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Complete Study Guide & Notes On CONTINUITY & DIFFERENTIAL CALCULUS

Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover!

IMPORTANT TERMS, DEFINITIONS & RESULTS

01. Formulae for Limits & Differential Calculus:

LIMITS FOR SOME STANDARD FORMS

a) 0

limcos 1

x

x

b) 0

sinlim 1x

x

x

c) 0

tanlim 1x

x

x

d) 1

0

sinlim 1x

x

x

e) 1

0

tanlim 1x

x

x

f) 0

1lim log , 0

xe

xaa a

x

g) 0

1lim 1x

xe

x

h)

0

log 1lim 1e

x

x

x

i) 1limx a

nn

nx ana

x a

j) 0

1/lim 1

k

x

xkx e , where k is any constant.

DERIVATIVE OF SOME STANDARD FUNCTIONS

a) 1n ndx n x

dx

b) ( ) 0,d

kdx

where k is any constant

c) log , 0 x xe

da a a a

dx

d) x xde e

dx

e) 1 1

log loglog

a ae

dx

dx x a xe

f) 1

loge

dx

dx x

g) sin cosd

x xdx

h) cos sind

x xdx

i) 2tan secd

x xdx

j) sec sec tand

x x xdx

k) 2cot cosecd

x xdx

l) cosec cosec cot d

x x xdx

m) 1

2

1sin , 1,1

1

dx x

dx x

n) 1

2cos , 1,1

1

1

dx x

dx x

o) 12

1tan , R

1

dx x

dx x

p) 1

2cot , R

1

1

dx x

dx x

q) 1

2

1sec

1

dx

dx x x

, where , 1 1, x

r) 1

2

1cosec

1

dx

dx x x

,where , 1 1, x

Do you know for trigonometric functions, angle ‘x’ is in Radians?

Following derivatives should also be memorized by you for quick use:

1

2

dx

dx x

2

1 1d

dx x x

1 logx xd

x x xdx

A Formulae Guide By OP Gupta (Indira Award Winner)

Continuity & Differentials By OP Gupta (INDIRA AWARD Winner, Elect. & Comm. Engineering)

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02. Important terms and facts about Limits and Continuity of a function:

For a function f x , limx m

f x exists iff lim limx m x m

f x f x

.

A function f x is continuous at a point x m iff lim limx m x m

f x f x f m

, where

limx m

f x

is Left Hand Limit of f x at x m and limx m

f x

is Right Hand Limit of

f x at x m . Also f m is the value of function f x at x m .

A function f x is continuous at x m (say) if, lim

mx

f m f x i.e., a function is continuous

at a point in its domain if the limit value of the function at that point equals the value of the function at the same point.

For a continuous function f x at x m , limx m

f x can be directly obtained by evaluating

f m .

Indeterminate forms or meaningless forms:

0 00, , 0. , , 1 , 0 ,

0.

The first two forms i.e. 0

,0

are the forms which are suitable for L’Hospital Rule.

03. Important terms and facts about Derivatives and Differentiability of a function:

Left Hand Derivative of f x at x m,

L lim

x m

f x f mf m

x m and,

Right Hand Derivative of f x at x m,

R lim

x m

f x f mf m

x m.

For a function to be differentiable at a point, the LHD and RHD at that point should be equal.

Derivative of y w.r.t. x: 0

lim

x

dy y

dx x.

Also for very- very small value h,

f x h f x

f xh

, 0as h .

04. Relation between Continuity and Differentiability:

a) If a function is differentiable at a point, it is continuous at that point as well. b) If a function is not differentiable at a point, it may or may not be continuous at that point. c) If a function is continuous at a point, it may or may not be differentiable at that point. d) If a function is discontinuous at a point, it is not differentiable at that point.

05. Rules of derivatives:

Product or Leibnitz Rule of derivatives: v v v d d d

u u udx dx dx

Quotient Rule of derivatives:

2 2

v v v v

v v v

d du ud u u udx dx

dx

.

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