ma8151 [regulation 2017] unit i differential calculus

Sri Hariganesh Institute of Mathematics (Ph: 9841168917 / 8939331876)

MA8151 [Regulation 2017]

Unit – I

Differential Calculus

1. Limit of a Function:

The limit of ( )f x , as x approaches a,

equals if we can make the value of

( )f x arbitrarily close to by taking

x to be sufficiently close to a but not

equal to a.

i.e. lim ( )x a

f x

2. Continuity:

A function f is continuous at a

number a if lim ( ) ( )x a

f x f a

.


3. Derivative:

The derivative of a function ( )f x at

x a denoted by ( )f a , is

0

( ) ( )( ) lim

h

f a h f af a

h

if this

limit exists.

(or) ( ) ( )

( ) limx a

f x f af a

x a

,

0

( ) ( )( ) lim

h

f x h f xf x

h

.

4. Table of derivative of the functions:

Sl.No. y dy

dx

1. Constant 0

2. nx 1nnx

3. x 1

4. 1

nx

1n

n

x


1

x

2

1

x

5. x

1

2 x

6. xe

xa

xe

logxa a

7.

log x

10log x

1

x

10

1log e

x

8. sin x cos x

9. cos x sin x

10. tan x 2sec x

11. cosecx cos cotecx x

12. sec x sec tanx x

13. cot x 2cosec x


14. 1sin x 2

1

1 x

15. 1cos x 2

1

1 x

16. 1tan x 2

1

1 x

5. Special Formulae:

(i) d dv du

uv u vdx dx dx

(ii) d dw du

uvw uv vwdx dx dx

dvuw

dx

(iii) 2

du dvv u

d u dx dx

dx v v


6. Equation of tangent line: 1 1( )y y m x x

7. Equation of normal line: 1 1

1( )y y x x

m

8. If the tangent line parallel to x-axis (horizontal) then 0

dy

dx .

9. If the tangent line parallel to y -axis (vertical) then 0

dx

dy .

10. Increasing and Decreasing Function

Let f be a function defined on the

interval [ , ]a b and have a finite derivative

inside the segment, then (i) f is increasing if and only if

( ) 0f x for all x in [ , ]a b .

(ii) f is decreasing if and only if

( ) 0f x for all x in [ , ]a b .


11. Monotonic Functions If a function f is completely increasing or

completely decreasing in an interval [ , ]a b

, then the function f is called monotonic

function in [ , ]a b . 12. Critical Number

A critical number of a function f is a

number c in the domain of f such that

( ) 0f c . 13. Maxima and Minima by First

Derivative Test Consider x a be a critical point of a

continuous function ( )f x .

(i) If ( )f x changes from positive to

negative at x a , then ( )f x has a

maximum at x a .

(ii) If ( )f x changes from negative to

positive at x a , then ( )f x has a

minimum at x a .


14. Maxima and Minima by Second

Derivative Test Consider x a be a critical point of a

continuous function ( )f x .

i) If ( ) 0f a , then ( )f x has a maximum

at x a .

ii) If ( ) 0f a , then ( )f x has a

minimum at x a . 15. Concavity Test

Suppose ( )f x is twice differentiable on an

interval I .

(i) If ( ) 0f x for all x in I , then the

graph of ( )f x is concave upward on I .

(ii) If ( ) 0f x for all x in I , then the

graph of ( )f x is concave downward on I .

16. Point of Inflection A point on a curve is called a point of P

inflection if the curve changes from

concave upward to concave downward or


from concave downward to concave

upward at . P

Unit – II

Functions of Several Variables

1. Euler’s Theorem:

If f is a homogeneous function of x

and y in degree n, then

(i) First Order

f fx y nf

x y

(ii) Second Order

2 2 2

2 2

2 22 1

f f fx xy y n n f

x x y y


2. Total Derivative:

If ( , , )u f x y z , 1 2( ), ( ),x g t y g t

3( )z g t then

du u dx u dy u dz

dt x dt y dt z dt

.

3. If 1 2( , ), ( , ), ( , )u f x y x g r y g r

then

(i) u u x u y

r x r y r

(ii) u u x u y

x y

4. Maxima and Minima:

Working Rules:

Step: 1 Find xf and yf . Put 0xf and

0yf . Find the value of x and y .


Step: 2 Calculate , ,xx xyr f s f yyt f .

Now 2rt s

Step: 3

i) If 0 , then the function have either

maximum or minimum.

1. If 0r f has maximum

2. If 0r f has minimum

ii) If 0, then the function is neither

Maximum nor Minimum, it is called

Saddle Point.

iii) If 0, then the test is inconclusive.

5. Maxima and Minima of a function using

Lagrange’s Multipliers:


Let ( , , )f x y z be given function and

( , , )g x y z be the subject to the

condition.

Form ( , , ) ( , , ) ( , , )F x y z f x y z g x y z ,

Putting 0x y zF F F F and then

find the value of , ,x y z .

6. Jacobian:

Jacobian of two dimensions:

, ( , )

, ( , )

u v u vJ

x y x y

u u

x y

v v

x y

7. The functions u and v are called

functionally dependent if ( , )

0( , )

u v

x y

.


8. ( , ) ( , )

1( , ) ( , )

u v x y

x y u v

9. Taylor’s Expansion:

1

( , ) ( , ) ( , ) ( , )1!

x yf x y f a b hf a b kf a b

2 21( , ) 2 ( , ) ( , )

2!xx xy yyh f a b hkf a b k f a b

3 2 2 31( , ) 3 ( , ) 3 ( , ) ( , ) ...

3!xxx xxy xyy yyyh f a b h kf a b hk f a b k f a b

where h x a and k y b

Unit – III

Integral Calculus

1.

1

1

nn x

x dx cn

2.

x xe dx e c

3.

1 1dx c

x x

1

1 1

1n ndx c

x n x


4.

3/22

3

xxdx c

5.

sin cosxdx x c

6.

cos sinxdx x c

7.

sec log sec tanxdx x x c

8.

cosec log cosec cotxdx x x c

9.

tan logsecxdx x c

10. cot logsinxdx x c

11. 1

2 2sin

dx xc

aa x

1

2sin

1

dxx c

x

12. 1

2 2cosh

dx xc

ax a

2 2( ) logor x x a c

13. 1

2 2sinh

dx xc

aa x

2 2( ) logor x x a c


14. 1

2 2

1tan

dx xc

a x a a

1

2tan

1

dxx c

x

15. 2 2

1log

2

dx x ac

x a a x a

16. 2 2

1log

2

dx a xc

a x a a x

17. 2

2 2 2 2 1sin2 2

x a xa x dx a x c

a

18. 2

2 2 2 2 1sinh2 2

x a xa x dx a x c

a

(or)

22 2 2 2log

2 2

x aa x x a x c

19. 2

2 2 2 2 1cosh2 2

x a xx a dx x a c

a

(or)

22 2 2 2log

2 2

x ax a x x a c


20. 2 2sin sin cos

axax e

e bxdx a bx b bxa b

21. 2 2cos cos sin

axax e

e bxdx a bx b bxa b

22. Reduction Formulae

2 2

0 0

cos (or) sin

1 3 5 2... .1 if is odd

2 4 3

n nxdx xdx

n n nn

n n n

1 3 5 1... . if is even

2 4 2 2

n n nn

n n n

2

0

sin cosm nx xdx

1 3 ... 1 3 ...

2 4 ...

m m n n

m n m n m n

1 3 ... 1 3 ...

2 4 ... 2

[ and are even]

m m n n

m n

m

m n n

n

m

23. 0

( ) 2 ( ) [if ( )isaneven function]

a a

a

f x dx f x dx f x

0 [if ( )isanoddfunction]f x


24. 0 0

( ) ( )

a a

f x dx f a x dx

25. ( ) ( )

b a

a b

f x dx f a x dx

26. Integration by Parts:

u dv uv vdu

27. Bernoulli’s Formulae:

1 2 3 4 ...uvdx uv u v u v u v

Unit – IV

Multiple Integrals

1.R

dxdy R

dydx Area (or)

To change into polar coordinate

cosx r siny r dxdy rdrd, and .

2.V

dxdydz V

dzdydx Volume (or)


Unit – V

Differential Equations

1. ODE with constant coefficients:

Solution C.F + P.Iy

Complementary functions:

Sl.No. Nature of

Roots C.F

1. 1 2m m ( )

mxAx B e

2. 1 2 3m m m 2 mx

Ax Bx c e

3. 1 2m m 1 2m x m x

Ae Be

4. 1 2 3m m m 31 2 m xm x m x

Ae Be Ce

5. 1 2 3, m m m 3( )

m xmxAx B e Ce

6. m i ( cos sin )x

e A x B x

7. m i cos sinA x B x


Particular Integral:

Type-I

If ( ) 0f x then . 0P I

Type-II

If ( )ax

f x e (or) sinhax (or) coshax

1.

( )

axP I e

D

Replace Dby a . If ( ) 0D , then it is P.I. If

( ) 0D , then diff. denominator w.r.t D and

multiply x in numerator. Again replace D

by a . If you get denominator again zero

then do the same procedure.

Type-III

Case: i If ( ) sin ( ) cosf x ax or ax

1. sin (or) cos

( )P I ax ax

D


Here you have to replace only for 2D not

for D . 2D is replaced by 2

a . If the

denominator is equal to zero, then

apply same procedure as in Type – I.

Case: ii If 2 2 3 3( ) (or) cos (or) sin (or) cosf x Sin x x x x

Use the following formulas 2 1 cos 2

2

xSin x

,

2 1 cos 2cos

2

xx

, x x x 3 3 1

sin sin sin 34 4

,

x x x 3 3 1cos cos cos 3

4 4 and separate 1 2

. & .P I P I

Case: iii If ( ) sin cos ( ) cos sinf x A B or A B ( ) cos cosor A B

( ) sin sinor A B

Use the following formulas:

1( ) in cos ( ) sin( )

2

1(ii) cos sin ( ) sin( )

2

1( ) cos cos cos( ) cos( )

2

1( ) sin sin cos( ) cos( )

2

i s A B sin A B A B

A B Sin A B A B

iii A B A B A B

iv A B A B A B


Type-IV

If ( ) mf x x

1.

( )

mP I xD

1

1 ( )

mxg D

1

1 ( ) mg D x

Here we can use Binomial formula as

follows:

i) 1 2 31 1 ...x x x x

ii) 1 2 31 1 ...x x x x

iii) 2 2 31 1 2 3 4 ...x x x x

iv) 2 2 31 1 2 3 4 ...x x x x

Type-V

If ( ) axf x e V where sin ,cos , mV ax ax x

1.

( )

axP I e VD

1

( )

axe VD a

Type-VI

If ( ) nf x x V where sin ,cosV ax ax

sin I.P of

cos R.P of

iax

iax

ax e

ax e


Type-VII

If ( ) sec (or) cosec (or) tanf x ax ax ax

1. ( ) ( )ax axP I f x e e f x dx

D a

1. ODE with variable co-efficient: (Euler’s Method) The equation is of the form

22

2( )

d y dyx x y f x

dx dx

Implies that 2 2( 1) ( )x D xD y f x

To convert the variable coefficients into

the constant coefficients

Put logz x implies zx e

2 2

3 3

( 1)

( 1)( 2)

xD D

x D D D

x D D D D

where dD

dx and

dD

dz


The above equation implies that

( 1) 1 ( )D D D y f x which is O.D.E with

constant coefficients.

2. Legendre’s Linear differential equation: The equation if of the form

22

2( ) ( ) ( )

d y dyax b ax b y f x

dx dx

Put log( )z ax b implies ( ) zax b e

2 2 2

3 3 3

( )

( ) ( 1)

( ) ( 1)( 2)

ax b D aD

ax b D a D D

ax b D a D D D

where dD

dx and

dD

dz

3. Method of Variation of Parameters: The equation is of the form

d y dya b cy f x

dx dx

2

2( )

1 2.C F Ay By and 1 2.P I Py Qy

where 2

1 2 1 2

( )y f xP dx

y y y y

and 1

1 2 1 2

( )y f xQ dx

y y y y


Textbook for Reference:

“ENGINEERING MATHEMATICS - I”

Publication: Sri Hariganesh Publications

Author: C. Ganesan

Mobile: 9841168917, 8939331876

To buy the book visit

www.hariganesh.com/textbook

----All the Best----

http://www.hariganesh.com/textbook

ma8151 [regulation 2017] unit i differential calculus

Documents