Formula Sheet for All subjects combined in ECE/TCE/IN/EEE departments... Very Useful for all Electronics Engineers...
Institute Of Engineering Studies (IES,Bangalore)
Formulae Sheet in ECE/TCE Department
Communication Systems Amplitude Modulation : DSBSC : 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 =
+
SSBAM : → 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 / 08032552008 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 / 08032552008 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 / 08032552008 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 =
=
MediumI Dielectric , Medium2 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 / 08032552008 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 / 08032552008 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 Xdirection 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 / 08032552008 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 / 08032552008 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 Ntype silicon n = N ; p = n / N Carrier concentration in Ptype 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 / 08032552008 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 / 08032552008 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 / 08032552008 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 psubstrate 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 / 08032552008 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 nchannel
i 
Value of Drain current obtained in saturation when
→ +ve → enhances channel . → ve → depletes channel is –ve for nchannel
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 / 08032552008 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 opamp . Gain Bandwidth product is constant . Closed loop voltage gain = β → feed back factor
⇒
dt → LPF acts as integrator ;
=
⇒
=
=
For Opamp 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 / 08032552008 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 RCphase 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 / 08032552008 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 opamp 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 / 08032552008 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 / 08032552008 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 / 08032552008 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