Assignment (To be done after studying the course material)
MTE-07
Course Code: MTE-07 Assignment Code: MTE-07/TMA/2008-09 Maximum Marks: 100
Note: Each question is worth the marks given against it. 1.
Which of the following statements are true? Give reasons for your answers. (10) [Note : Suppose the given statement is lim ( x + y) = 2 . Simply saying that this is false is not ( x , y ) →( 0, 0 )
enough. You have to give reasons for saying so. The reasons may be a proof or an example. In this case you should show that the limit is not 2 it is zero. Similarly, if you believe that a particular statement is true, then you have to give a reason for saying so. For example, if the statement is like the one given below. “A function of two variables which is continuous at a point need not have any of the partial derivatives at that point.” This statement is true. Here you can give an example of a function f which is continuous at some point and which does not have any of the partial derivatives at that point.] a)
x2 −1 Zero is a lower bound of 0 < x < 1 . 2x
b)
The domain of the function f, defined by f ( x , y, z) = 1 − x 2 − y 2 − z 2 , consists of all those
c) d)
2.
points in R 3 whose distance from the origin is less than or equal to 1. If both the partial derivatives of f(x, y) are zero on a subset A of R 2 , then f(x, y) is a constant on A. ( x − 1)( y − 1) The function f, defined by f ( x , y) = , is a homogeneous function on its domain. xy
e)
The function f, defined by f ( x , y) = x 3 + xy + y , is integrable on [1, 2] × [1, 3] .
a)
Find the following limits: (i)
(ii)
b)
x2 lim 2 x → +∞ 8 x − 3 sin x lim+ (sin x )
(3)
x →0
Using only the definitions find f xy (0, 0) and f yx (0, 0) , if they exist, for the function x2y , f ( x , y) = x 2 + y 2 0 ,
c)
1/ 3
( x , y) ≠ (0, 0) otherwise
Find the domain and range of the function f, defined by, f ( x , y) = level curves of this function.
3.
a)
(4)
Let the function f be defined by 3x 2 y 4 , ( x , y) ≠ (0, 0) f ( x , y) = x 4 + y 8 0, ( x , y) = (0, 0)
2 xy . Also find two x + y2 (3) 2
Show that f has directional derivatives in all directions at (0, 0). b)
Let a = (1, 2, 3), b = (−5, 3, − 2), c = (2, − 4, 1) be three points in R 3 . Find 2b − a + 3c . (2)
c)
Let x = e r cos θ, y = e r sin θ and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that 2 ∂ 2f ∂ 2f ∂ 2f 2 2 ∂ f + = ( x + y ) + ∂x 2 ∂y 2 ∂r 2 ∂θ 2
4.
5.
(5)
a)
Find the extreme values of the function f ( x , y) = x 2 + y on the surface x 2 + 2 y 2 = 1.
b)
Find the value of ‘t’ for which
c)
Using the Implicit Function Theorem, show that there exists a differentiable function g defined in a neighbourhood of 1 such that g(1) = 2 and F(g(y), y) = 0 in a neighbourhood of (2, 1), where F is given by F( x , y) = x 5 + y 5 − 16 xy 3 − 1 = 0 Also find g ′( y) . (2)
a)
Check the continuity and differentiability of the function at (0, 0), where 2x 3 y , ( x , y) ≠ (0, 0) (6) f ( x , y) = x 2 + y 2 0, otherwise State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of R 2 . Verify the condition for the functions f and g, defined by
b)
sin 2 x + t sin x x3
is finite. Hence evaluate the limit
y−x x , g ( x , y) = . y+x y ∂ ( u , v, w ) Find the Jacobian at (1, 1, − 1) if ∂ ( x , y, z ) f ( x , y) =
6.
(3)
a)
(5) (3)
(4)
u = xy − xy 2 + xz 2 v = xyz + xy 2 − 1 w = x 2 + y 2 + z 2 − 2 xyz Further, deduce that the function F given by F( x , y, z) = ( xy − xy 2 + xz 2 , xyz + xy 2 − 1, x 2 + y 2 + z 2 − 2xyz) is locally invertible at (1, 1, − 1) . (5) b)
Check whether f yz or f zy exist at (0, 0, 0) for the function f, defined by 2x y + , y ≠ 0, z ≠ 0 f ( x , y, z) = y 3z 0 , otherwise
(5)
2
7.
a)
Evaluate
∫∫ x
3
y 3 dx dy , where D is the region bounded by
D
y 2 = 2 x , y 2 = 4 x , x 2 = 3y, x 2 = 6 y using the transformation u = b)
y2 x2 and v = . x y
Find the volume of the cylinder x 2 + y 2 = 16 lying between the planes y + z = 5 and z =0.
(5)
(5)
2x f ( x , y)dy dx and express the integral with the order 0 x2 (3) 2
8.
a)
Sketch the region of integration in
of integration changed. b) c)
∫ ∫
Find the moment of inertia of a solid sphere of radius 2 and of density given by the function ρ defined by ρ( x , y, z) = 10 , about the z-axis. (5) Use Green’s theorem to evaluate
∫ ( 3x
2
− 4 y)dx + (2x + y 3 )dy
C
where C is the ellipse 4 x 2 + 9 y 2 = 36 . 9.
a)
(2)
If z = f(x, y), x = ln u, y = ln v, show that ∂ 2z ∂2z = uv . ∂x ∂y ∂u ∂v
b)
Find the third Taylor polynomial of e x + y at (1, 0).
c)
Show that the function
(3) (2)
y x F = 2 xy + + ln y, x 2 + ln x + x y
10.
is conservative. Find f, such that F = ∇f .
(3)
d)
Verify Euler’s relation for the function f, defined by f ( x , y) = e x / y .
(2)
a)
Find the stationary points of the function f, defined by f ( x , y) = x 4 + y 4 − x 2 − y 2 + 1 , and check whether they are extreme points or not. (5)
b)
Roughly sketch the level surfaces of the following: i) ii) iii)
c)
f ( x , y, z ) = x 2 + z 2 h ( x , y, z) = yz g ( x , y, z ) = x + 3 y − 5z
Check whether lim
x →∞
(3)
sin x is equal to 1 or not. x
3
(2)
