Practice Sheet BMAS 0104

1. Find the nth differential coefficient of log(ax + x2 ).

h i
Answer: (−1)n−1 (n − 1)! x1n + (x+a) 1
n .

2. If y = cos3 x find yn .
Answer: 14 3n cos 3x + nπ nπ



2 + 3 cos x + 2 .
3. If y = sin 2x sin 3x, find yn .
Answer:     
1 nπ n nπ
yn = cos x + − 5 cos 5x + .
2 2 2

4. Find n-th derivative of (sin2 x)(cos3 x).

Answer:       
1 n nπ n nπ n nπ
yn = 2 cos x + − 5 cos 5x + − 3 cos 3x + .
16 2 2 2
1
5. If y = 1−5x+6x2 , find yn .
Answer:
(2)n + 1 (3)n + 1
 
yn = (−1)n n! + .
(2x − 1)n+1 (3x − 1)n+1
1
6. Find the n-th derivative of x2 −a2 .
Answer:  
1 1
yn = (−1)n n! + .
(x − a)n+1 (x + a)n+1
x2
7. If y = (x−1)2 (x+2) , find the n-th derivative of y.
Answer:
(n + 1)! n! n!
yn = (−1)n + 5(−1)n + 4(−1)n .
3(x − 1)n+2 9(x + 2)n+1 9(x + 2)n+1
 
8. Find the n-th derivative of tan−1 1−x 2x
2 .

Answer  
1
yn = 2(−1)n−1 (n − 1)! · sin nθ · sin θ, where θ = tan−1 .
x
x3
9. Prove that the value of the n-th differential coefficient of x2 −1 for x = 0 is zero if n is even; and is
(−n) if n is odd and greater than 1.
x−1
10. If y = x log x+1 , show that
 
n−2 x−n x+n
yn = (−1) (n − 2)! − .
(x − 1)n (x + 1)n

11. If y = sin m sin−1 x , then prove that

(1 − x2 )y2 − xy1 + m2 y = 0

and
(1 − x2 )yn+2 − (2n + 1)xyn+1 − n2 − m2 yn = 0.

1
12. If y 1/m + y −1/m = 2x, prove that

(x2 − 1)yn+2 + (2n + 1)xyn+1 + (n2 − m2 )yn = 0.

13. If y = a cos(log x) + b sin(log x), prove that:

(i) x2 y2 + xy1 + y = 0,
(ii) x2 yn+2 + (2n + 1)xyn+1 + (n2 + 1)yn = 0.
14. If y = (x2 − 1)n , prove that:

(x2 − 1)yn+2 + 2xyn+1 − n(n + 1)yn = 0.

15. If x = tan(log y), prove that:

(1 + x2 )yn+1 + (2nx − 1)yn + n(n − 1)yn−1 = 0.

16. If y = sin loge (x2 + 2x + 1), prove that:

(1 + x2 )yn+2 + (2n + 1)(1 + x)yn+1 + (n2 + 4)yn = 0.

17. If y = sin−1 x, prove that:

(1 − x2 )yn+2 − (2n + 1)xyn+1 − n2 yn = 0.

−1
18. If y = em sin x
, prove that:

(i) (1 − x2 )y2 − xy1 = m2 y,

(ii) (1 − x2 )yn+2 − (2n + 1)xyn+1 − (n2 + m2 )yn = 0.
19. Find the first-order partial derivatives of the following functions:

(i) u = y x
(ii) u = log(x2 + y 2 )
(iii) u = x2 sin xy
 
y
(iv) u = x
y tan−1 x

Answers:
(i) xx log y, xy x−1
2x 2y
(ii) x2 +y 2 , x2 +y 2
(iii) 2x sin xy − y cos xy , −x cos xy
x2
(iv) −x
x2 +y 2 + 1
y tan−1 xy , y(x2 +y 2 ) − x
y2 tan−1 y
x
 
x2 +y 2
20. Find the first-order partial derivatives of u = tan−1 x+y

Answer:
∂u x2 + 2xy − y 2 ∂u y 2 + 2xy − x2
= , = .
∂x (x + y)2 + (x2 + y 2 )2 ∂y (x + y)2 + (x2 + y 2 )2

2
  
21. Find the first-order partial derivatives of u = cos−1 x
y .

Answer:
∂u −1 ∂u −y
=p , =p .
∂x y 2 − x2 ∂y y 2 − x2

22. If z = eax+by f (ax − by), show that:

∂z ∂z
b +a = 2abz.
∂x ∂y

23. If u = log(x2 + y 2 ) + tan−1 xy , prove that:

∂2u ∂2u
+ 2 = 0.
∂x2 ∂y

24. If z = f (x + ay) + ϕ(x − ay), prove that:

∂2z 2
2∂ z
= a .
∂y 2 ∂x2
x2 x3 x1 x3 x1 x2
25. If y1 = x1 , y2 = x2 , y3 = x3 , then show that

∂(y1 , y2 , y3 )
= 4.
∂(x1 , x2 , x3 )

26. If y1 = 1 − x1 , y2 = x1 (1 − x2 ), y3 = x1 x2 (1 − x3 ), find the value of

∂(y1 , y2 , y3 )
.
∂(x1 , x2 , x3 )

27. If x = u(1 + v), y = v(1 + u), then find the value of

∂(x, y)
.
∂(u, v)

u+v
28. If x = uv, y = u−v , find
∂(u, v)
.
∂(x, y)

29. If x = r cos θ, y = r sin θ, z = z, find

∂(x, y, z) ∂(r, θ, z)
and .
∂(r, θ, z) ∂(x, y, z)

∂(u,v,w)
30. Calculate the Jacobian ∂(x,y,z) of the following:

u = x + 2y + z, v = x + 2y + 3z, w = 2x + 3y + 5z.

31. If u = xyz, v = x + y + zx, w = x + y + z, then compute the Jacobian

∂(u, v, w)
.
∂(x, y, z)

3
32. If u3 + v 3 + w3 = x + y + z, u2 + v 2 + w2 = x2 + y 2 + z 2 , and u + v + w = x2 + y 2 + z 2 , then show
that
∂(u, v, w) (x − y)(y − z)(z − x)
= .
∂(x, y, z) (u − v)(v − w)(w − u)

33. If u1 = x1 + x2 + x3 + x4 , u2 = x2 + x3 + x4 , u3 = x3 + x4 , u4 = x4 , then show that

∂(x1 , x2 , x3 , x4 )
= 1.
∂(u1 , u2 , u3 , u4 )

34. If u = cos xy+yz+zx

x2 +y 2 +z 2 , prove that
∂u ∂u ∂u
x +y +z = 0.
∂x ∂y ∂z
35. Show that
∂u ∂u ∂u x3 + y 3 + z 3
+y x
+z = 2 tan u, where u = sin−1 .
∂x ∂y ∂z ax + by + cz
√ √
36. If u(x, y) = ( x + y)5 , find the value of

∂2u ∂2u 2
2∂ u
x2 + 2xy + y .
∂x2 ∂x∂y ∂y 2
 
37. If x = x4 y 2 sin−1 x
y + log x − log y, show that
 
∂ ∂ x
x +y = 6x4 y 2 sin−1 .
∂x ∂y y

y
, prove that x ∂u ∂u


38. (i) If u = f x ∂x + y ∂y = 0.

(ii) If u = xf xy , prove that x ∂u ∂u

∂x + y ∂y = u.
x3 y 3
(i) If V = x3 +y 3 , prove that x ∂V ∂V
∂x + y ∂y = 3V .

(ii) If u = x3 + y 3 + z 3 + 3xyz, show that x ∂u ∂u ∂u

∂x + y ∂y + z ∂z = 3u.

39. Verify Euler’s theorem for the following functions:

(i) f (x, y) = ax2 + 2hxy + by 2
x y z
(ii) u = y + z + x
−x2 (x2 −y 2 )3
(iii) f (x, y) = (x2 +y 2 )3
x1/3 +y 1/3
(iv) z = x1/2 +y 1/2

x3 y 3
40. If V = x3 +y 3 , prove that x ∂V ∂V
∂x + y ∂y = 3V .

41. If u = x3 + y 3 + z 3 + 3xyz, show that x ∂u ∂u ∂u

∂x + y ∂y + z ∂z = 3u.

42. If  
xy + yz + zx
u = cos ,
x2 + y 2 + z 2
prove that
∂u ∂u ∂u
x +y +z = 0.
∂x ∂y ∂z

4
43. If u = (x1/4 + y 1/4 )(x1/5 + y 1/5 ), apply Euler’s theorem to find the value of

∂u ∂u
x +y .
∂x ∂y

44. If  
4 2 −1 x
z = x y sin + log x − log y,
y
show that  
∂z ∂z x
x +y = 6x4 y 2 sin−1 .
∂x ∂y y

45. If u = sin−1 (x − y), x = 3t, y = 4t3 , show that

du 3
=√ .
dt 1 − t2

du
46. If u = x2 − y 2 + sin yz, where y = ex and z = log x, find dx .

47. If f (x, y) = 0, ϕ(y, z) = 0, show that

∂f ∂ϕ dz ∂f ∂ϕ
· · = · .
∂y ∂z dx ∂x ∂y

48. If u = f (2x − 3y, 3y − 4z, 4z − 2x), prove that

1 ∂u 1 ∂u 1 ∂u
+ + = 0.
2 ∂x 3 ∂y 4 ∂z

49. If ϕ(cx − az, cy − bz) = 0, show that

∂z ∂z
a +b = c.
∂x ∂y
50. If  
y−x z−x
u=u , ,
xy xz
show that
∂u ∂u ∂u
x2 + y2 + z2 = 0.
∂x ∂y ∂z
du
51. If u = x log(xy), where x3 + y 3 + 3xy = 1, find dx .

52. If u = f (r, s) and r = x + y, s = x − y, show that

∂u ∂u ∂u
+ =2 .
∂x ∂y ∂r

5
 Prove that
R π/2 √ R π/2 √ π
1. 0 tan θdθ = 0 cot θdθ = √
2
.

1√
R∞ 2
2. 0
xe−x dx = 4 π.

R∞√ 3
√
π
3. 0
xe−x dx = 3 .

R∞√ 3
√
π
4. 0
xe−x dx = 3 .
R∞ xc Γ(c+1)
5. 0 ex dx = (log c)c+1 , c > 1.

1 n−1
R∞

6. 0
log x dx = Γ(n), n > 0.
R∞ 2 Γ(n)
7. 0
x2n−1 e−ax dx = 2an .
R∞ (−1)n n!
8. 0
xm (log x)n dx = (m+1)n+1 , where n is a positive integer and m > −1.
R∞√ 2 R∞ 2
9. 0
xe−x dx = 0
e−x dx = π
√
2 2
.
R∞ −x 2 R∞ 4
10. 0
e√
x
dx × 0
x2 e−x dx = π
√
4 2
.
R2
(8 − x3 )−1/3 dx = 13 Γ 1 2



11. 0 3 Γ 3 .
R1 1
12. 0
x5 (1 − x3 )3 dx = 60 .
R π/2
13. 0
sin3 x cos5/2 x dx = 8
77 .

1. Find ∇ϕ when ϕ is given by ϕ = 3x2 y − y 3 z 2 at the point (1, −2, −1).

Answer:
∇ϕ = −12î − 9ĵ − 16k̂ at the point (1, −2, −1).

2. If θ is the acute angle between the surfaces xy 2 z = 3x + z 2 and 3x2 − y 2 + 2z = 1 at the point
(1, −2, 1), show that
3
cos θ = √ .
7 6
3. Find the angle between the surfaces x2 + y 2 + z 2 = 9 and z = x2 + y 2 − 3 at the point (2, −1, 2).
Answer:  
−1 8
θ = cos √ .
3 21
4. Find the directional derivative of the function f (x, y, z) = xy 2 + yz 3 at the point (2, −1, 1) in the
direction of the vector î + 2ĵ + 2k̂.
Answer:
−11
.
3
5. Find the directional derivative of ϕ = (x2 + y 2 + z 2 )−1/2 at the point P (3, 1, 2) in the direction of
the vector yz î + zxĵ + xy k̂.
Answer:
−9
√ .
49 14

6
 6. Find the divergence and curl of the vector field F, where

F = grad(x3 + y 3 + z 3 − 3xyz).

Answer:
∇ · F = 6(x + y + z).

∇ × F = 0.

7. Find the divergence and curl of the vector field V, where

V = (xyz)î + (3x2 y)ĵ + (xz 2 − y 2 )k̂.

at the point (2, −1, 1).

Answer:
∇ · V = 14 at (2, −1, 1).

∇ × V = 2î − 3ĵ − 14k̂ at (2, −1, 1).

8. If r = xî + y ĵ + z k̂, show that:

(a) div r = 3
(b) curl r = 0

9. If F⃗ = 3xy î − y 2 ĵ, evaluate Z

F⃗ · d⃗r,
C

where C is the arc of the parabola y = 2x2 from (0, 0) to (1, 2).
Answer: Z
7
F⃗ · d⃗r = .
C 6

10. If u = x+y +z, v = x2 +y 2 +z 2 , w = yz +zx+xy, prove that ∇u, ∇v, and ∇w are coplanar vectors.

11. Find ∇ϕ when ϕ is given by

ϕ = 3x2 y − y 3 z 2
at the point (1, −2, −1).

Answer:
∇ϕ = −12î − 9ĵ − 16k̂ at (1, −2, −1).

12. Find the divergence and curl of the vector field

F = (3x2 y + 5z 2 )î + (4xy 3 + 8z)ĵ + (6x + 3y 2 z)k̂.

Answer
Divergence: ∇ · F = 6xy + 12xy 2 + 3y 2 .

Curl: ∇ × F = (6x + 3y 2 )î + (6x2 y − 6)ĵ + (4y 3 − 3x2 )k̂.

7
13. Evaluate by Green’s theorem I
(ex sin y dx + ex cos y dy)
C

and hence verify Green’s theorem, where C is the rectangle with vertices (0, 0), (π, 0), (π, π2 ), and
(0, π2 ).
Answer:
2(e−π − 1).

14. State Green’s Theorem.

15. State Gauss Divergence Theorem.

16. State Stoke’s Theorem.