b g EXPANSIONS OF FUNCTIONS

FE / GQ/ MATHS I / EXPANSION & SD
EXPANSIONS OF FUNCTIONS
Maclaurian’s Theorem:
Let f (x) be a function of x then
bg
b g
f x
= f 0
x2
+
f 
2!
b g
+ x f 0
b0g
x3
+
f '
3!
b0g +
....
Taylor’s Theorem:
b
i. f x + h
g
b g
ii. f x
b
iii. f x + h
g
b g + x f' b h g +
= f h
x2
f''
2!
3
b h g + x3! f' '' b h g + ......
b x - a g f' b a g + b x - a g f' ' b a g + b x - a g
2
b g
= f a
+
2!
2
b g + h f' b x g
h
+
f''
2!
= f x
bxg
3!
3
h
+
f' ' '
3!
3
f'' '
b a g....
b x g + .......
Standard Results:
2
3
4
5
x
x
x
x
+
+
+
+ ..........
2!
3!
4!
5!
1.
ex = 1 + x +
2.
sin x = x -
3.
cos x = 1 -
4.
x
x
sinh x = x +
+
+ ...........
3!
5!
5.
x
x
x
cosh x = 1 +
+
+
+ ........
2!
4!
6!
6.
log 1 + x
7.
log
8.
tanh-1 x = x +
9.
x
x
x
tan x = x +
+ ........
3
5
7
10.
x
2x
tan x = x +
+
3
15
11.
x
2x
tanh x = x +
3
15
3
5
2
b
g
b1 - xg
-1
7
x
x
x
+
+ ............
3!
5!
7!
4
6
x
x
x
+
+ ..........
2!
4!
6!
3
5
2
4
6
2
= x -
3
2
= - x 3
3
3
…..Progress through quality education
5
3
4
5
x
x
x
x
........
2
3
4
5
5
x
x
+
3
5
3
4
x
x
x
x
+
+
 .........
2
3
4
5
5
+ .........
7
5
+ ..........
5
 .......
1
FE / GQ/ MATHS I / EXPANSION & SD
12.
13.
14.
b1 + xg
b1 + xg
b1 - xg
b
m m -1
2!
gx
b
= 1 + mx +
1
= 1 - x + x2 - x3 + x4 .........
1
2
gb
m m -1
m-2
3!
m
+
g
x3 + ....
= 1 + x + x2 + x3 + x4 + ......
TYPE -I : Method of using Maclaurin’s Series
e
x
j
Pr ove that log 1 + e
2.
Find the Maclaurian's series expansion of
3.
= log 2 +
1
1 2
1 4
x +
x x + .....
2
8
192
1.
b
LM FG
N H
log tan
IJ OP upto x
KQ
F 1 + x IJ .
log G
H 1-x K

+ x
4
g
Obtain the series of log 1 + x . Hence find the series of
Use this expansion to find log
FG 11 IJ
H9K
4.
Expand log (sec x + tan x) as a series in ascending powers of x
5.
ex
Find the series expansion of
1  ex
6.
Proe that log log 1 + x
7.
Pr ove that
8.
Prove that e
9.
LM b
OP =
g
N
Q
b1 + xg = 1 + x
1
x
x
Prove that e
x sin x
ax
-1
5 2 1 3
251 4
x +
x x +
x + .....
2
24
8
2880
2
-
1 3
5 4
x +
x .......
2
6
1 4
1
6
x +
x + .......
3
120
2
= 1 + x +
2
a - b
cos bx = 1 + a x +
2!
e
x
sin e - 1
j
10.
Prove that
11.
Prove that
12.
Expand log 1 + x + x + x
= x +
13.
Prove that
2
LM
N
b1 + xg
1x
x
3
Expand log 1- x + x - x
…..Progress through quality education
j
3!
jx
3
+ ......
OP
Q
x
11 2
+
x ......
2
24
15.
3
+
2
8
14.
2
2
1 2
5 4
x x + .......
2
24
x
Expand x
upto x . Hence show that
2
e 1
e
x
e
a a - 3b
2
j upto a term in x
= e 1 4
2
1
1 2
1 3
1
4
x x x +
x + .......
2
8
48
384
1 + sin x = 1 +
e
5
F
GH
x
I
J
- 1 K
e + 1
e
x
2
4
x
x
= 1 +
+ .........
2
720
8
upto a term in x
2
FE / GQ/ MATHS I / EXPANSION & SD
16.
17.
18.
b
Prove that log 1 + sin x
g
= x -
1 2
1 3
x +
x + .......
2
6
FG 1 + x + x + 5 x + 5 x + ........IJ
H
K
6
8
b - 1g 2 x
Prove that cos x = 1 + 
b 2n g !
1
2
Prove that log b 1 + tan x g = x - x +
x +.........
2
3
Prove that e
e
x
2
= e

2
3
n
4
2n-1
2n
n=1
19.
2
3
1 2
7
4
x +
x + ......
6
360
20.
Prove that x cosec x = 1 +
21.
Prove that log
23.
Expand tan-1 x in ascending powers of x
24.
Expand sin-1 x in ascending powers of x
25.
Expand cos-1 x in ascending powers of x
26.
Prove that log sec x =
27.
Prove that sinh x = x -
F1+e I
GH e JK
2x
= log2 +
x
1 2
1 4
1 6
x x +
x + ......
2
12
45
1 2
1 4
1 6
x +
x +
x + .........
2
12
45
1 3
3 5
x +
x + .......
6
40
-1
Type-II : Expansion of function using Substitution
28.
29.
30.
31.
32.
33.
34.
FG 2 x IJ = 2 L x - 1 x + 1 x .......O
MN 3
PQ
H 1-x K
5
F 2 x IJ in ascending powers of x
Expand sin G
H1+x K
L 1 x + 1 x ......OP
Prove that cos tanh b log x g =  - 2 M x 5
N 3
Q
F 1 IJ = 2 L x + 1 x + 3 x + ....... O
Prove that sec G
PQ
H 1 - 2 x K MN 6
40
L 1 x + 3 x .......OP
Show that sinh e 3 x + 4 x j = 3 M x 40
N 6
Q
LM 1 + x  1 OP
Expand tan
x
MN
PQ in ascending powers of x
Pr ove that tan
-1
3
5
2
-1
2
-1
3
-1
3
5
5
2
-1
3
-1
5
2
-1
Expand tan
3
1 - x
in ascending powers of x
1 + x
…..Progress through quality education
3
FE / GQ/ MATHS I / EXPANSION & SD
FG
H
35.
Expand tan-1
36.
Prove that sin
37.
Prove that tan
IJ in ascending powers of x
K
e 3 x + 4 x j = 3 LMN x + 16 x + 403 x ....... OPQ
F 3 x - x I = 3 L x - 1 x + 1 x .......... O
GH 1  3 x JK
MN 3
PQ
5
p - qx
q + px
-1
-1
3
3
3
5
3
5
2
EXPS - 3
2
3
38.
1 2
1 3 1 4
x
x
If x = y y +
y y ......., then prove that y = x +
+
+ ......
2
3
4
2!
3!
39.
x
x
x
y
y
y
If y = x +
+
+
+ ....... then prove that x = y +
+ .....
2!
3!
4!
2
3
4
40.
If x = y -
41.
y
y
y
If x = 1 +
...... find y in a series of x
2!
4!
6!
42.
y = tan -1 x
2
3
2
4
x cos 
b
sin x sin 
y3
2y5
+
+ .......
3
15

g
=
xn
Sin n 
n!

n=1
x
44.
45.
46.
47.
Expand
Expand
Expand
Expand
48.
If x + y + xy - 1 = 0
e cosx in ascending powers of x
sin x sinh x in ascending power of x
ex sin x in ascending powers of x
eax sin bx in ascending powers of x
3
3
prove that
y = 1 -
b g b g
b 1 - y g b 1 - 2y g prove that
y e 1 + y j, prove that y =
2
50.
If x =
51.
If x
52.
If y3 + y - x = 0, prove that y = x - x3 + 3 x5 .......
54.
2
y = 1 + x - 2 x .......
3
5
x - x + 3 x ......
3
+ 2xy - y + x = 1 prove that y = -1 + x -
e sin x j
Prove that log F x +
H
Prove that
-1
…..Progress through quality education
2
b g
1
26 3
x x ........
3
81
If x =
2
b g
then by obtaining y 0 , y1 0 , y2 0 , and y3 0
49.
53.
4
6
prove that x = y +
Pr ove that e
3
3
y3
y5
y7
x3
3x5
+
+ ....... then prove that y = x +
+
+ ......
3!
5!
7!
6
40
2
43.
4
2
4
1 2
x .....
3
6
x
x
x
=
2 +
2.22 +
2.22  42 + .....
2!
4!
6!
1 + x2
IK
= x -
x3 2
x5 2 2
x7 2 2 2
1 +
3 1 5 .3 .1 + ......
3!
5!
7!
4
FE / GQ/ MATHS I / EXPANSION & SD
55.
sin -1x
If y =
1 x
57.
,
show that
2 3
2.4 5
246 7
x +
x +
x + ......
3
3.5
3 5 7
y = x +
56.
2
Sinh-1x
Show that
= x -
1 + x2
Prove that e
cos-1 x

e2
=
2 3
8 5 16 7
x +
x x + .......
3
15
35
LM 1 - x +
N
OP
Q
x2
x3
+ ........
2
3
EXPS-4: Taylor’s Theorem
58.
59.
60.
61.
Expand e
b
x
in powers of x - 1
g
bx - 1g
F  IJ
Expand tan x in powers of G x H 4K
x - 2
bx - 2g
Prove that log x = log2 +
-1
Expand tan x in powers of
-1
2
2
+
8
FG x -  IJ
H 2K
bx - 2g
21
62.
Expand cos x in powers of
63.
Expresses x - 5 x + 6 x - 7 x + 8 x - 9 in powers of
64.
65.
66.
67.
68.
69.
5
4
3
3
2
.......
bx - 1g
bx + 2g
Expand x - 3 x + 2 x - x + 1 in powers of b x - 3 g
Express f bxg = 2 x + 3 x - 8 x + 7 in terms of b x - 2 g
Expand 7 x - 3 x + x + 2 in powers of b x - 1 g
Expand x - 3 x + 4 x + 3 in powers of b x - 2 g
Expand 2 x + 7 x + x - 6 in powers of b x - 2 g
3
2
Expand 3 x - 2 x + x - 4 in powers of
4
3
2
3
6
5
3
2
2
2
3
2
70. By using Taylor’s Theorem arrange in powers of x
7 +
bx + 2g + 3bx + 2g + bx + 2g - bx + 2g
3
4
5
71. By using Taylor’s theorem arrange in power of x
b
17 + 6 x + 2
g + 3 b x + 2 g + b x + 2g - b x + 2 g
3
4
5
72. Using Taylor’s theorem express
bx - 2g
4
73.
b
- 3 x - 2
g
3
Arrange in powers of x:
…..Progress through quality education
b
+ 4 x - 2
bx - 3g
4
g
2
+ 5
b
in powers of x
- 3 x - 3
g + bx - 3g
3
2
+ 5
5
FE / GQ/ MATHS I / EXPANSION & SD
LM FG
N H
74.
Arrange log tan
75.
Arrange sin
FG
H
IJ OP
KQ

+ x
4
IJ
K

+ x
6
in power of x by Taylor's theorem
in powers of x
76. Using Taylor’s theorem show that
1 + x + 2x
77.
Expand
2
1
7 2
7 3
x +
x x .....
2
8
16
= 1 +
e1 + x + 2x j
2
1
2
b
in powers of x - 1
g
78. If sinh 1.5 = 2.1293 & cosh 1.5 = 2.3524, Using Taylor’s series calculate sinh 1.505
79.
Using Taylor's series find
9.12 correct to five decimal places
80. Using Taylor’s theorem obtain tan-1 ( 1. 003 ) to four decimal places where  = 3.1416
FG 11 IJ where f b x g
H 10 K
= x3 + 3x2 + 15x - 10 using Taylor's series
81.
Find the value of f
82.
Using Taylors theorem obtain the expansion of
tan
FG x +  IJ
H 4K
in ascending
powers of x upto a term in x4 and hence find approximately
The values of tan (43) and tan (46 30’)
83. Using Taylor’s theorem find sin (30, 30)
84. Using Taylor’s theorem find cos (64)
85.
Calculate the value of
10 correct to four decimal places by using Taylors theorem
ANSWERS
FG 1 + x IJ
H 1-x K
3.
log
4.
x +
6.
1
4
12.
x +
LM 4 - 2
MN e
5
2
+ 4
2
2
+ 6
3
j
I
JK
7
x
x
x
+
+
+ ...... ;
3
5
7
FG 11 IJ
H 10 K
= 0.20067
1
1
1 3
+ x x + ......
2
4
48
5.
2
log
e
j
4
e
j
6
OP
PQ
x
4
4
4 x
6
6
6 x
+ 2 + 4 + 6
- 2 + 4 + 6
....
2!
4!
6!
5
6
7
x
x
3 4
x
x
x
3 8
+
x +
+
+
x + ......
2
3
4
5
6
7
8
x
x
Fx+
GH
1 3
1 5
x +
x + ........
6
24
2
14.
= 2
e - 1
= 1 -
x
x2
x4
+
+ .....
2
12
720
…..Progress through quality education
6
FE / GQ/ MATHS I / EXPANSION & SD
2
3
5
-x +
23.
x
x
x
x +
+ ......
3
5
7
25.

2
3
33.
35.
6
7
x
x
3 4
x
x
x
3 8
x 
x ......
2
3
4
5
6
7
8
15.
5
LM x +
MN
7
OP
PQ
3
x
3 5
+
x + .....
6
40
LM x - x + x ......OP
MN 3 5 PQ
F p IJ - FG x - x +
tan G
HqK H 3
3
1
2
3
x
3 5
x +
+
x + ......
6
40
24.
5
3
-1
58.
59.
x
45.
x
x
x - 8
+ 32
.......
6!
10 !
1 3
3 5
x +
x .....
6
40
I
JK
5
6
2
10
b g
3
x
x
+
.........
2
3
46. 1 +
e
OP
Q

1
4
2
x
..........
5
1 3 1 4
x x ........
44. 1 + x 3
6
2
LM x +
N
34.
LM
MN
= e 1 +
b
x - 1
g b x -2!1 g
2
+

1
+
4
2
+
bx - 1g
OP
PQ
3
3!
......
b x - 1 g - 41 b x - 1 g + 121 b x - 1 g .......
F x -  I FG x -  IJ
F  IJ + FG x -  IJ GG 4 JJ   H 4 K + .......
tan G
H 4 K H 4 K GG 1 +  JJ 4 F  I
H 16 K GH1 + 16 JK
F  IJ + 1 FG x -  IJ - 1 FG x -  IJ .......
G x H 2 K 3! H 2 K 5! H 2 K
6 - 3 b x - 1 g - 9 b x - 1 g - 4 b x - 1 g + b x - 1 g
38 + 45 b x + 2 g - 20 b x + 2 g + 3 b x + 2 g
16 + 38 b x - 3 g + 29 b x - 3 g + 9 b x - 3 g + b x - 3 g
19 + 28 b x - 2 g + 15 b x - 2 g + 2 b x - 2 g
7 + 29 b x -1 g + 76 b x -1 g + 110 b x -1 g + 90 b x - 1 g + 39 b x -1 g
-1
tan x =
2
3
2
-1
60.
2
2
3
62
63.
64.
65.
66.
67.
…..Progress through quality education
5
2
3
5
2
2
2
3
2
3
2
3
3
4
4
5
b
+ 7 x -1
g
6
7
FE / GQ/ MATHS I / EXPANSION & SD
68.
69.
b
g + 3bx - 2g + bx - 2g
40 + 53 b x - 2 g + 19 b x - 2 g + 2 b x - 2 g
2
7 + 4 x - 2
3
2
3
70. 17 - 11 x - 38 x2 - 29 x3 - 9 x4 - x5
71. 37 - 6 x - 38 x2 - 29 x3 - 9 x4 - x5
72. 61 - 84 x + 46 x2 - 11 x3 + x4
73. 185 - 51 x + 85 x2 - 15 x3 + x4
4 3
4 5
x +
x
74. 2 x +
3
3
75.
1
+
2
77.
2 +
3
1 x2
x 2
2 2!
5
4
bx - 1g +
78. 2.1411
80. 0.78690
82.
b
tan 43
g
7
64
3 x3
1 x4
+
+
2 3!
2 4!
bx - 1g
79.
81.
= 0.9326 ;
83. 0.50735
85. 3.1623
…..Progress through quality education
b
2
-
505
3072
3 x5
........
2 5!
b x - 1 g .....
3
3.0146
3.511
tan 46 30'
g
= 1.05378
84. 0.4384
8
FE / GQ/ MATHS I / EXPANSION & SD
Successive Differentiation
STANDRED RESULTS
y n  a n e ax
1. y = e ax
a
f
a
f
2. y = sin ax + b
3. y = cos ax + b
a
f
a
f
5. y = e ax cos bx  c
a
fm
7. y =
a
1
m
ax + b
a
a
f
n/2
e
j
e
n/2
y n  a 2  b2
y n  a 2  b2
j
e ax sin
ebx  c  n tan-1ab / afj
e ax cos
ebx  c  n tan-1ab / afj
f a f
n
1f a m + n -1f! a n
a
yn 
am  1f! aax  bfmn
a1fn1a n ann  1f!
yn 
aax  bf
yn 
f
8. y = log ax + b
f
y n  a n cos ax  b  n / 2
4. y = e ax sin bx  c
6. y = ax + b
a
y n  a n sin ax  b  n / 2
f
a
(N - 95)
m!
m n
a n ax  b
if m > n
m n !
(M - 98)
TYPE-SD – I
Find the n th derivative of the following functions
1. sin2x cos3x
2. cos3x sin2 x (June - 04)
3. sin5x cos3 x
4. cosx cos2x cos3x
5. e 3x sin 4 x
6. e 2x sin 2 x cosx
7. e ax cos2 x sin x (N - 94)
8. e 3x sin x cosx (N - 95)
a
9. 2 x sin 2 x cos3 x M - 97
f
11. sin 2 x sin3x cos4x (D - 04)
a
f
10. e xcos cos x sin (N - 97)
3
12. sin 3x ( Jan  03)
TYPE-SD – II
1.
3.
x
ax -1f ax - 2f ax - 3f
1
a3x - 2f ax - 3f3
(M - 96)
(M - 95)
…..Progress through quality education
2.
4.
x2  x  1
x 3  6x 2  11x  6
x2
ax  2f a2x + 3f aN - 95f
9
FE / GQ/ MATHS I / EXPANSION & SD
5.
x
6.
x2  a 2
4x
ax  1f ax + 1f
7.
2
c2 x
x2
2
h
(m - 06)
 7x  6
TYPE-SD – III
Find the n th derivative of the following functions
3. tan -1
F 1  x2  1I
GGH x JJK
F 2x IJ (D - 05)
4. sin -1G
H 1  x2 K
aM - 99f
1. tan -1x
FG 1  x IJ aN - 95, Jan - 02, M - 05f
H 1 xK
F x  x1 I (M  04)
5. cos-1G
H x  x1 JK
2. tan -1
FG 2x IJ
H 1  x2 K
3. y = tan -1
Examples based on Leibniz’s Theorem
1. If y  x 4 cos 3x , find y n
e
2. If y = sin -1x
sin -1x
3 If y =
1 x
2
j
2
then prove that
then prove that
e1- x2 j y n2  b2n  1 g xy n1  n2 y n  0 aM - 97f
e1- x2 j y n1  b2n  1 g xy n  n2 y n1 = 0 aN - 95f
a f
a f
b g
e j
-1
5. If y = e tan x then prove that e x 2  1jy n2  b2( n  1) x  1 gy n  n b n  1 gy n  0
6. If x = sin , y = sin2 then prove that e1- x 2 jy n2  b2 n  1 gxy n1  e n 2  4jy n  0
7. If y = sin e m sin -1xj then prove that e1- x 2 j y n2  b2 n  1 g xy n1  e n 2  m2 j y n  0
y
F xI n
8. If cos-1
 log G J then prove that x 2 y n2  b2 n  1 g xy n1  2 n 2 y n  0
H nK
b
4. If y = a cos logx  b sin logx then prove that x 2 y n2  2 n  1 xy n1  n 2  1 y n  0
9. If
FG
H
y
1
m
IJ
K
+ y
FG
H
1
m
IJ
K
= 2x,
aM - 96f
prove that
a 2n + 1 f xy n + 1 + e n2 - m2 j y n = 0 and hence find y n a 0 f
2
10. If Y = log e x +
x + 1j
prove that
Y a 0 f = 0 and Y
a 0 f = a - 1 fn 12 . 32 . 52 ..... a 2 n - 1 f2
ex
2
j
- 1 yn + 2 +
2n
11. If
2n+1
f ( x ) = cot x, show that
n
n-1
n
n-3
C1 f
0 - C3 f
a f
a f = tan x ,
12. If f x
P. T. f
…..Progress through quality education
n
a0f
+
n
C5 f
n-5
a0f
- ..... = cos
n
2
a0f - nC2 f n - 2 a0f + nC4 f n - 4 a0f - ........ = sin n2
10
`