Department of Mathematics
Motilal Nehru National Institute of Technology Allahabad
Mathematics-I (MAN11101), Semester-I
Course Instructor: Dr. Naren Bag
Assignment 1 (Unit I)
1. Using Lagrange Mean Value theorem, show that |tan−1 x1 − tan−1 x2 | ≤ |x1 − x2 |.
√ 1
2. Verify Cauchy Mean Value for f (x) = x and g(x) = √ in [a, b], a > 0.
x
1+x 2x
3. Using Mean Value theorem show that 2x < log < 1−x 2 , 0 < x < 1.
1−x
4. Using Lagrange’s Mean Value Theorem, prove that |cos b − cos a| ≤ |b − a|.
sin α − sin β π
5. Show that = cot θ, where 0 < α < θ < β < 2.
cos β − cos α
6. If a function f is such that its derivative f ′ is continuous on [a, b] and differentiable on (a, b), then
show that there exist a number c between a and b such that f (b) = f (a)+(b−a)f ′ (a)+ 21 (b−a)2 f ′′ (c).
tan x x π
7. Show that > , for 0 < x < 2.
x sin x
8. Examine the validity of Rolle’s theorem-
(a) f (x) = |x| on [-1, 1].
(b) f (x) = x3 − 4x on [-2, 2].
9. Examine the hypothesis and conclusion of Lagrange’s Mean Value Theorem-
(a) f (x) = |x| on [-1, 1]
(b) f (x) = log x on [1/2, 2]
1
(c) f (x) = x 3 on [-1, 1]
(d) f (x) = 2x2 − 7x + 10 on [2, 5].
10. Let f ′ (x) and g ′ (x) be continuous and differentiable functions on [a, b], then show that there exists
f (b) − f (a) − (b − a)f ′ (a) f ′′ (c) ′′
c, a < c < b, such that = , g (c) ̸= 0.
g(b) − g(a) − (b − a)g ′ (a) g ′′ (c)
11. Prove that between two real roots of ex sin x = 1, there exists atleast one real root of ex cos x+1 = 0.
12. Find a point on the parabola y = (x + 2)2 , where the tangent is parallel to chord joining (−2, 0)
and (0, 4).
13. Prove that there is no real number k for which the equation x3 − 3x + k = 0 has two distinct real
roots in [0, 1].
x log10 e
14. Show that log10 (x + 1) = , for some t, 0 < t < 1, x > 0.
1 + tx
15. Using ϵ − δ approach establish the following limits -
(a) lim(x,y)→(2,1) (3x + 2y) = 10
(b) lim(x,y)→(1,1) (x2 + y 2 − 1) = 1
(c) lim(x,y)→(0,0) (y + xcos( y1 ) + 2y) = 0
1
(d) lim(x,y)→(0,0) (x2 + y 2 )sin( xy )=0
2 2
(e) lim(x,y)→(0,0) xy xx2 −y
+y 2 = 0
16. Show that the limit does not exist in each case
x2
(a) lim(x,y)→(0,0) x2 +y 2
2xy
(b) lim(x,y)→(0,0) x2 +y 2
x3 +y 3
(c) lim(x,y)→(0,0) x−y [Hint : put y = x − mx3 ]
x4 y 2
(d) lim(x,y)→(0,0) (x4 +y 2 )2
z
(e) lim(x,y,z)→(0,0,0) log( xy )
17. Determine the following limits if they exist.
(a) lim(x,y)→(0,0) tan−1 ( xy )
x3 −y 3
(b) lim(x,y)→(1,1) x−y
(x−1)siny
(c) lim(x,y)→(1,0) ylogx
2
x − xy
(d) lim(x,y)→(0,0) √ √
x− y
18. Show that
xsin(x2 +y 2 )
(a) lim(x,y)→(0,0) x2 +y 2 =0
−1
sin (xy−2) 1
(b) lim(x,y)→(2,1) tan−1 (3xy−6) = 3
√
(c) lim(x,y)→(4,π) x2 sin y8 = 8 2
(d) lim(x,y)→(α,0) (1 + xy )y = eα
19. Investigate the continuity at (0, 0) of
2xy
if (x, y) ̸= (0, 0)
(a) f (x, y) = x2 + y 2
0 if (x, y) = (0, 0)
xy
p if (x, y) ̸= (0, 0)
(b) f (x, y) = x2 + y 2
0 if (x, y) = (0, 0)
p p
sin p|xy| − |xy|
if (x, y) ̸= (0, 0)
(c) f (x, y) = x2 + y 2
0 if (x, y) = (0, 0)
2 2
x +y
if (x, y) ̸= (0, 0)
(d) f (x, y) = tanxy
0 if (x, y) = (0, 0)
2 2
2x + y
if (x, y) ̸= (0, 0)
(e) f (x, y) = 3 + sinx
0 if (x, y) = (0, 0)
2xy
if (x, y) ̸= (0, 0)
(f) f (x, y, z) = x2 − 3Z 2
0 if (x, y) = (0, 0)
20. Show that the function
2xy
if (x, y) ̸= (0, 0)
f (x, y) = x2 + y 2
0 if (x, y) = (0, 0)
is continuous every where except at (0, 0).
21. Let f : R2 −→ R be defined by
xy 2
if x ̸= 0
f (x, y) = x2 + y4
0 if x = 0,
Then show that f is discontinuous at (0, 0) but directional derivatives exist at (0, 0) along each
non-zero vector u ∈ R2 .
22. Let f : R2 −→ R be defined by
x2 y 2
if (x, y) ̸= (0, 0)
f (x, y) = x4 + y 4
0 if (x, y) = (0, 0)
Then show that
(a) Limit of f (x, y) does not exist at (0, 0).
(b) Partial derivatives D1 f , D2 f exist.
(c) For u = (u1 , u2 ), u1 u2 ̸= 0, the directional derivative Du (0, 0) does not exist.
(d) f is differentiable every where except at (0, 0).
23. Let f : R2 −→ R be defined by
(
0 if xy ̸= 0
f (x, y) =
y+1 if otherwise
Then show that f is discontinuous at (0, 0)
1
(
(x + y)sin x+y if x + y ̸= 0
24. Show that the function f (x, y) = is continuous at (0, 0) but
0 if x + y = 0
partial derivatives do not exist at (0, 0).
2 2
x +y
if (x, y) ̸= 0
25. Show that the function f (x, y) = |x| + |y| is continuous at (0, 0) but partial
0 if (x, y) = 0
derivatives do not exist at (0, 0).
xy
2 + 2y 2
if (x, y) ̸= 0
26. Show that the function f (x, y) = x is discontinuous at (0, 0) but
0 if (x, y) = 0
partial derivatives exist at (0, 0).
p
27. Let f : R2 −→ R be defined by f (x, y) = |xy|. Show that-
(a) f is continuous at (0, 0).
(b) Partial derivatives exist at (0, 0).
(c) f is not differentiable at (0, 0).
