LAC MID-2 Question Bank

1. Verify Cauchy’s Mean Value theorem for

(a) 𝑓(𝑥 ) = 𝑒 𝑥 , 𝑔(𝑥 ) = 𝑒 −𝑥 𝑖𝑛 [3,7]
1
(b) 𝑓(𝑥 ) = √𝑥, 𝑔(𝑥 ) = 𝑥 𝑖𝑛 [𝑎, 𝑏]
√
1
(c) 𝑓(𝑥 ) = 𝑙𝑜𝑔𝑥 , 𝑔(𝑥 ) = 𝑥 𝑖𝑛 [1, 𝑒]
1 1
(d) 𝑓(𝑥 ) = 𝑥 2 , 𝑔(𝑥 ) = 𝑥 𝑜𝑛 [𝑎, 𝑏]
(e) Verify Cauchy Mean Value theorem for sin x, cos x in a, b
𝜋
2. Obtain Taylor’s series expansion of 𝑆𝑖𝑛 𝑥 in powers of 𝑥 − 4
5
3. Verify Maclaurin’s theorem for 𝑓𝑥) = (1 − 𝑥 )2 with Lagrange’s form of
remainder up to two terms in [0, 1].
5
4. Verify Taylor’s theorem for 𝑓𝑥) = (1 − 𝑥 )2 with Lagrange’s form of remainder up
to two terms in [0, 1].
𝑥 𝑥2 𝑥4
5. Show that log(1 + 𝑒 𝑥 ) = log 2 + + − +…. And hence deduce that
2 8 192
𝑒𝑥 1 𝑥 𝑥3
= + − +⋯
1+𝑒 𝑥 2 4 48
1
6. Verify Taylor’s theorem for 𝑓 (𝑥 ) = 𝑥 3 − 3𝑥 2 + 2𝑥 in (0, ) .
2
−1 𝜋
7. Express 𝑡𝑎𝑛 𝑥 in the powers of 𝑥 −
4
sin 𝑥
8. Expand 𝑒 by Maclaurin’s series up to the term containing 𝑥 4
9. Expand 𝑙𝑜𝑔𝑒 𝑥 in powers of (𝑥 − 1) and hence evaluate 𝑙𝑜𝑔𝑒 1.1 correct
to 4 decimal places.
10. Find the Maclaurin’s theorem with Lagrange’s form of remainder for
𝑓 (𝑥 ) = cos 𝑥
UNIT-4
𝜕 𝜕 𝜕 2
1. 1. If 𝑢 = log(𝑥 3 + 𝑦 3 + 𝑧 3 − 3𝑥𝑦𝑧), show that ( + + ) 𝑢=
𝜕𝑥 𝜕𝑦 𝜕𝑧
−9
(𝑥+𝑦+𝑧)2
𝜕 3𝑢
2. If 𝑢 = 𝑒 𝑥𝑦𝑧 , find the value of 𝜕𝑥𝜕𝑦𝜕𝑧
−1
𝜕2𝑢 𝜕2𝑢 𝜕2𝑢
3. If 𝑢 = (𝑥 2 + 𝑦 2 + 𝑧 2 ) 2 , prove that 2 + 2 + =0
𝜕𝑥 𝜕𝑦 𝜕𝑧 2
 𝜕2𝑧 𝜕2𝑧
4. If 𝑧 = 𝑥 3 + 𝑦 3 − 3𝑎𝑥𝑦, show that 𝜕𝑦𝜕𝑥 = 𝜕𝑥𝜕𝑦 = −3𝑎

𝑑𝑢
6. If 𝑢 = 𝑥 2 + 𝑦 2 + 𝑧 2 and 𝑥 = 𝑒 2𝑡 , 𝑦 = 𝑒 2𝑡 cos 3𝑡, 𝑧 = 𝑒 2𝑡 sin 3𝑡 find
𝑑𝑡

𝑑𝑢
7. If 𝑢 = 𝑥 log 𝑥𝑦 where 𝑥 3 + 𝑦 3 + 3𝑥𝑦 = 1, find 𝑑𝑥

𝑥 𝑑𝑢
8. Given 𝑢 = 𝑠𝑖𝑛 (𝑦) , 𝑥 = 𝑒 𝑡 & 𝑦 = 𝑡 2 , find as a function of t. Verify your result
𝑑𝑡
by direct substitution
𝜕𝑢 𝜕𝑢 𝜕𝑢
9. If 𝑢 = 𝐹(𝑥 − 𝑦, 𝑦 − 𝑧, 𝑧 − 𝑥 ), prove that 𝜕𝑥 + 𝜕𝑦 + 𝜕𝑧 =0

1 𝜕𝑢 1 𝜕𝑢 1 𝜕𝑢
10. If 𝑢 = 𝑓 (2𝑥 − 3𝑦, 3𝑦 − 4𝑧, 4𝑧 − 2𝑥 ) then prove that + 3 𝜕𝑦 + 4 𝜕𝑧 = 0
2 𝜕𝑥
(2023)
𝑑𝑧
10. Given 𝑧 = √𝑥 2 + 𝑦 2 and 𝑥 3 + 𝑦 3 + 3𝑎𝑥𝑦 = 5𝑎2 find the value of at 𝑥 =
𝑑𝑥
𝑎&𝑦 =𝑎
𝜕(𝑥,𝑦,𝑧)
11. If 𝑥 + 𝑦 + 𝑧 = 𝑢, 𝑦 + 𝑧 = 𝑢𝑣, 𝑧 = 𝑢𝑣𝑤 then evaluate
𝜕(𝑢,𝑣,𝑤)

𝜕(𝑢,𝑣)
12. If 𝑢 = 𝑥 2 − 𝑦 2 , 𝑣 = 2𝑥𝑦 where 𝑥 = 𝑟 cos 𝜃, 𝑦 = 𝑟 𝑠𝑖𝑛𝜃 show that 𝜕(𝑟,𝜃) = 4𝑟 3

𝜕(𝑥,𝑦) 𝜕(𝑢,𝑣)
13. If 𝑥 = 𝑢(1 − 𝑣), 𝑦 = 𝑢𝑣 then prove that 𝜕(𝑢,𝑣) ∗ 𝜕(𝑥,𝑦) = 1

14. Verify 𝑢 = 𝑥 + 𝑦 − 𝑧, 𝑣 = 𝑥 − 𝑦 + 𝑧, 𝑤 = 𝑥 2 + 𝑦 2 + 𝑧 2 − 2𝑦𝑧 are functionally

dependent, if so find the relation between them.

15. Verify if 𝑢 = 2𝑥 − 𝑦 + 3𝑧, 𝑣 = 2𝑥 − 𝑦 − 𝑧, 𝑤 = 2𝑥 − 𝑦 + 𝑧 are functionally

dependent and if so, find the relation between them.
𝑥 𝑥+𝑦
16. Verify 𝑢 = 𝑦 , 𝑣 = 𝑥−𝑦 are functionally dependent, if so find the relation between
them
𝑥+𝑦
17. If 𝑢 = 1−𝑥𝑦 , 𝑣 = 𝑡𝑎𝑛−1 𝑥 + 𝑡𝑎𝑛−1 𝑦, 𝑠𝑡𝑎𝑡𝑒 where 𝑈&𝑉 are functionally
dependent, if so obtain the relation between them.

18. If 𝑢 = 𝑠𝑖𝑛−1 𝑥 + 𝑠𝑖𝑛−1 𝑦, 𝑦 = 𝑥√1 − 𝑦 2 + 𝑦√1 − 𝑥 2 find out if they are

functionally dependent and if so find the relation between them
𝑦𝑧 𝑧𝑥 𝑥𝑦 𝜕(𝑢,𝑣,𝑤)
19. If 𝑢 = ,𝑣 = ,𝑤 = , show that =4
𝑥 𝑦 𝑧 𝜕(𝑥,𝑦,𝑧)
 20. If 𝑢 = 𝑥 2 − 2𝑦 2 𝑎𝑛𝑑 𝑣 = 2𝑥 2 − 𝑦 2 where 𝑥 = 𝑟 𝑐𝑜𝑠𝜃, 𝑦 = 𝑟 𝑠𝑖𝑛𝜃 then show
𝜕(𝑢,𝑣)
that 𝜕(𝑟,𝜃) = 6𝑟 3 sin 2𝜃

𝑦 𝜕(𝑟,𝜃) 1
21. If 𝑟 = √𝑥 2 + 𝑦 2 , 𝜃 = 𝑡𝑎𝑛−1 𝑥 , then show that 𝜕(𝑥,𝑦) = 𝑟

22. Discuss the maxima and minima 𝑓 (𝑥, 𝑦) = 𝑥 2 𝑦 + 𝑥𝑦 2 − 𝑎𝑥𝑦

23. Examine the function 𝑥 3 + 𝑦 3 − 3𝑎𝑥𝑦 for maxima and minima

24. Discuss the maxima and minima of 𝑓 (𝑥, 𝑦) = 𝑥 3 𝑦 2 (1 − 𝑥 − 𝑦)

1 1
25. Find the maximum and minimum values of 𝑓(𝑥, 𝑦) = 𝑥𝑦 + 𝑎3 (𝑥 + 𝑦)

26. Find the maximum and minimum values of 𝑓(𝑥, 𝑦) = 𝑥 3 + 3𝑥𝑦 2 − 15𝑥 2 −
15𝑦 2 + 72𝑥

27. Discuss the maxima of 𝑓(𝑥, 𝑦) = 𝑥 3 + 𝑦 3 − 63(𝑥 + 𝑦) + 12𝑥𝑦

28. Find the dimensions of the rectangular box, open at the top of maximum capacity
surface is 432 sq.cm

29. Given 𝑥 + 𝑦 + 𝑧 = 𝑎, find the maximum value of 𝑥 𝑚 𝑦 𝑛 𝑧 𝑝

1 1 1
30. Find the maximum and minimum values of 𝑥 + 𝑦 + 𝑧 subject to 𝑥 + 𝑦 + 𝑧 = 1

31. Find the maximum value of 𝑥 2 + 𝑦 2 + 𝑧 2 under the condition 𝑥 + 𝑦 + 𝑧 = 𝑎

32. Find the maximum and minimum of the function 𝑥 3 + 3𝑥𝑦 2 + 3𝑥 2 − 3𝑦 2 + 4

33. Expand 𝑒 𝑥 log(1 + 𝑦) in powers 𝑥 & 𝑦 upto terms of third degree

Expand 𝑥 2 𝑦 + 3𝑦 − 𝑧 in powers of (𝑥 − 1) and (𝑦 + 2) using Taylor’s theorem

𝑦
34. Expand 𝑓(𝑥, 𝑦) = 𝑡𝑎𝑛−1 (𝑥 ) in powers of (𝑥 − 1) & (𝑦 − 1) upto third degree
terms. Hence evaluate 𝑓(1.1, 0.9) approximately.

35. In a plane triangle, find the maximum value of 𝐶𝑜𝑠 𝐴 𝐶𝑜𝑠 𝐵 𝐶𝑜𝑠 𝐶

2M Questions

1. Find the first and second partial derivatives of 𝑧 = 𝑥 3 + 𝑦 3 − 3𝑎𝑥𝑦

𝜕(𝑢,𝑣)
2. Find if 𝑢 = 𝑒 𝑥 and 𝑣 = 𝑒 𝑦
𝜕(𝑥,𝑦)
3. Define total derivative of a function
4. Write properties of Jacobian
5. Define functional dependence and functional independence
6. State Taylor’s theorem for functions of two variables
𝜕(𝑟,𝜃,𝑧) 𝜕(𝑥,𝑦,𝑧)
7. If 𝑥 = 𝑟 𝑐𝑜𝑠𝜃, 𝑦 = 𝑟 𝑠𝑖𝑛𝜃, 𝑧 = 𝑧 find given that =𝑟
𝜕(𝑥,𝑦,𝑧) 𝜕(𝑟,𝜃,𝑧)
𝜕(𝑟,𝜃,∅)
8. If 𝑥 = 𝑟 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠∅, 𝑦 = 𝑟 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛∅, 𝑧 = 𝑟 𝑐𝑜𝑠𝜃 then find given that
𝜕(𝑥,𝑦,𝑧)
𝜕(𝑥,𝑦,𝑧)
= 𝑟 2𝑠𝑖𝑛𝜃
𝜕(𝑟,𝜃,∅)
9. Write a condition for maxima and minima of 𝑓(𝑥, 𝑦)
𝜕𝑧 𝜕𝑧
10. Evaluate 𝜕𝑥 and 𝜕𝑥 if 𝑧 = 𝑥 2 𝑦 − 𝑥 sin 𝑥𝑦
𝑢2 𝑣2 𝜕(𝑢,𝑣)
11. If 𝑥 = ,𝑦 = find
𝑣 𝑢 𝜕(𝑥,𝑦)
UNIT-V
Multiple integrals
2M
2 3
1. Evaluate ∫0 ∫0 (𝑥 + 𝑦) 𝑑𝑥 𝑑𝑦
3 2 1
2. Evaluate ∫2 ∫1 𝑑𝑥 𝑑𝑦
𝑥𝑦
2 1
3. Evaluate ∫0 ∫0 4𝑥𝑦 𝑑𝑥 𝑑𝑦
2 3
4. Evaluate ∫1 ∫1 𝑥𝑦 2 𝑑𝑥 𝑑𝑦
2 𝑥
5. Evaluate ∫0 ∫0 𝑒 𝑥+𝑦 𝑑𝑦 𝑑𝑥
1 1 1
6. Evaluate ∫0 ∫0 ∫0 𝑒 𝑥+𝑦+𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧
1 𝑥 𝑥+𝑦
7. Find ∫0 ∫0 ∫0 𝑑𝑧 𝑑𝑦 𝑑𝑥
2 𝑥+2
8. Evaluate ∫−1 ∫𝑥 2 𝑑𝑦 𝑑𝑥
𝜋⁄ 𝜋
9. Evaluate ∫0 2 ∫𝜋⁄ cos(𝑥 + 𝑦) 𝑑𝑥 𝑑𝑦
2
2 𝑥 √𝑥+𝑦
10. Evaluate ∫0 ∫0 ∫0 𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧

Essay Type Questions

5 𝑥2
1. Evaluate ∫0 ∫0 𝑥(𝑥 2 + 𝑦 2 ) 𝑑𝑥 𝑑𝑦
1 √𝑥
2. Evaluate ∫0 ∫𝑥 (𝑥 2 + 𝑦 2 ) 𝑑𝑥 𝑑𝑦
𝑥
1 𝑥
3. Evaluate ∫0 ∫0 𝑒 𝑦 𝑑𝑥 𝑑𝑦
𝑥
4 𝑥2
4. Evaluate ∫0 ∫0 𝑒 𝑦 𝑑𝑦 𝑑𝑥
𝑥2 𝑦2
5. Evaluate ∫ ∫ 𝑥𝑦 𝑑𝑥 𝑑𝑦 taken over the positive quadrant of the ellipse + =1
𝑎2 𝑏2
6. Evaluate ∫ ∫ 𝑦 𝑑𝑥 𝑑𝑦, where R is the region bounded by the parabolas 𝑦 2 = 4𝑥 and
𝑥 2 = 4𝑦
7. Evaluate ∫ ∫ 𝑥𝑦(𝑥 + 𝑦)𝑑𝑥 𝑑𝑦 over the region R bounded by the curves curves 𝑦 =
𝑥 2 and 𝑦 = 𝑥
∞ ∞
8. By change of order of integration of ∫0 ∫0 𝑒 −𝑥𝑦 sin 𝑝𝑥 𝑑𝑥 𝑑𝑦 , show that
∞ sin 𝑝𝑥 𝜋
∫0 𝑑𝑥 =
𝑥 2
𝑥
𝑎 √
𝑎
9. Evaluate ∫0 ∫ (𝑥 2 + 𝑦 2 )𝑑𝑥𝑑𝑦 by changing the order of the integration
𝑥
𝑎

∞ ∞ 𝑒 −𝑦
10. Evaluate ∫0 ∫𝑥 𝑑𝑥 𝑑𝑦 by change of order of integration.
𝑦

4𝑎 2√𝑎𝑥
11. Change the order of integration in 𝐼 = ∫0 ∫𝑥2 𝑑𝑦 𝑑𝑥 and hence evaluate.
4𝑎

1 𝑒 𝑑𝑦 𝑑𝑥
12. Evaluate ∫0 ∫𝑒 𝑥 by changing the order of integration
log 𝑦
16𝑎 2
13. Show that the area between the parabolas 𝑦 2 = 4𝑎𝑥 and 𝑥 2 = 4𝑎𝑦 is .
3
∞ ∞ 2 +𝑦 2 )
14. Evaluate ∫0 ∫0 𝑒 −(𝑥 𝑑𝑥 𝑑𝑦 by changing into polar coordinators.
15. Find ∫ ∫ 𝑥𝑦 𝑑𝑥 𝑑𝑦 over the positive quadrant of the circle 𝑥 2 + 𝑦 2 = 𝑎2 by changing
into polar coordinates.
𝑎 √𝑎 2−𝑥 2 2+𝑦 2 )
16. Evaluate ∫0 ∫0 𝑒 −(𝑥 𝑑𝑥 𝑑𝑦 by changing into polar coordinators.
𝑎 𝑥 𝑥+𝑦
17. Evaluate ∫0 ∫0 ∫0 𝑒 𝑥+𝑦+𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥
log 𝑎 log 𝑏 log 𝑐
18. Evaluate ∫0 ∫0 ∫0 𝑒 𝑥+𝑦+𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧
1 𝑧 𝑥+𝑧
19. Evaluate ∫−1 ∫0 ∫𝑥−𝑧 (𝑥 + 𝑦 + 𝑧) 𝑑𝑥 𝑑𝑦 𝑑𝑧
1 √1−𝑥 2 √1−𝑥 2−𝑦 2
20. Evaluate ∫0 ∫0 ∫0 𝑥𝑦𝑧 𝑑𝑧 𝑑𝑦 𝑑𝑥
21. Evaluate ∫ ∫ ∫ 𝑥𝑦𝑧 𝑑𝑥 𝑑𝑦 𝑑𝑧, over the positive octant of the sphere
𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑎2
22. Calculate the volume of the solid bounded the planes 𝑥 = 0, 𝑦 = 0, 𝑥 + 𝑦 + 𝑧 =
𝑎 and 𝑧 = 0
𝑥2 𝑦2 𝑧2
23. Find the volume of the ellipsoid
𝑎 2 + 𝑏 2 + 𝑐2
=1