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
Read More