Visvesvaraya National Institute of Technology, Nagpur
Department of Mathematics
Mathematics-II (MAL-102)
Assignment 1 on Differential Calculus
(Limit, continuity, differentiability, partial derivatives)
1. Using ϵ-δ definition, find the following limits (if exist):
( 2 ) x3 3x2 y
(i) lim x + 2y , (ii) lim , (iii) lim .
(x,y)→(1,2) (x,y)→(0,0) x2 + y 2 (x,y)→(0,0) x2 + y 2
2. Show that the following limits do not exist:
xy xy 2 x2
(i) lim (ii) lim , (iii) lim .
(x,y)→(0,0) x2 + y 2 (x,y)→(0,0) x2 + y 4 (x,y)→(0,0) x2 + y 2
sin−1 (x + 2y)
if (x, y) ̸= (0, 0)
tan−1 (2x + 4y)
3. Show that the function f (x, y) = ,
1
if (x, y) = (0, 0)
2
is continuous at the point (0, 0).
4
x − y
2
if (x, y) ̸= (0, 0)
4. Discuss about the continuity of the function f (x, y) = x4 + y 2 ,
0 if (x, y) = (0, 0)
at the point (0, 0).
x2 y 2
if (x, y) ̸= (0, 0)
5. Consider the function f (x, y) = x2 y 2 + (x − y)2 .
0 if (x, y) = (0, 0)
Show that the function satisfies the following:
( ) ( )
(a) The iterated limits lim lim f (x, y) and lim lim f (x, y) exist and equal to zero.
x→0 y→0 y→0 x→0
(b) lim f (x, y) does not exist.
(x,y)→(0,0)
(c) f (x, y) is not continuous at (0, 0).
(d) The partial derivatives exist at (0, 0).
xy x − y
2 2
if (x, y) ̸= (0, 0)
6. Let f (x, y) = x2 + y 2 .
0 if (x, y) = (0, 0)
Prove that
(a) f (x, y) is continuous function at (0,0) using ϵ-δ definition.
(b) fx (0, y) = −y and fy (x, 0) = x for all x and y.
(c) fxy (0, 0) = −1 and fyx (0, 0) = 1 and
(d) f (x, y) is differentiable at (0, 0).
2 2
x + y for (x, y) ̸= (0, 0)
7. Prove that the function f (x, y) = x−y is not continuous at (0, 0).
0 for (x, y) = (0, 0)
Do you think the first order partial derivatives exists for this function at the point (0, 0)? Justify
your answer.
( )
1
8. Let f (x, y) = (x2 + y 2 ) sin if (x, y) ̸= (0, 0) and 0, otherwise. Show that the
x2 + y 2
partial derivatives are not continuous at (0, 0).
9. Prove or disprove
√ that the functions √
i)f (x, y) = x2 + y 2 , ii)f (x, y) = |xy| and iii)f (x, y) = |xy| are differentiable at the
origin.
10. If f (x, y) = xey , find the rate of change of f at the point P (2, 0) in the direction from P to
Q( 12 , 2). In what direction does f have the maximum rate of change? What is the value of
maximum rate of change?
80
11. The temperature at a point (x, y, z) in space is given by T (x, y, z) = ,
1 + + 2y 2 + 3z 2
x2
where T is measured in centigrade and x, y, z in meters. In which direction does the tempera-
ture increase fastest at the point (1, 1, −2)? What is the maximum rate of change?
12. Is there a function f (x, y) whose partial derivatives are fx (x, y) = x + 4y and fy (x, y) =
3x − y? Explain.
13. If u = log(x3 + y 3 + z 3 − 3xyz) then prove that
( )2
∂u ∂u ∂u 3 ∂ ∂ ∂ 9
i) + + = ii) + + u=− .
∂x ∂y ∂z x+y+z ∂x ∂y ∂z (x + y + z)2
x2 y2 z2
14. Prove that if 2 + 2 + 2 = 1, then
( )2 ( a + )2u (b +)u2 c (+ u )
∂u ∂u ∂u ∂u ∂u ∂u
+ + =2 x +y +z
∂x ∂y ∂z ∂x ∂y ∂z
∂u ∂u ∂u
15. If u = log(tan x + tan y + tan z) then show that sin 2x + sin 2y + sin 2z = 2.
∂x ∂y ∂z
