Electronics Engineering Formula Sheets

Institute Of Engineering Studies (IES,Bangalore)

Formulae Sheet in ECE/TCE Department

Communication Systems Amplitude Modulation : DSB-SC : u (t) = m(t) cos 2π t Power P = Conventioanal AM : u (t) = [1 + m(t)] Cos 2π t . as long as |m(t)| ≤ 1 demodulation is simple . Practically m(t) = a m (t) . () () Modulation index a = ( ) , m (t) = | ( )| Power =

+

SSB-AM : → Square law Detector SNR =

()

Square law modulator ↓ = 2a / a → amplitude Sensitivity Envelope Detector R C (i/p) < < 1 /

R C (o/P) >> 1/

R C x} = 1 – P { X ≤ x} = 1- (x)

PDF :Pdf =

7

(x) =

(x)

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)

Formulae Sheet in ECE/TCE Department

= x } δ(x = x )

Pmf = (x) = Properties: (x) ≥ 0 

(x) =



(∞) =



(x) * u(x) =

(x) dx

(x) dx =1 so, area under PDF = 1 (x)dx

P{x 1 (over damped) :-

11

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)

Formulae Sheet in ECE/TCE Department

C(t) = 1 T= >

>

>

Time Domain Specifications :∅



Rise time t =



Peak time t =

  

/ Max over shoot % =e × 100 Settling time t = 3T 5% tolerance = 4T 2% tolerance . Delay time t =



Damping actor



Time period of oscillations T =



No of oscillations =

 

t ≈ 1.5 t t = 2.2 T Resonant peak =



Bandwidth ω = ω (1

∅ = tan

=

/

(

) (

)

= ; ω =ω 2

1

2

+

ω > ω ) → good conductor Complex permittivity = 1 = -j .



Tan

e

cos(ωt – βz) ;

=

/η.

|η | < /

tan 2

/

=

=

= σ/ω .

η=

/

∠ .

E & H are in phase in lossless dielectric

Free space :- (σ = 0,

15

= 8.686 dB

.

Plane wave in loss less dielectric :- ( σ ≈ 0)  α=0;β=ω ω= λ = 2π/β 

| N | = 20 log

=

, =

)

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)

 α=0,β=ω ; u = 1/ Here also E & H in phase .

Formulae Sheet in ECE/TCE Department

, λ = 2π/β

Good Conductor :σ >>ω σ/ω → ∞ ⇒ σ = ∞

=

η=

/

< = 12 π ∠

=



α = β = π σ ; u = 2ω/ σ ; λ = 2π / β ; η =

 

Skin depth δ = 1/α η= 2e / =



Skin resistance R =



R

=



R

=

∠

=

.

.

Poynting Vector :( ) ds =  S

+

[

] dv – σ

v



δ



Total time avge power crossing given area

(z) =

| |

e

dv

cos

a (s) ds

= S

Direction of propagation :- ( a ×a =a

)

a ×a =a → Both E & H are normal to direction of propagation → Means they form EM wave that has no E or H component along direction of propagation . Reflection of plane wave :(a) Normal incidence Reflection coefficient Γ = = coefficient Τ =

=

Medium-I Dielectric , Medium-2 Conductor :> :Γ there is a standing wave in medium Max values of | | occurs = - nπ/β = n = 1 2…. =

16

(

)

=

(

wave in medium ‘2’.

)

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)

<

occurs @ β

:-

β

(

=

= nπ ⇒

)

=

⇒

Formulae Sheet in ECE/TCE Department =

(

)

=

(

)

=

min occurs when there is |t |max | | | | | | S=| | = | | = | |;|Γ|=

Since |Γ| < 1 ⇒ 1 ≤ δ ≤ ∞

Transmission Lines : Supports only TEM mode  LC = ; G/C = σ / .   

-r

-r

=0;

=0

Γ = (R + ω )( + ωC) = α + jβ V(z, t) = e cos (ωt- βz) + e



=

=

=

cos (ωt + βz)

=

Lossless Line : (R = 0 =G; σ = 0) → γ = α + jβ = jω C α = β = w = /C

C

λ = 1/

C , u = 1/

C

Distortion less :(R/L = G/C) →α= R →

=

β=ω =

= ωC λ = 1/

i/p impedance :=

=ω

C; u=

C =

for lossless line

;u

= 1/C , u /

= 1/L

γ = jβ ⇒ tan hjβl = j tan βl

= 

VSWR = Γ =

  

CSWR = - Γ Transmission coefficient S = 1 + Γ | | SWR = = = = |

|

( 

|

|

=

=S



|

|

=

=

) (

<

)

=j

tan βl

/S

Shorted line :- Γ = -1 , S = ∞

17

>

=

=

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)



Γ = -1 , S = ∞



=

Formulae Sheet in ECE/TCE Department

tan βl.

=j

may be inductive or capacitive based on length ‘0’ If l < λ / 4 → inductive ( +ve) < l < λ/2 → capacitive ( -ve)

Open circuited line := = -j cot βl Γ =1 s=∞

l < λ / 4 capacitive < l < λ/2 inductive

= Matched line : ( = ) = Γ = 0 ; s =1 No reflection . Total wave

. So, max power transfer possible .

Behaviour of Transmission Line for Different lengths :l = λ /4 → l = λ /2 :

→ impedance inverter @ l = λ /4 =

⇒

impedance reflector @ l = λ /2

Wave Guides :TM modes : ( = ) = sin x sin h =k +k

ye

∴γ=

+

ω

where k = ω

m→ no. of half cycle variation in X-direction n→ no. of half cycle variation in Y- direction . Cut off frequency ω =

+

γ = 0; α = 0 = β



k <

+

→ Evanscent mode ; γ = α ; β = 0



k >

+

→ Propegation mode γ = β α =

β= k 

18

=

+

u = phase velocity =

is lossless dielectric medium

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)

Formulae Sheet in ECE/TCE Department



λ =u/ =



β=β



u = ω/β λ = 2π/β = u /f → phase velocity & wave length in side wave guide



η

=

η

=η

=

( )

1

β = ω/ W

=-

=

β = phase constant in dielectric medium.

1

=

1

TE Modes :- ( →

( )

η → impedance of UPW in medium

= 0)

cos

e

cos

→η

=

= η / 1

→ η

>η

→

Dominant mode

Antennas :Hertzian Dipole :-

=

sin

e

=η

Half wave Dipole :=

=η

;

EDC & Analog      

19

Energy gap =

.

/ /

- KT ln

. .

.

=

.

Energy gap depending on temperature

.

+ KT ln

)/ No. of electrons n = N e ( )/ No. of holes p = N e ( Mass action law n = n = N N e Drift velocity = E (for si ≤1

(KT in ev) /

cm/sec)

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore) .



Hall voltage

 

Conductivity σ = ρ ; = σR . Max value of electric field @ junction



Charge storage @ junction

=

Formulae Sheet in ECE/TCE Department

. Hall coefficient R = 1/ρ .

=-

= -

ρ → charge density = qN = ne …

N .n

= -

N .n

.

= qA x N = qA x N EDC



Diffusion current densities J = - q D

 

Drift current Densities = q(p + n )E , decrease with increasing doping concentration .



=

J =-qD

= KT/q ≈ 25 mv @ 300 K

 

Carrier concentration in N-type silicon n = N ; p = n / N Carrier concentration in P-type silicon p = N ; n = n / N



Junction built in voltage



Width of Depletion region

ln = x +x =

+

(

+

)

= 12.9

* 

=

=



Charge stored in depletion region q =



Depletion capacitance C =

.

; C =

.A. /

C =C / 1+ C = 2C 

(for forward Bias)

Forward current I =

+

;

= Aq n = Aq n



Saturation Current

 

Minority carrier life time = Minority carrier charge storage Q= + = I



Diffusion capacitance C =

 



20

= Aq n

/ /

1 1

+ /D ; = /D = , = = mean transist time I = .g ⇒ C ∝ I.

→ carrier life time , g = conductance = I / )/ = 2( Junction Barrier Voltage = = (open condition) = - V (forward Bias) = + V (Reverse Bias) Probability of filled states above ‘E’ f(E) = ( )/

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)



Drift velocity of e



Poisson equation

Transistor : = +  = –α  =–α +

≤1

cm/sec

=

⇒

=

=E=

→ Active region (1- e / )

Common Emitter : = (1+ β) +β   

Formulae Sheet in ECE/TCE Department

β=

= → Collector current when base open → Collector current when = 0 > or → - 2.5 mv / C ; →



Large signal Current gain β =



D.C current gain β



(β



Small signal current gain β =



Over drive factor =

=h

=

=h

) ≈ β when

>

Conversion formula :CC ↔ CE  h =h ; h =1; CB ↔ CE  h =

; h =

C R

= h =

= - 0.25 mv / C

(

)

∵

→

h = - (1+ h ) ;

-h

.

;h =

h

; h

h C

=β

=h

=

CE parameters in terms of CB can be obtained by interchanging B & E . Specifications of An amplifier :

= =

=h +h

=

=h -

=

.

.

= =

.

=

.

.

Choice of Transistor Configuration : For intermediate stages CC can’t be used as 10 R

; R = - 2 (R + R )] [ 0.07

. S] < 1/

;

<

Hybrid –pi(π)- Model :g =|

|/

r = h /g r =h -r r =r /h g = h - (1+ h ) g For CE : = 

=h

(

)

=

(

;

)

=

C = C + C (1 + g R )

=

= S.C current gain Bandwidth product = Upper cutoff frequency For CC :

For CB: =

22

≈

=

(

=

)

=

= (1 + h )

(

)

= (1 + β)

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)



=

>

Ebress moll model :=-α + (1- e =-α α

/

>

)

/

(1- e

+

Formulae Sheet in ECE/TCE Department

)

=α

Multistage Amplifiers : * = 2 / 1 ; .



Rise time t =



t = 1.1 t

+t

+



= 1.1

+

+



= 1.1

+

+

=

=

/

. .

Differential Amplifier :

= h + (1 + h ) 2R = 2 h R ≈ 2βR |



g =



CMRR =

|

= g of BJT/4

=

;

R ↑,→

Darlington Pair : = (1 + β ) (1 + β ) ; 

=

(

)



R =(

)



g = (1 + β ) g

Ω

α → DC value of α ↑

↑

C RR ↑

≈ 1 ( < 1) [ if

&

have same type ] =

R

+

Tuned Amplifiers : (Parallel Resonant ckts used ) : 

23

=

Q → ‘Q’ factor of resonant ckt which is very high

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)



B.W =



=

-



=

+



Formulae Sheet in ECE/TCE Department

/Q ∆ ∆

For double tuned amplifier 2 tank circuits with same

used .

=

.

MOSFET (Enhancement) [ Channel will be induced by applying voltage]    

NMOSFET formed in p-substrate If ≥ channel will be induced & i (Drain → source ) → +ve for NMOS i ∝( - ) for small



↑ → channel width @ drain reduces . =

channel width ≈ 0 → pinch off further increase no effect

-



For every



i =

>

[(

there will be

-

)

-

→ triode region (

]

= 

i =



r

[

=

(

   

24

[(

)

] → saturation

)

→ Drain to source resistance in triode region

→ induced channel

i =

-

C

PMOS : Device operates in similar manner except , ,  i enters @ source terminal & leaves through Drain . ≤

<

) -

≥ ]

→ Continuous channel

=

are –ve

C

≤ → Pinched off channel . NMOS Devices can be made smaller & thus operate faster . Require low power supply . Saturation region → Amplifier For switching operation Cutoff & triode regions are used NMOS

PMOS

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)

≥

≤

Formulae Sheet in ECE/TCE Department

→ induced channel

-

>

-

<

→ Continuous channel(Triode region)

≥

-

≤

-

→ Pinchoff (Saturation)

Depletion Type MOSFET :- [ channel is physically implanted . i flows with 

For n-channel



i -



Value of Drain current obtained in saturation when

→ +ve → enhances channel . → -ve → depletes channel is –ve for n-channel

characteristics are same except that

∴

=0]

=

=0⇒

.

.

MOSFET as Amplifier : 

For saturation > To reduce non linear distortion



i =

 

(

)

⇒

g =

) (

)

=-g R Unity gain frequency

JFET : ≤  ≤

≤

=

(

)

⇒ i = 0 → Cut off ≤ 0, ≤ i



< < 2(

≤0 ,

=

2 1 ≥

→ Triode

⇒ |

→ Saturation

|

|

|

Zener Regulators :

25

For satisfactory operation

≥

+

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)



R

Formulae Sheet in ECE/TCE Department

=



Load regulation = - (r || R )



Line Regulation =



For finding min R

. take

&

,

(knee values (min)) calculate according to that .

Operational Amplifier:- (VCVS)     

Fabricated with VLSI by using epitaxial method High i/p impedance , Low o/p impedance , High gain , Bandwidth , slew rate . FET is having high i/p impedance compared to op-amp . Gain Bandwidth product is constant . Closed loop voltage gain = β → feed back factor



⇒



dt → LPF acts as integrator ;

=

⇒

=

=



For Op-amp integrator



Slew rate SR =



Max operating frequency



In voltage follower Voltage series feedback



In non inverting mode voltage series feedback



In inverting mode voltage shunt feed back



= -η



== -η



26

dt ;

∆ ∆

=

∆ ∆

= .

∆ ∆

dt ; = A. =

(HPF)

Differentiator

=-

∆ ∆

=

. ∆

∆

.

ln

ln

Error in differential % error =

× 100 %

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)

Formulae Sheet in ECE/TCE Department

Power Amplifiers :

Fundamental power delivered to load



Total Harmonic power delivered to load

R

=

=

=

+

=

1+

+

R .. +

+ ……

= [ 1+ D ] Where D =

+D +

. . +D

D =

D = total harmonic Distortion . Class A operation : o/p flows for entire  ‘Q’ point located @ centre of DC load line i.e., = / 2 ; η = 25 %  Min Distortion , min noise interference , eliminates thermal run way  Lowest power conversion efficiency & introduce power drain  = -i if i = 0, it will consume more power  is dissipated in single transistors only (single ended) Class B:   

flows for 18 ; ‘Q’ located @ cutoff ; η = 78.5% ; eliminates power drain Higher Distortion , more noise interference , introduce cross over distortion Double ended . i.e ., 2 transistors . = 0 [ transistors are connected in that way ] =i = 0.4 → power dissipated by 2 transistors .

=i

Class AB operation :   

flows for more than 18 & less than ‘Q’ located in active region but near to cutoff ; η = 60% Distortion & Noise interference less compared to class ‘B’ but more in compared to class ‘A’ Eliminates cross over Distortion

Class ‘C’ operation : flows for < 180 ; ‘Q’ located just below cutoff ; η = 87.5%  Very rich in Distortion ; noise interference is high . Oscillators : For RC-phase shift oscillator f = f=

27

h ≥ 4k + 23 +

where k = R /R

> 29

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)



For op-amp RC oscillator f =

|

Formulae Sheet in ECE/TCE Department

| ≥ 29 ⇒ R ≥ 29 R

Wein Bridge Oscillator :h ≥3 ≥3 A≥3⇒ R ≥2R

f=

Hartley Oscillator :f=

(

)

|h | ≥ | | ≥ |A| ≥ ↓

Colpits Oscillator :f=

|h | ≥ |

| ≥

|A| ≥

28

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)

Formulae Sheet in ECE/TCE Department

MatheMatics Matrix : If |A| = 0 → Singular matrix ; |A| ≠ 0 Non singular matrix  Scalar Matrix is a Diagonal matrix with all diagonal elements are equal  Unitary Matrix is a scalar matrix with Diagonal element as ‘1’ ( = ( ) = )  If the product of 2 matrices are zero matrix then at least one of the matrix has det zero  Orthogonal Matrix if A = .A = I ⇒ =  A= → Symmetric A=→ Skew symmetric Properties :- (if A & B are symmetrical )  A + B symmetric  KA is symmetric  AB + BA symmetric  AB is symmetric iff AB = BA  For any ‘A’ → A + symmetric ; A skew symmetric.  Diagonal elements of skew symmetric matrix are zero  If A skew symmetric → symmetric matrix ; → skew symmetric  If ‘A’ is null matrix then Rank of A = 0. Consistency of Equations : r(A, B) ≠ r(A) is consistent  r(A, B) = r(A) consistent & if r(A) = no. of unknowns then unique solution r(A) < no. of unknowns then ∞ solutions . Hermition , Skew Hermition , Unitary & Orthogonal Matrices :-

29

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)

     

Formulae Sheet in ECE/TCE Department

= → then Hermition = → then Hermition Diagonal elements of Skew Hermition Matrix must be purely imaginary or zero Diagonal elements of Hermition matrix always real . A real Hermition matrix is a symmetric matrix. |KA| = |A|

Eigen Values & Vectors : Char. Equation |A – λI| = 0. Roots of characteristic equation are called eigen values . Each eigen value corresponds to non zero solution X such that (A – λI)X = 0 . X is called Eigen vector .  Sum of Eigen values is sum of Diagonal elements (trace)  Product of Eigen values equal to Determinent of Matrix .  Eigen values of & A are same | |  λ is igen value o then 1/ λ → & is Eigen value of adj A. 

λ , λ …… λ are Eigen values of A then →

λ , K λ …….. λ

→ λ , λ ………….. λ . A + KI → λ + k , λ + k , …….. λ + k ( ) → (λ k) , ……… (λ k)     

Eigen values of orthogonal matrix have absolute value of ‘1’ . Eigen values of symmetric matrix also purely real . Eigen values of skew symmetric matrix are purely imaginary or zero . λ , λ , …… λ distinct eigen values of A then corresponding eigen vectors linearly independent set . adj (adj A) = | | ; | adj (adj A) | = | |( )

,

, .. …

for

Complex Algebra :

Cauchy Rieman equations Neccessary & Sufficient Conditions for f(z) to be analytic

 



30

( )/(

a)

f(z) = f( ) + ( )

dz = (

)

[

(a) ] if f(z) is analytic in region ‘C’ & Z =a is single point ( )

(

)

(

)

+ …… + ( ) + ………. Taylor Series ⇓ ( ) if = 0 then it is called Mclauren Series f(z) = a ( ) ; when a = If f(z) analytic in closed curve ‘C’ except @ finite no. of poles then +

No.1 Training center for GATE/IES/JTO/PSUs in Bangalore @ Malleshwaram & Jayanagar, Bangalore. Ph: 0 99003 99699/ 0 97419 00225 / 080-32552008 Email : [email protected] Site: www.onlineIES.com Google+: http://bit.ly/gplus_iesgate FB: www.facebook.com/onlineies

Institute Of Engineering Studies (IES,Bangalore)

Formulae Sheet in ECE/TCE Department

( )d = 2πi (sum of Residues @ singular points within ‘C’ ) Res f(a) = lim

(

→

= Φ(a) / = lim

( ) (a)

→ (

)

((

a) f(z) )

Calculus :Rolle’s theorem :If f(x) is (a) Continuous in [a, b] (b) Differentiable in (a, b) (c) f(a) = f(b) then there exists at least one value C (a, b) such that

(c) = 0 .

Langrange’s Mean Value Theorem :If f(x) is continuous in [a, b] and differentiable in (a, b) then there exists atleast one value ‘C’ in (a, b) such that

(c) =

( )

( )

Cauchy’s Mean value theorem :If f(x) & g(x) are two function such that (a) f(x) & g(x) continuous in [a, b] (b) f(x) & g(x) differentiable in (a, b) (c) g (x) ≠ 0 ∀ x in (a, b) Then there exist atleast one value C in (a, b) such that (c) / g (c) =

( ) ( )

( ) ( )

Properties of Definite integrals : 

31

(x). dx = (x). dx + a