EM 2 Group - 4 engineering mathematics pptx

1. 1

2. 2
Que.1) Using Taylor’s series find expression for sin x in
powers of x.
Consider the derivative of f (x) = sin x.
f’(x) = cos x , f’’(x) = - sin x , f’’’(x) = - cos x , f ’’’’(x) = sin x , …….
From the Taylor series expansion we have :
f (x) = f (0) + f ′(0)
𝑥
1!
+ f ′′(0)
𝑥2
2!
+ f ′′′(0)
𝑥3
3!
+ ⋯
In this example we have f (x) = sin x.
Since ,sin 0 = 0 all even powers of x will equal to 0 in this series expansion.
Thus,
f (x) = sin x = cos 0
𝑥
1!
− cos 0
𝑥2
3!
+ cos 0
𝑥3
5!
− ⋯
Ans:-

3. 3
Now , since cos 0 = 1 the series reduces to:
Sin x =
𝑥
1!
−
𝑥3
3!
+
𝑥5
5!
− ⋯
The series is infinite in odd powers of x with alternating sign and thus can be
written as :
Sin x = 𝒏=𝟎
∞
(−𝟏)𝒏 𝒙𝟐𝒏+𝟏
(𝟐𝒏+𝟏)!

4. 4
Que.2) Visit any Dome shape place measure radius take
picture of that shape. Find volume of that Dome shape
using Triple Integration.
Ans:-
V= 0
𝜋/2
0
2𝜋
0
𝑟
𝑟2. 𝑠𝑖𝑛∅. 𝑑𝑟. 𝑑𝜃. 𝑑∅
V= 0
𝜋/2
0
2𝜋
(
𝑟3
3
)0
𝑟. 𝑠𝑖𝑛∅. 𝑑𝜃. 𝑑∅
Taking r=12m
V= 0
𝜋/2
0
2𝜋
(
123
3
)0
𝑟. 𝑠𝑖𝑛∅. 𝑑𝜃. 𝑑∅
V=
123
3 0
𝜋/2
0
2𝜋
( 𝑠𝑖𝑛∅. 𝑑𝜃. 𝑑∅

5. 5
V=
123
3 0
𝜋/2
(𝜃)0
2𝜋
𝑠𝑖𝑛∅. 𝑑∅
V=
123
3
∗ 2𝜋 0
𝜋/2
𝑠𝑖𝑛∅. 𝑑∅
V=
123
3
∗ 2𝜋 ∗ (−𝑐𝑜𝑠∅)0
𝜋/2
V=
123
3
∗ 2𝜋 ∗ (−
cosπ
2
+ cos 0)
V=2 *3.14* 123
3
V=3617.28

6. 6
Que :-3) Solve.
Find y , z at x=1.4
Ans:-
By R K Method :-
o
𝑑𝑦
𝑑𝑥
= 𝑥 + 𝑦 𝑦 1 = 1 𝑥0 = 1 , 𝑦0 = 1
h = 0.4 , x = 1.4
By R K Method ,
K =
1
6
[𝐾1 + 2𝐾2 + 2𝐾3 + 𝐾4 ]
K1 = h f (x0 , y0 )
=0.4 (2)

7. 7
K1 = 0.8
K2 = h f (x0 + h/2, y0 + K1/2 )
=0.4 [2.6]
=1.04
K3 = h f (x0 + h/2 , y0 + K2/2 )
=0.4 [2.72]
=1.088
K4 = h f (x0 + h , y0 + K3 )
=0.4 [3.488]
=1.3952
K =
1
6
[0.8 + 2 ∗ 1.04 + 2 ∗ 1.088 + 1.3952 ]
K = 1.0752
y1 = y0 + k
=1 + 1.0752
y1 = 2.0752
o
𝑑𝑦
𝑑𝑥
=
𝑥
𝑧
𝑧 1 = 2 𝑧0 = 2 , 𝑥0 = 1
h = 0.4
By R K Method ,
z1 = z0 + k
K =
1
6
[𝐾1 + 2𝐾2 + 2𝐾3 + 𝐾4 ]
K1 = h f (x0 , z0 )
=0.4 (0.5)
= 0.2
K2 = h f (x0 + h/2, z0 + K1/2 )
=0.4 [0.5714]
=0.2286
K3 = h f (x0 + h/2 , z0 + K2/2 )
=0.4 [0.5676]
=0.2270
K4 = h f (x0 + h , z0 + K3 )
=0.4 [0.6286]
=0.2515

8. 8
K =
1
6
[0.2 + 2 ∗ 0.2286 + 2 ∗ 0.2270 + 0.2515]
K =
1
6
[1.3627]
K = 0.2271
z1 = z0 + k
=2 + 0.2271
z1 = 2.2271

9. 9
Que.4) Take picture of D shape curve you observed in
Sanjivani college, take approximate values find double
integral on that shape if f(x,y)=x + y2
2

10. 10
∫ .∫ (x²+y²)dxdy
Area = ∫ [x²y+ y3/3]dx limits 0 to x²
= ∫ [x4+x6/3]dx
= 0.2 + 0.0476
=0.2476 * 2
Area =0.4952
X=1
X=0
y=x²
y=0
X=0
X=1
X=0
X=1
=
15
5
+
17
21
=
𝑥5
5
+
𝑥7
21]
[ Limits from 0 to 1

11. 11
References:-
1) Higher Engineering Mathematics by Dr. B. S. Grewal(42nd Edition).
2) Numericals Methods by M. K. Jain , S. R. K. Iyengar , R K Jain .
3) Higher Engineering Mathematics by B. V. Ramana .
 All Books In E-Book Available on:
http://tutorial.math.lamar.edu.

12. 12

13. 13