Mathematics.

DIFFERENTIAL AND INTEGRAL CALCULUS.

1. y = x. ey .. y Yx

X=0

=0

dxy=ey + x dye.. d = oy = 1

X

dx2y 2 dxy e +x dxo y ey + x (dxy)2 e .. dx2y = 2 dx3y = 3 dx3y ey + 3 e1 (dxy)2 + 3 x dxy dx'y e

.. dx1 = o y = 27 + 6 + 12 + 3 + 6 + 9 + 1

2. If u = f (x y) = o be an =a in which y is considered a func

tion of x, then the first derived" is

(1) d(x) u + d(y) u. dxy = o

Now from (i)

=u, suppose then d(x) u, + d(y) u, dxy = o

d(x) u, = d(x)2 u + d(x) d(y) u dxy

+ d(y) u dx2y

and d(y) u, = d(x) d(y) u + d(y)2 u dxy .. d(x) u, + d(y) u, dxy

(2) = d(x)2 u + 2 d(x) d(y) u dxy + d(y)2 u (dx y )2

+d(y) u dx3y = 0 = u2 suppose) then d(x) ug + d(y) u。 dxy = o and from (2)

d(x) u。 = d(x)3 u + 2 d(x)o d(y) u dxy + 2 d(x) d(y) u dx2y

+d(x) d(y)2 u (dxy)2 + 2 d(y)2 u dx2y dxy

+d(x) d(y) u dx2y + d(y) u dx3y

and dy) ug = d(x)2 d(y) u + 2 d(x) d (y)2 u dxy

2

+d(y) u (dxy) + d(y) u dx2y

.. d(y) u。 dxy = d(x)2 d(y) u dxy

+ 2 d (x) d (y)o u (dxy)2 + d (y)3 u (dxy)3
+ d(y)2 u dx2y dxy

.. the third derived ="

0 = d(x) u2 + d(y) u。 dxy

= d(x)3 u + 3 d(x)2 d(y) u dxy

+ 3 d(x) d(y) u dx3y + 3 d(x) d (y)2 u (dxy)? +3d (y) u dx2y dxy + d(y)3 u (dxy)3

.. the at which the curve cuts the axis of x = 0.

Let AA' and BB' be the axes of an ellipse.

.. The radius of curvature is

{ 1 + (dxy)° } } { a*y? + b* x® }!

-dx3y

and

a2 b2

which is greatest when x and y = b, and least when x = a, Hence the portion b, a of the evolute belongs to the portion B A of the ellipse, a b to A B,' b a1 to B' A' and a, b, to A' B'

y = 0.

n

4. (1) The = to the curve is y + x + 1 = (1

As x increases to 1, y increases on the negative side and has two

- 2

values, and when x = 1, y =

When x is greater than 1, y is impossible.

ve and increases, y has two values, one +ve and
αly=α

When x is

when x

ve

1

15

and dx2 y = (1 — x)3 which

Y

M

-X

P

Let X'OX and YOY be the rect angular axes. Take O M=1 and M P, then the curve is represented in the figure, the brach O P is concave to the axis of x, and O Q and O T convex.

..

(2.) y3 = a x2

When x =o and = a, y = 0; as x increases from zero to a y increases and decreases on the positive side, and when x 7 a is ve, and increases as x increases and to infinity when x = infinity.

y

When x is negative, y is +

ve and increases as x increases, and

.. there are two branches at the origin cutting at 90°,

P