Short Notes: Calculus | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

 Page 1


 
 
 
 
 
CALCULUS 
 
  Important Series Expansion  
a. ? ?
n
n
n
r  0
r
r
1 x C x
?
??
?
    
b. ? ?
1
2
1 x 1 x x ............
?
? ? ? ? ?   
c. ? ? ? ?
23
23
x
xx
a 1 x log a xloga xloga ................
2! 3!
? ? ? ? ?  
d. 
35
xx
sinx x   .................
3! 5!
? ? ?    
e. ??
24
xx
cosx 1 +  ......................
2! 4!
      
f. tan x = 
3
5
2
x
x + x + .........
3!
15
?    
g. log (1 + x) = 
23
xx
x +  + ............, x < 1
23
?   
 
  Important Limits  
 
a. 
lt
sinx
    1
x0 x
?
?
     
b.  
lt
tanx
     1
x0 x
?
?
       
c. ? ?
1
n
x
lt
  1 nx e
x0
??
?
 
d. 
lt
 cos x  1
x0
?
?
             
e. ? ?
1
x
lt
 1 x  e
x0
??
?
     
f. 
? ?
??
??
x
lt
1
 1 e
x
x
   
 
L – Hospitals Rule  
 If f (x) and g(x) are to function such that  
     ? ?
lt
 f x 0
xa
?
?
    and   ? ?
lt
 g x 0
xa
?
?
   
Page 2


 
 
 
 
 
CALCULUS 
 
  Important Series Expansion  
a. ? ?
n
n
n
r  0
r
r
1 x C x
?
??
?
    
b. ? ?
1
2
1 x 1 x x ............
?
? ? ? ? ?   
c. ? ? ? ?
23
23
x
xx
a 1 x log a xloga xloga ................
2! 3!
? ? ? ? ?  
d. 
35
xx
sinx x   .................
3! 5!
? ? ?    
e. ??
24
xx
cosx 1 +  ......................
2! 4!
      
f. tan x = 
3
5
2
x
x + x + .........
3!
15
?    
g. log (1 + x) = 
23
xx
x +  + ............, x < 1
23
?   
 
  Important Limits  
 
a. 
lt
sinx
    1
x0 x
?
?
     
b.  
lt
tanx
     1
x0 x
?
?
       
c. ? ?
1
n
x
lt
  1 nx e
x0
??
?
 
d. 
lt
 cos x  1
x0
?
?
             
e. ? ?
1
x
lt
 1 x  e
x0
??
?
     
f. 
? ?
??
??
x
lt
1
 1 e
x
x
   
 
L – Hospitals Rule  
 If f (x) and g(x) are to function such that  
     ? ?
lt
 f x 0
xa
?
?
    and   ? ?
lt
 g x 0
xa
?
?
   
 
 
 
 
    
 Then  
? ?
? ?
? ?
? ?
lt lt f x f' x
    
x a x a g x g' x
?
??
    
 If f’(x) and g’(x) are also zero asxa ? , then we can take successive derivatives till this  
 condition is violated.  
 
 For continuity, ? ? ? ?
lim
f x =f a
xa ?
   
 For differentiability,  
? ? ? ?
0 0
f x h f x lim
h0 h
??
??
??
?
??
??
  exists and is equal to ? ?
0
f' x  
If a function is differentiable at some point then it is continuous at that point but converse 
may not be true.  
 
Mean Value Theorems  
? Rolle’s Theorem  
If there is a function f(x) such that f(x) is continuous in closed interval a = x = b and f’(x) 
is existing at every point in open interval a < x < b and f(a) = f(b).  
     Then, there exists a point ‘c’ such that f’(c) = 0 and a < c < b. 
 
? Lagrange’s Mean value Theorem 
If there is a function f(x) such that, f(x) is continuous in closed interval  a = x = b; and f(x) is 
differentiable in open interval (a, b) i.e., a < x < b,  
Then there exists a point ‘c’,  such that 
     ? ?
? ? ? ?
? ?
f b f a
f' c
ba
?
?
?
   
  
Differentiation  
  
 Properties:  (f + g)’ = f’ + g’ ;  (f – g)’ = f’ – g’ ; (f g)’ = f’ g + f g’  
  
 Important derivatives  
a. 
n
x
 ? n 
n  1
x
?
        
b. 
1
nx
x
?
    
c. 
? ?
?
aa
1
log x  (log e)
x
   
d. 
xx
e  e ?
    
Page 3


 
 
 
 
 
CALCULUS 
 
  Important Series Expansion  
a. ? ?
n
n
n
r  0
r
r
1 x C x
?
??
?
    
b. ? ?
1
2
1 x 1 x x ............
?
? ? ? ? ?   
c. ? ? ? ?
23
23
x
xx
a 1 x log a xloga xloga ................
2! 3!
? ? ? ? ?  
d. 
35
xx
sinx x   .................
3! 5!
? ? ?    
e. ??
24
xx
cosx 1 +  ......................
2! 4!
      
f. tan x = 
3
5
2
x
x + x + .........
3!
15
?    
g. log (1 + x) = 
23
xx
x +  + ............, x < 1
23
?   
 
  Important Limits  
 
a. 
lt
sinx
    1
x0 x
?
?
     
b.  
lt
tanx
     1
x0 x
?
?
       
c. ? ?
1
n
x
lt
  1 nx e
x0
??
?
 
d. 
lt
 cos x  1
x0
?
?
             
e. ? ?
1
x
lt
 1 x  e
x0
??
?
     
f. 
? ?
??
??
x
lt
1
 1 e
x
x
   
 
L – Hospitals Rule  
 If f (x) and g(x) are to function such that  
     ? ?
lt
 f x 0
xa
?
?
    and   ? ?
lt
 g x 0
xa
?
?
   
 
 
 
 
    
 Then  
? ?
? ?
? ?
? ?
lt lt f x f' x
    
x a x a g x g' x
?
??
    
 If f’(x) and g’(x) are also zero asxa ? , then we can take successive derivatives till this  
 condition is violated.  
 
 For continuity, ? ? ? ?
lim
f x =f a
xa ?
   
 For differentiability,  
? ? ? ?
0 0
f x h f x lim
h0 h
??
??
??
?
??
??
  exists and is equal to ? ?
0
f' x  
If a function is differentiable at some point then it is continuous at that point but converse 
may not be true.  
 
Mean Value Theorems  
? Rolle’s Theorem  
If there is a function f(x) such that f(x) is continuous in closed interval a = x = b and f’(x) 
is existing at every point in open interval a < x < b and f(a) = f(b).  
     Then, there exists a point ‘c’ such that f’(c) = 0 and a < c < b. 
 
? Lagrange’s Mean value Theorem 
If there is a function f(x) such that, f(x) is continuous in closed interval  a = x = b; and f(x) is 
differentiable in open interval (a, b) i.e., a < x < b,  
Then there exists a point ‘c’,  such that 
     ? ?
? ? ? ?
? ?
f b f a
f' c
ba
?
?
?
   
  
Differentiation  
  
 Properties:  (f + g)’ = f’ + g’ ;  (f – g)’ = f’ – g’ ; (f g)’ = f’ g + f g’  
  
 Important derivatives  
a. 
n
x
 ? n 
n  1
x
?
        
b. 
1
nx
x
?
    
c. 
? ?
?
aa
1
log x  (log e)
x
   
d. 
xx
e  e ?
    
 
 
 
 
 
e. 
xx
e
a  a log a ?
  
f. sin x  ? cos x  
g. cos x ? -sin x  
h. tan x ?  
2
sec x
  
i. sec x ?   sec x tan x  
j. cosec x ?  - cosec x cot x  
k. cot x ?  - cosec
2
 x  
l. sin h x ? cos h x  
m. cos h x ?  sin h x  
n. 
?
?
1
2
1
sin x  
1 - x
 
o. 
2
1
-1
cos x  
1x
?
?
?
 
   
p. 
?
?
?
2
1
1
tan x  
1x
 
 
q. 
2
1
-1
cosec x  
x x 1
?
?
?
  
r. 
?
?
?
2
1
1
sec x  
x x 1
   
s. 
1
2
-1
cot x  
1x
?
?
?
      
 
 
Increasing & Decreasing Functions  
? ? ? f' x 0 V ? ? ? x a, b ? , then f is increasing in [a, b] 
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly increasing in [a, b] 
? ? ? f' x 0 V ? ? ? x a, b ? , then f is decreasing in [a, b] 
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly decreasing in [a, b] 
 
 
 
 
Page 4


 
 
 
 
 
CALCULUS 
 
  Important Series Expansion  
a. ? ?
n
n
n
r  0
r
r
1 x C x
?
??
?
    
b. ? ?
1
2
1 x 1 x x ............
?
? ? ? ? ?   
c. ? ? ? ?
23
23
x
xx
a 1 x log a xloga xloga ................
2! 3!
? ? ? ? ?  
d. 
35
xx
sinx x   .................
3! 5!
? ? ?    
e. ??
24
xx
cosx 1 +  ......................
2! 4!
      
f. tan x = 
3
5
2
x
x + x + .........
3!
15
?    
g. log (1 + x) = 
23
xx
x +  + ............, x < 1
23
?   
 
  Important Limits  
 
a. 
lt
sinx
    1
x0 x
?
?
     
b.  
lt
tanx
     1
x0 x
?
?
       
c. ? ?
1
n
x
lt
  1 nx e
x0
??
?
 
d. 
lt
 cos x  1
x0
?
?
             
e. ? ?
1
x
lt
 1 x  e
x0
??
?
     
f. 
? ?
??
??
x
lt
1
 1 e
x
x
   
 
L – Hospitals Rule  
 If f (x) and g(x) are to function such that  
     ? ?
lt
 f x 0
xa
?
?
    and   ? ?
lt
 g x 0
xa
?
?
   
 
 
 
 
    
 Then  
? ?
? ?
? ?
? ?
lt lt f x f' x
    
x a x a g x g' x
?
??
    
 If f’(x) and g’(x) are also zero asxa ? , then we can take successive derivatives till this  
 condition is violated.  
 
 For continuity, ? ? ? ?
lim
f x =f a
xa ?
   
 For differentiability,  
? ? ? ?
0 0
f x h f x lim
h0 h
??
??
??
?
??
??
  exists and is equal to ? ?
0
f' x  
If a function is differentiable at some point then it is continuous at that point but converse 
may not be true.  
 
Mean Value Theorems  
? Rolle’s Theorem  
If there is a function f(x) such that f(x) is continuous in closed interval a = x = b and f’(x) 
is existing at every point in open interval a < x < b and f(a) = f(b).  
     Then, there exists a point ‘c’ such that f’(c) = 0 and a < c < b. 
 
? Lagrange’s Mean value Theorem 
If there is a function f(x) such that, f(x) is continuous in closed interval  a = x = b; and f(x) is 
differentiable in open interval (a, b) i.e., a < x < b,  
Then there exists a point ‘c’,  such that 
     ? ?
? ? ? ?
? ?
f b f a
f' c
ba
?
?
?
   
  
Differentiation  
  
 Properties:  (f + g)’ = f’ + g’ ;  (f – g)’ = f’ – g’ ; (f g)’ = f’ g + f g’  
  
 Important derivatives  
a. 
n
x
 ? n 
n  1
x
?
        
b. 
1
nx
x
?
    
c. 
? ?
?
aa
1
log x  (log e)
x
   
d. 
xx
e  e ?
    
 
 
 
 
 
e. 
xx
e
a  a log a ?
  
f. sin x  ? cos x  
g. cos x ? -sin x  
h. tan x ?  
2
sec x
  
i. sec x ?   sec x tan x  
j. cosec x ?  - cosec x cot x  
k. cot x ?  - cosec
2
 x  
l. sin h x ? cos h x  
m. cos h x ?  sin h x  
n. 
?
?
1
2
1
sin x  
1 - x
 
o. 
2
1
-1
cos x  
1x
?
?
?
 
   
p. 
?
?
?
2
1
1
tan x  
1x
 
 
q. 
2
1
-1
cosec x  
x x 1
?
?
?
  
r. 
?
?
?
2
1
1
sec x  
x x 1
   
s. 
1
2
-1
cot x  
1x
?
?
?
      
 
 
Increasing & Decreasing Functions  
? ? ? f' x 0 V ? ? ? x a, b ? , then f is increasing in [a, b] 
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly increasing in [a, b] 
? ? ? f' x 0 V ? ? ? x a, b ? , then f is decreasing in [a, b] 
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly decreasing in [a, b] 
 
 
 
 
 
 
 
 
 
Maxima & Minima  
  Local maxima or minima    
 There is a maximum of f(x) at x = a if f’(a) = 0 and f”(a) is negative.  
 There is a minimum of f (x) at x = a, if f’(a) = 0 and f” (a) is positive.  
To calculate maximum or minima, we find the point ‘a’ such that f’(a) = 0 and then decide 
if it is maximum or minima by judging the sign of f”(a).  
 
  Global maxima & minima  
 We first find local maxima & minima & then calculate the value of ‘f’ at boundary points of 
 interval given eg. [a, b], we find f(a) & f(b) & compare it with the values of local maxima &  
 minima. The absolute maxima & minima can be decided then.  
 
Taylor & Maclaurin series  
? Taylor series  
f(a + h) = f(a) + h f’(a) + 
2
h
2
 f”(a) + ………………..   
? Maclaurin 
f(x) = f(0) + x f’(0) + 
2
x
2
 f“(0)+……………..   
 
Partial Derivative  
If a derivative of a function of several independent variables be found with respect to any 
one of them, keeping the others as constant, it is said to be a partial derivative.  
 
Homogenous Function  
 
 
n 2 2 n n  1 n
n
0 1 2
a x a x y a x y ............. a y
? ?
? ? ? ?  is a homogenous function  
  of x & y,  of degree ‘n’  
  = 
? ? ? ? ? ?
2n
n
0 1 2
n y y y
x a a a .................... a
x x x
??
? ? ? ?
??
??
   
 
Euler’s Theorem  
 If u is a homogenous function of x & y of degree n, then  
   
uu
x y nu
xy
?? ??
??
??
??
??
   
Page 5


 
 
 
 
 
CALCULUS 
 
  Important Series Expansion  
a. ? ?
n
n
n
r  0
r
r
1 x C x
?
??
?
    
b. ? ?
1
2
1 x 1 x x ............
?
? ? ? ? ?   
c. ? ? ? ?
23
23
x
xx
a 1 x log a xloga xloga ................
2! 3!
? ? ? ? ?  
d. 
35
xx
sinx x   .................
3! 5!
? ? ?    
e. ??
24
xx
cosx 1 +  ......................
2! 4!
      
f. tan x = 
3
5
2
x
x + x + .........
3!
15
?    
g. log (1 + x) = 
23
xx
x +  + ............, x < 1
23
?   
 
  Important Limits  
 
a. 
lt
sinx
    1
x0 x
?
?
     
b.  
lt
tanx
     1
x0 x
?
?
       
c. ? ?
1
n
x
lt
  1 nx e
x0
??
?
 
d. 
lt
 cos x  1
x0
?
?
             
e. ? ?
1
x
lt
 1 x  e
x0
??
?
     
f. 
? ?
??
??
x
lt
1
 1 e
x
x
   
 
L – Hospitals Rule  
 If f (x) and g(x) are to function such that  
     ? ?
lt
 f x 0
xa
?
?
    and   ? ?
lt
 g x 0
xa
?
?
   
 
 
 
 
    
 Then  
? ?
? ?
? ?
? ?
lt lt f x f' x
    
x a x a g x g' x
?
??
    
 If f’(x) and g’(x) are also zero asxa ? , then we can take successive derivatives till this  
 condition is violated.  
 
 For continuity, ? ? ? ?
lim
f x =f a
xa ?
   
 For differentiability,  
? ? ? ?
0 0
f x h f x lim
h0 h
??
??
??
?
??
??
  exists and is equal to ? ?
0
f' x  
If a function is differentiable at some point then it is continuous at that point but converse 
may not be true.  
 
Mean Value Theorems  
? Rolle’s Theorem  
If there is a function f(x) such that f(x) is continuous in closed interval a = x = b and f’(x) 
is existing at every point in open interval a < x < b and f(a) = f(b).  
     Then, there exists a point ‘c’ such that f’(c) = 0 and a < c < b. 
 
? Lagrange’s Mean value Theorem 
If there is a function f(x) such that, f(x) is continuous in closed interval  a = x = b; and f(x) is 
differentiable in open interval (a, b) i.e., a < x < b,  
Then there exists a point ‘c’,  such that 
     ? ?
? ? ? ?
? ?
f b f a
f' c
ba
?
?
?
   
  
Differentiation  
  
 Properties:  (f + g)’ = f’ + g’ ;  (f – g)’ = f’ – g’ ; (f g)’ = f’ g + f g’  
  
 Important derivatives  
a. 
n
x
 ? n 
n  1
x
?
        
b. 
1
nx
x
?
    
c. 
? ?
?
aa
1
log x  (log e)
x
   
d. 
xx
e  e ?
    
 
 
 
 
 
e. 
xx
e
a  a log a ?
  
f. sin x  ? cos x  
g. cos x ? -sin x  
h. tan x ?  
2
sec x
  
i. sec x ?   sec x tan x  
j. cosec x ?  - cosec x cot x  
k. cot x ?  - cosec
2
 x  
l. sin h x ? cos h x  
m. cos h x ?  sin h x  
n. 
?
?
1
2
1
sin x  
1 - x
 
o. 
2
1
-1
cos x  
1x
?
?
?
 
   
p. 
?
?
?
2
1
1
tan x  
1x
 
 
q. 
2
1
-1
cosec x  
x x 1
?
?
?
  
r. 
?
?
?
2
1
1
sec x  
x x 1
   
s. 
1
2
-1
cot x  
1x
?
?
?
      
 
 
Increasing & Decreasing Functions  
? ? ? f' x 0 V ? ? ? x a, b ? , then f is increasing in [a, b] 
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly increasing in [a, b] 
? ? ? f' x 0 V ? ? ? x a, b ? , then f is decreasing in [a, b] 
? ? ? f' x 0 V ? ? ? x a, b ? , then f is strictly decreasing in [a, b] 
 
 
 
 
 
 
 
 
 
Maxima & Minima  
  Local maxima or minima    
 There is a maximum of f(x) at x = a if f’(a) = 0 and f”(a) is negative.  
 There is a minimum of f (x) at x = a, if f’(a) = 0 and f” (a) is positive.  
To calculate maximum or minima, we find the point ‘a’ such that f’(a) = 0 and then decide 
if it is maximum or minima by judging the sign of f”(a).  
 
  Global maxima & minima  
 We first find local maxima & minima & then calculate the value of ‘f’ at boundary points of 
 interval given eg. [a, b], we find f(a) & f(b) & compare it with the values of local maxima &  
 minima. The absolute maxima & minima can be decided then.  
 
Taylor & Maclaurin series  
? Taylor series  
f(a + h) = f(a) + h f’(a) + 
2
h
2
 f”(a) + ………………..   
? Maclaurin 
f(x) = f(0) + x f’(0) + 
2
x
2
 f“(0)+……………..   
 
Partial Derivative  
If a derivative of a function of several independent variables be found with respect to any 
one of them, keeping the others as constant, it is said to be a partial derivative.  
 
Homogenous Function  
 
 
n 2 2 n n  1 n
n
0 1 2
a x a x y a x y ............. a y
? ?
? ? ? ?  is a homogenous function  
  of x & y,  of degree ‘n’  
  = 
? ? ? ? ? ?
2n
n
0 1 2
n y y y
x a a a .................... a
x x x
??
? ? ? ?
??
??
   
 
Euler’s Theorem  
 If u is a homogenous function of x & y of degree n, then  
   
uu
x y nu
xy
?? ??
??
??
??
??
   
 
 
 
  
 
Maxima & minima of multi-variable function  
  
2
2
xa
yb
f
let r
x
?
?
??
?
?
??
??
?
??
   ;       
2
xa
yb
f
s  
xy ?
?
??
?
?
??
??
??
??
       ;   
2
2
xa
yb
f
t  
y
?
?
??
?
?
??
??
?
??
      
? Maxima  
rt > 
2
s  ;    r < 0  
? Minima 
rt > 
2
s ;     r > 0  
? Saddle point  
rt < 
2
s   
 
Integration  
Indefinite integrals are just opposite of derivatives and hence important derivatives must 
always be remembered. 
 
 Properties of definite integral  
a. 
? ? ? ?
?
??
bb
aa
f x dx f t dt   
b. ? ? ? ?
ba
a b
f x dx f x dx ??
??
   
c. 
? ? ? ? ? ?
??
? ? ?
b cb
a a c
f x dx f x dx f x dx   
d. ? ? ? ?
b b
aa
f x dx f a b x dx ? ? ?
??
  
e.  ? ?
? ?
? ?
? ? ? ? ? ? ? ? ? ? ? ?
t
t
d
f x dx f t ' t f t ' t
dt
?
?
? ? ? ? ? ?
?
    
  
Vectors  
? Addition of vector  
ab ?  of two vector a = 
1 2 3
a ,a ,a ??
??
 and  b = 
1 2 3
b ,b ,b ??
??
  
   ?? ? ? ? ?
?? 1 1 2 2 3 3
 a + b = a b ,a b ,a b   
? Scalar Multiplication  
  ??
?? 1 2 3
ca = ca , ca , ca