AMITY UNIVERSITY UTTAR PRADESH
B.Tech (Ist Semester)
APPLIED MATHEMATICS - I
1.
a)
b)
c)
d)
e)
f)
g)
h)
2.
a)
b)
c)
d)
TUTORIAL SHEET-3
Find the fifth order derivative of the functions given below by using Leibnitz rule:
𝑥 3 𝑒 𝑎𝑥
𝑥 3 sin 𝑥
𝑥 3 log 𝑥
𝑒 𝑥 log 𝑥
𝑥 2 (2𝑥 + 3)10
𝑒 4𝑥 (3𝑥 + 5)2
𝑒 𝑥 cos 2𝑥
sin 𝑥 sin 2𝑥
Find the 𝑛𝑡ℎ order derivative of the following functions by using Leibnitz rule:
𝑙𝑜𝑔𝑥
𝑥
𝑥 𝑛−1 𝑙𝑜𝑔𝑥
𝑒 𝑥 𝑙𝑜𝑔𝑥
𝑥 4 sin(2𝑥 + 3)
3. If 𝑦 = a cos (log 𝑥) + 𝑏 sin(log 𝑥) , then prove that
𝑥 2 𝑦 ′′ + 𝑥𝑦 ′ + 𝑦 = 0.
Hence show that
𝑥 2 𝑦 (𝑛+2) + (2𝑛 + 1)𝑥𝑦 (𝑛+1) + (𝑛2 + 1)𝑦 (𝑛) = 0
4. If 𝑦 = log[𝑥 + √(𝑎2 + 𝑥 2 )], then show that
(𝑎2 + 𝑥 2 )𝑦 ′′ + 𝑥𝑦 ′ = 0
Hence, deduce that
(𝑎2 + 𝑥 2 )𝑦 (𝑛+2) + 𝑥(2𝑛 + 1)𝑦 (𝑛+1) + 𝑛2 𝑦 (𝑛) = 0.
5. If 𝑦 = tan−1 𝑥, then prove that
(1 + 𝑥 2 )𝑦 ′′ + 2𝑥𝑦 ′ = 0.
Use Leibnitz rule to deduce that
(1 + 𝑥 2 )𝑦 (𝑛+2) + 2𝑥(𝑛 + 1)𝑦 (𝑛+1) + 𝑛(𝑛 + 1)𝑦 (𝑛) = 0.
6. If 𝑦 = sin−1 𝑥, then prove that
(1 − 𝑥 2 )𝑦 ′′ − 𝑥𝑦 ′ = 0.
Use Leibnitz rule to deduce that
(1 − 𝑥 2 )𝑦 (𝑛+2) − (2𝑛 + 1)𝑥𝑦 (𝑛+1) − 𝑛2 𝑦 (𝑛) = 0.
7. If = cos−1 𝑥 , then prove that
(1 − 𝑥 2 )𝑦 ′′ − 𝑥𝑦 ′ − 𝛼 2 𝑦 = 0
Use Leibnitz rule to deduce that
(1 − 𝑥 2 )𝑦 (𝑛+2) − (2𝑛 + 1)𝑥𝑦 (𝑛+1) − (𝑛2 + 𝛼 2 ) 𝑦 (𝑛) = 0
−1
8. If 𝑦 = 𝑒 ∝ cos 𝑥 ,then prove that
(1 − 𝑥 2 )𝑦 ′′ − 𝑥𝑦 ′ − 𝛼 2 𝑦 = 0
Use Leibnitz rule to deduce that
(1 − 𝑥 2 )𝑦 (𝑛+2) − (2𝑛 + 1)𝑥𝑦 (𝑛+1) − (𝑛2 + 𝛼 2 ) 𝑦 (𝑛) = 0
−1
9. If 𝑦 = 𝑒 ∝ tan 𝑥 ,then show that
(1 + 𝑥 2 )𝑦 ′′ + (2𝑥 − 𝑎)𝑦 ′ = 0.
Hence deduce that
(1 + 𝑥 2 )𝑦 (𝑛+2) + (2𝑛𝑥 + 2𝑥 − 𝑎)𝑦 (𝑛+1) + 𝑛(𝑛 + 1)𝑦 (𝑛) = 0.
10. If 𝑦 = sin(𝑎 sin−1 𝑥),then prove that
(1 − 𝑥 2 )𝑦 ′′ − 𝑥𝑦 ′ + 𝑎2 𝑦 = 0.
Deduce that
(1 − 𝑥 2 )𝑦 (𝑛+2) − (2𝑛 + 1)𝑥𝑦 (𝑛+1) − (𝑛2 − 𝑎2 )𝑦 (𝑛) = 0.
Hence or otherwise expand sin 𝑚𝜃 in the powers of sin 𝜃. Find (𝑦𝑛 )0 .
11. If 𝑦 = (𝑥 + √(1 + 𝑥 2 ))𝑚 , then prove that
(1 + 𝑥 2 )𝑦 ′′ + 𝑥𝑦 ′ − 𝑚2 𝑦 = 0.
Hence deduce that
(1 + 𝑥 2 )𝑦 (𝑛+2) + (2𝑛 + 1)𝑥𝑦 (𝑛+1) + (𝑛2 − 𝑚2 )𝑦 = 0.
𝑥 𝑚
𝑦
12. If cos−1 (𝑏 ) = log (𝑚) , prove that
𝑥 2 𝑦 (𝑛+2) + (2𝑛 + 1)𝑥𝑦 (𝑛+1) + (𝑛2 + 𝑚2 )𝑦 (𝑛) = 0.
13. If 𝑥 + 𝑦 = 1, prove that
2
2
𝑑𝑛
(𝑥 𝑛 𝑦 𝑛 ) = 𝑛! {𝑦 𝑛 − (𝑛𝐶 1 ) 𝑦 𝑛−1 𝑥 + (𝑛𝐶 2 ) 𝑦 𝑛−2 . 𝑥 2 −. . +(−1)𝑛 𝑥 𝑛 }.
𝑛
𝑑𝑥
14. If 𝑦 = 𝑥 𝑛 𝑙𝑜𝑔𝑥, prove that 𝑥𝑦 (𝑛+1) = 𝑛 !
15. If 𝑦 = 𝑒 𝑎𝑥 sin 𝑏𝑥, show that
𝑦 (𝑛+1) = 2𝑎𝑦 (𝑛) − (𝑎2 + 𝑏 2 )𝑦 (𝑛−1)
1
16. If 𝑥 = sin{𝑎 𝑙𝑜𝑔𝑦}, show that
(1 − 𝑥 2 )𝑦 ′′ − 𝑥𝑦 ′ − 𝑎2 𝑦 = 0.
Hence prove that
(1 − 𝑥 2 )𝑦 (𝑛+2) − (2𝑛 + 1)𝑥𝑦 (𝑛+1) − (𝑛2 + 𝑎2 )𝑦 (𝑛) = 0.
1
17. If = cosh(𝑚 log 𝑦) , prove that
(𝑥 2 − 1)𝑦 ′′ + 𝑥𝑦 ′ − 𝑚2 𝑦 = 0.
and (𝑥 2 − 1)𝑦 (𝑛+2) + (2𝑛 + 1)𝑥𝑦 (𝑛+1) + (𝑛2 − 𝑚2 )𝑦 (𝑛) = 0.
18. If = (𝑥 2 − 1)𝑚 , prove that 𝑦 (2𝑚) = 2𝑚!
19. Evaluate lim (𝑥 2 + 𝑦 2 ).
(𝑥,𝑦)→(0,0)
𝑦 2 −𝑥2
20. If 𝑓(𝑥, 𝑦) = 𝑥 2 +𝑦 2, then show that lim{lim 𝑓(𝑥, 𝑦)} ≠ lim {lim 𝑓(𝑥, 𝑦)}.
𝑥→0 𝑦→0
21. Show that the
22. Evaluate
lim
(𝑥,𝑦)→(0,0) 𝑥 4 +𝑦 2
𝑥 2 +𝑥−𝑥𝑦−𝑦
lim
(𝑥,𝑦)→(0,0)
23. Evaluate (a).
(
𝑥2𝑦
(
lim
𝑥−𝑦
) does not exist.
) , 𝑥 ≠ 𝑦.
(𝑥 2 + 𝑥𝑦 + 𝑦 2 )
(𝑥,𝑦)→(3,−4)
𝑦→0 𝑥→0
(b).
lim
(𝑥,𝑦)→(1,2)
24. Discuss the continuity at (0, 0) of the following functions:
2𝑥𝑦
(𝑥 2 +𝑦2 )
𝑥 3 −𝑦 3
(a). 𝑓(𝑥, 𝑦) = {
(𝑥 2 +𝑦 2)
(𝑥, 𝑦) ≠ (0,0)
0
(𝑥, 𝑦) = (0,0)
(b). 𝑓(𝑥, 𝑦) = {
(𝑥/√𝑥 2
+ 𝑦2 )
2
}
(𝑥, 𝑦) ≠ (0,0)
}
(𝑥, 𝑦) = (0,0)
(𝑥 2 /√𝑥 2 + 𝑦 2 ) (𝑥, 𝑦) ≠ (0,0)
}
(𝑥, 𝑦) = (0,0)
3
u
u
 x
 y
y
 0.
25. If u  sin 1    tan 1   , show that x
x
y
 x
 y
(c). 𝑓(𝑥, 𝑦) = {
2
 u u 
 u u 
26. If u( x  y)  x  y , prove that     41 
 
 x y 
 x y 
z
z
 2abz
27. If z  e axby f ax  by  , prove that b  a
x
y
2
2
28. If x x y y z z  c , show that

29. If u  1  2 xy  y 2

1 / 2
2z
1
 x log( ex)
xy
, prove that
 
u    2 u 
0
1 x2
 y

x 
x  y  y 
 x3  y3 
 , prove that 𝑥 2
30. If u  tan 1 
 x y 
2 cos 3𝑢 sin 𝑢
 x1 / 2  y 1 / 2 

31. If u  cos ec  1 / 3
1/ 3 
x y 
1
32. If u  log

𝜕2 𝑢
𝜕𝑥 2

+ 2𝑥𝑦
1/ 2
, prove that x 2
𝜕2 𝑢
𝜕2 𝑢
+ 𝑦 2 𝜕𝑦 2 = sin 4𝑢 − sin 2𝑢 =
𝜕𝑥𝜕𝑦
2
 2u
 2u
tan u 13 tan 2 u 
2  u

2
xy

y

 

xy
12 12
12 
x 2
y 2
x2  y2
 2u
 2u
 2u
 y2 2
, find the value of x 2 2  2 xy
xy
x
y
x y
du
3

dt
1 t 2
u u u


0.
34. If u = f(y-z, z-x, x-y), show that
x y z
33. If u  sin 1 ( x  y) , x = 3t, y = 4t3 , show that
𝑥 4 +𝑦 4
35.If 𝑢 = log (
𝑥+𝑦
36. Show that x
𝜕𝑢
𝜕𝑢
), show that 𝑥 𝜕𝑥 + 𝑦 𝜕𝑦 = 3.
 x y 
u
u 1
1
.
y
 cot u  0 where u  cos 
 x y
x
y 2


𝜕𝑤 2
1
𝜕𝑤 2
𝜕𝑓 2
𝜕𝑓 2
37. If 𝑤 = 𝑓(𝑥, 𝑦), 𝑥 = 𝑟 𝑐𝑜𝑠𝜃, 𝑦 = 𝑟 𝑠𝑖𝑛𝜃, show that ( 𝜕𝑟 ) + 𝑟 2 ( 𝜕𝜃 ) = (𝜕𝑥 ) + (𝜕𝑦) .
TUTORIAL SHEET-4
 ( x, y , z ) 2
= r sin  .
 ( r , ,  )
2. If u3+ v3+ w3 = x+ y +z, u2 + v2 +w2 = x3+ y3 + z3, u+ v+ w= x2+ y2 + z2 ,
u, v, w)  ( y  z )( z  x)( x  y )

then prove that
.
( x, y, z ) (u  v)(v  w)( w  u )
1. If x = rsin  cos  , y= rsin  sin  ,z = rcos  then show that
3. If u= x+y+z+t, v= x+y-z-t, w= xy- zt, r = x2+ y2 - z2- t2 show that
4. Verify JJ’=1 if x= sin  cos  , y= sin  sin  where J =
5. If 𝑥 = 𝑟𝑐𝑜𝑠𝜃, 𝑦 = 𝑟𝑠𝑖𝑛𝜃, show that
6.
𝜕(𝑥,𝑦)
𝜕(𝑟,𝜃)
u, v, w, r ) 
 0.
 ( x, y , z , t )
 ( x, y )
 ( ,  )
and J’=
 ( ,  )
 ( x, y )
= 𝑟.
Find the maximum and minimum distances of the point
3, 4,12
from the sphere
x 2  y 2  z 2  1.
7. Find the maximum and minimum values of 𝑓(𝑥, 𝑦) = x3+ y3- 3axy.
8. The sum of three number is constant. Prove that their product is maximum when they are
equal.
9. Find the dimension of the rectangular box, open at the top of maximum capacity whose
surface area is 108 square units.
10. A rectangular box, open at the top and having volume 32 cubic units. Find the dimension of
the box requiring least material of its construction.
10. Find the points at which the absolute extreme values of the function f(x,y) = 2+ 2x+ 2y – x2
– y2 on the triangular plate in the first quadrant bounded by the lines x = 0, y = 0, y = 9 – x exist.
Also write values of the function at these points
11. Use Lagrange’s method of undetermined multipliers to solve the problem given below:
(i)
Find the point P(x,y,z) closest to the origin on (i) the plane 2x+y-z-5=0 and
the (ii) hyperbolic cylinder x2 – z2 – 1=0.
(ii)
Find the points on the curve of intersection of the plane x+y+z=1 and the
cylinder x2+y2=1 closest or farthest from the origin.
(iii) The temperature at a point (x,y) on a metal plate is T(x,y) = 4x2-4xy+y2. An
ant on the plate walks around the circle of radius 5 centered at the origin.
What are the highest and lowest temperature encountered by the ant?
−1
12. Expand tan (𝑦/𝑥) in the neighbourhood of (1, 1) by Taylor’s theorem.
13. Expand 𝑒 𝑥 𝑐𝑜𝑠𝑦 in powers of 𝑥 and 𝑦 up to second degree.