1

2 2

0

sin cos 1( , )

( sin cos ) 2

m n

m n m nd B m n

a b a b

5) Evaluate


0

1x

axee dx

x

6) Prove that

1

0

1log

log 1

a bx x adx

x b

; a > 0, b > 0

7) Verify the rule of differentiation under integral sign for the

integral

2

1

0

tan

ax

dxa

8) Define error function and state and prove any two properties of error function

Batch III 1) Prove that 𝛤(n+1) = nΓn


1)

1

0 1log

xdx

x

2)

0

n axx e dx

3) 24

0

xa dx

4) 2 2h xe dx


1.

1

3 5

0

(1 )x x dx 2.

1 25

2

0

1

1

xx dx

x

4) Prove that

/2 /2

0 0sin

dsin d

Prove that 2 1

1.3.5.7...(2 1)2

n

n n

Assignment No. 5

Assignment Title Differential equation of 1st order & 1

st degree CO5

Batch I Solve the following differential equations

1. 2(sin .cos ) cos .sin tan 0xx y e dx x y y dy

2. 2 1 2 1 0x y dx y x dy

3. 2 2 2 2 2 2 0x y a xdx x y b ydy

4. 22 22 0a xy y dx x y dy


5. 2 sin 0yx e dy y x x dx

6. 11 cos log sin 0y y dx x x x y dyx

7. 2 log

dy y

dx y y y x

8. 2 2( ) 2 1

ydx xdy dx

x y x

9. 2 3 2 2( 2 ) 2 3 0yxe xy y dx a x y xy dy

10. 22 0xy y dx xdy

11. ( ) 0yx xxe dx dy e dx ye dy

12. 2 21 ( 1)dy

x x y x xdx

13. x y yxdy

e e edx

14. 2 2 2 3 33 1 (2 1)dy

x x y x y axdx

2cos tandy

x y xdx

Batch II Solve the following differential equations

1.

2.

3.

4.

5.

2 3 xdyy e

dx

2 2(1 ) (1 ) 0y xy dx x xy x y dy

2 3(3 ) log 0y

x y dx x x dyx

22 22 0a xy y dx x y dy

1dx

x ydy


6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

tan(1 ) sec

1

xdy yx e y

dx x

2

1

4 33 xdy

x y x e ydx

2 2( ) 2 1

ydx xdy dx

x y x

4 2 2 3 2 43 2(5 3 2 ) 2 3 5 0x x y xy dx x y x y y dy

2secy ydx xdy e dy

2

2

log (log )dy y y y y

dx x x

2 21 ( 1)dy

x x y x xdx

x y yxdye e e

dx

2 2 2 3 33 1 (2 1)dy

x x y x y axdx

2 2 2(1 2 cos 2 ) (sin ) 0xy x xy dx x x dy

2 2

3

1 3

1 2

dy y x y

dx xy x


Batch III Solve the following differential equations

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Assignment No 6

Assignment Title Expansion of functions CO6

Batch I

1. Prove that 2 3 5 71 1 11tan

2 3 5 7

x x x xx

x

2. Expand 5 4 3 25 6 7 8 9x x x x x in powers of ( x - 1)

2 3 xdyy e

dx

2 2(1 ) (1 ) 0y xy dx x xy x y dy

22 22 0a xy y dx x y dy

tan(1 ) sec

1

xdy yx e y

dx x

2 2( ) 2 1

ydx xdy dx

x y x

4 2 2 3 2 43 2(5 3 2 ) 2 3 5 0x x y xy dx x y x y y dy

2

2

log (log )dy y y y y

dx x x

2 21 ( 1)dy

x x y x xdx

x y yxdye e e

dx

2 2 2 3 33 1 (2 1)dy

x x y x y axdx

2 2 2(1 2 cos 2 ) (sin ) 0xy x xy dx x x dy

2 2

3

1 3

1 2

dy y x y

dx xy x


3. Expand 1

log log 1 xx

up to 3x

4. If 3 2 32 1x xy y x then expand y in ascending powers of x

5. Find approximate value of tan 43 correct up to four places of decimals

6. Obtain expansion of sinxe x in powers of x up to 6x

7. Using Taylor’s theorem expand 4 3

2 3 2 4( 2) 3x x x in of x

8. Evaluate limx o

1

2

112

x xx e e

x

9. If limx o 3

sin 2 sinx p x

x

is finite; find the value of p and limit

Find (i) limx a

7tan2

2ax

a

(ii) lim

x

1

1 1 12 3 5

3

xx x x

Batch II 1. Expand loge x in powers of (x-1) and hence evaluate log 1.1e correct up to

four decimal places

2. Prove that 2 3

log(1 )2 3

x x xe x x

3. Prove that If 3 2 32 1x xy y x then expand y in ascending powers of x

4. Find approximate value of 30'sin30 correct up to four places of decimals

5. Obtain expansion of sin xe in powers of x up to

4x

6. Using Maclaurin’s series prove that

42 2 3sin ...........

6

x xe x x x

7. Using Taylor’s theorem expand 3 4 517 6( 2) 3( 2) ( 2) ( 2)x x x x in

powers of x

8. Evaluate following limits

1. limx o

2

2

sin

1

xe x x x

x xlog x

2. log(2 )cot( 1)

1lim x xx


3. limx a

tan2

2

xax

a 4. lim

x o

2 2(1 )

log(1 )

xe x

x x

Batch III 1. Prove that

2 3 5 71 1....

2 3 5 7

11tan

x x x xn x

x

2. Expand loge x in powers of (x-1) and hence evaluate log 1.1e correct up to

four decimal places

3. Prove that 2 3

log(1 )2 3

x x xe x x

4. Prove that 3

21 5

.....2 24 8

log log 1

x xx xx

5. Find approximate value of 30'sin30 correct up to four places of decimals

6. Using Maclaurin’s series prove that

42 2 3sin ...........

6

x xe x x x

7. Using Taylor’s theorem expand 3 4 57 ( 2) 3( 2) ( 2) ( 2)x x x x in

powers of x

8. Evaluate following limits

1)0

limx

2

1

tan xx

x

2) limx a

1

2 2

sin

sin

a x

a x

a x

3) lim

x a

cot( )

log 2

x ax

a

If limx o 3

sin 2 sinx p x

x

is finite; find the value of p and limit

List of Tutorials - At the end of the tutorial students should be able to:

T1 Solve Examples on De Moivre’s theorem. Find roots of complex numbers.

T2 Obtain real & imaginary parts of a complex number.

T3 Relate circular & hyperbolic functions

T4 Solve examples on, hyperbolic functions & inverse hyperbolic functions

T5 Discuss convergence of series and use various tests of convergence.

T6 Use Cauchy root test for deciding the convergence of alternating series.

T7 Solve improper integral and use Gamma and beta functions.


T8 Solve ODE of 1st order and 1

st degree and apply the knowledge for orthogonal trajectories

T9 Expand given functions as power series.

T10 Evaluate of indeterminate forms

List of open ended experiments/assignments/ activities

Assignment

1. Solve above given assignments by using scilab and verify your answer

2. Trace the curves mentioned in the curriculum by using software’s like function

plotter

✽ ✽ ✽

fy b. tech. semester ii complex numbers and calculus · sanjay ghodawat university, atigre. course...

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