LAC IMPORTANT QUESTIONS
UNIT-3
1. Verify Rolle’s theorem for f (x) = x 2 − 2x − 3 in [−1, 3]
2. Verify Rolle’s theorem for f (x) = 𝑒 𝑥 𝑠𝑖𝑛 𝑥 in [0, π ]
𝑠𝑖𝑛 𝑥
3. Verify Rolle’s theorem for f (x) = in [0, π ]
𝑒𝑥
4. Using Lagrange’s mean value theorem, find the value of c for the function 𝑓 (𝑥 ) = 𝑒 𝑥 in
[0,1]
𝑎 𝑏 𝑏
5. Using Lagrange’s Mean value theorem, for 0 < a < b, Prove that 1 − < log 𝑎 < −1
𝑏 𝑎
1 6 1
and hence show that < log 5 <
6 5
6. Verify Lagrange’s Mean value theorem for f(x) =log 𝑒 𝑥 in [1, e]
7. If f(x) = e𝑥 , g(x) = e−𝑥 in [a, b] then show that ‘c’ is the average of ‘a’ and ‘b’ using
Cauchy’s Mean value theorem
log 𝑏−𝑙𝑜𝑔𝑎 𝑎+𝑏
8. If f(x) = log 𝑥, g(x) = x 2 in [a, b] with b > a> 1 then show that 𝑏−𝑎
= 2𝑐 2
using
Cauchy’s Mean value theorem
𝜋
9. Verify Cauchy’s Mean value theorem for f(x) = sinx , g(x) = cos x in [0, 2 ]
10. Obtain Taylor’s series expansion of e𝑥 about x = -1
11. Obtain Maclaurin’s series expansion of (i) e𝑥 (ii) sin x
UNIT-4
1. Find first and second order partial derivatives of f(x, y) = x3+y3-3axy and also
𝜕 2𝑓 𝜕 2𝑓
verify = 𝜕𝑦𝜕𝑥.
𝜕𝑥𝜕𝑦
𝑥 3+ 𝑦 3 𝜕𝑢 𝜕𝑢
2. If u = tan-1( ), prove that x 𝜕𝑥 + y = sin 2u.
𝑥+𝑦 𝜕𝑦
𝑦 𝑍 𝜕𝑢 𝜕𝑢 𝜕𝑢
3. If u = 𝑧 + 𝑥 , then prove that x𝜕𝑥 + 𝑦 𝜕𝑦 + 𝑧 𝜕𝑧 = 0
𝜕2𝑢 𝜕 2𝑢 𝜕 2𝑢
4. If r2 = x2 + y2 + z2 and u=rm then prove that ( 𝜕𝑥 2 + 𝜕𝑦 2 + 𝜕𝑧 2 ) = 𝑚(𝑚 + 1)𝑟 𝑚−2
1 𝜕 2 𝑢 𝜕 2𝑢 𝜕2𝑢
5. If 𝑢 = ,𝑥 2 + 𝑦 2 + 𝑧 2 ≠ 0 then prove that + + 𝜕𝑧 2 =0
√𝑥 2 +𝑦 2 +𝑧 2 𝜕𝑥 2 𝜕𝑦 2
𝜕 2𝑢 𝜕2𝑢 𝜕2𝑢
6. If u = log(𝑥 2 + 𝑦 2 + 𝑧 2 ) then prove that (𝑥 2 + 𝑦 2 + 𝑧 2 ) ( 𝜕𝑥 2 + 𝜕𝑦 2 + 𝜕𝑧 2 ) = 2
𝜕 𝜕 𝜕 2 −9
7. If u = log(𝑥 3 + 𝑦 3 + 𝑧 3 − 3𝑥𝑦𝑧) then prove that [𝜕𝑥 + + ] 𝑢=
𝜕𝑦 𝜕𝑧 (𝑥+𝑦+𝑧)2
( x, y )
8. If x = u(1 + v), y = v(1+u) then prove that 1 u v
(u , v)
𝜕(𝑥,𝑦,𝑧)
9. If 𝑥 = 𝑟𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜑, 𝑦 = 𝑟 𝑠𝑖𝑛𝜃 𝑠𝑖𝑛𝜑 , 𝑧 = 𝑟 𝑐𝑜𝑠𝜃 then show that = 𝑟 2 𝑠𝑖𝑛𝜃 .
𝜕(𝑟,𝜃,𝜑)
𝜕(𝑟,𝜃,𝜑)
Also find .
𝜕(𝑥,𝑦,𝑧)
𝜕(𝑥,𝑦,𝑧)
10. If x + y + z = u , y + z = uv , z = uvw , then Show that = 𝑢2 𝑣
𝜕(𝑢,𝑣,𝑤)
11. Show that the functions u = x + y + z , v = xy + yz + xz and w = x2 + y2 + z2 are
functionally dependent. Also find the relation between them.
𝜋
12. Expand 𝑒 𝑥 cos 𝑦 near (1, 4 ) by Taylor’s series method.
13. A rectangular box open at the top is to have a volume of 32 cubic feets. Find the dimensions
of the box requiring least material for its construction.
UNIT-5
5 𝑥2
1. Evaluate∫0 ∫0 𝑥(𝑥 2 + 𝑦 2 )𝑑𝑥𝑑𝑦
2. Evaluate ∬𝑅 𝑦 𝑑𝑥𝑑𝑦 where R is the region bounded by the parabolas 𝑦 2 = 4x and 𝑥 2 =
4y
3. Evaluate∬𝑅 𝑥𝑦 𝑑𝑥𝑑𝑦where R is the region bounded by x-axix and the ordinate x = 2a
and the curve 𝑥 2 =4ay
𝜋 𝑎𝑠𝑖𝑛𝜃
4. Evaluate ∫0 ∫0 𝑟𝑑𝑟𝑑𝜃
5. Evaluate∬ 𝑟 sin 𝜃 𝑑𝑟𝑑𝜃 over the cardioids r = a(1-cos 𝜃) above the initial line
∞ ∞ 2+𝑦 2 )
6. Evaluate∫0 ∫0 𝑒 −(𝑥 𝑑𝑥𝑑𝑦 by changing into polar coordinates and hence show that
∞ 2 √𝜋
∫0 𝑒 −𝑥 𝑑𝑥 = 2
1 1 1
7. Evaluate ∫0 ∫0 ∫0 𝑒 𝑥+𝑦+𝑧 𝑑𝑧𝑑𝑥𝑑𝑦.
log 2 𝑥 𝑥+𝑦
8. Evaluate∫0 ∫0 ∫0 𝑒 𝑥+𝑦+𝑧 𝑑 𝑧𝑑𝑦𝑑𝑥.
1 √1−𝑥 2
9. Evaluate ∫0 ∫0 𝑦 2 𝑑𝑦𝑑𝑥 by changing the order of integration.
10. Evaluate ∭𝑅 (𝑥 + 𝑦 + 𝑧)𝑑𝑧𝑑𝑦𝑑𝑥 where R is the region of space bounded by planes
𝑥 = 0, 𝑥 = 1, 𝑦 = 0, 𝑦 = 1, 𝑧 = 0, 𝑧 = 1
