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Question:

Evaluate $\oint_C(2x^2+3y^2)\,dx,$ where C is the curve given by

$x(t)=at^2,\quad y(t)=2at,\quad 0\leq t\leq 1.$

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Question:

Find the domain and range of the function f , defined by $f(x,y)=\frac{2xy}{x^2+y^2}.\text{Also}$

find two level curves of this function. Give a rough sketch of them

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Question:

Check the continuity and differentiability of the function at (0,0) where

$f(x,y)=\begin{cases}\frac{2x^3 y}{x^2+y^2},&(x,y)\neq(0,0)\\0,&\text{otherwise}\end{cases}$

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Question:

Check the local inevitability of the function f defined by $f(x,y)=(x^2-y^2,2xy)$ at ,(1 − .1) Find a domain for the function f in which f is invertible.

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Question:

Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that $g(1)=2\text{and}F(g(y),y)=0.$ in a neighbourhood of (,1,2) where

$F(x,y)=x^5+y^5-16xy^3-1=0$

defines the function F. Also find g′( y).

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Question:

State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . R²Verify this theorem for the functions f and g, defined by

$f(x,y)=\frac{y-x}{y+x},\quad g(x,y)=\frac{x}{y}.$

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Question:

Find the extreme values of the function

$f(x,y)=x^2+y\text{on the surface}x^2+2y^2=1.$

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Question:

State Green’s theorem, and apply it to evaluate

$\oint_C(3x^2-4y)\,dx-(2x+y^3)\,dy,$

Where C is the ellipse $4x^2+9y^2=36.$

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Question:

Find the mass of the solid bounded by $z=1\text{and}z=x^2+y^2,$ the density function being δ (z,y,x )= .|x|

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Question:

Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the

curves $y=4x^2\text{and}x=4.$

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Question:

$\text{Let}a=(1,2,3),b=(-5,3,-2),c=(2,-4,1)$ )1,4 be three points in . R³

Find |2 b − a + 3c l.

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Question:

$\text{Let}x=e^r\cos\theta,y=e^r\sin\theta\text{and}$ be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that

$\frac{\partial^2 f}{\partial r^2}+\frac{\partial^2 f}{\partial\theta^2}=(x^2+y^2)\left(\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}\right)$

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Question:

Let the function f be defined by

$f(x,y)=\begin{cases}\frac{3x^2 y^4}{x^4+y^8},&(x,y)\neq(0,0)\\0,&(x,y)=(0,0)\end{cases}$

Show that f has directional derivatives in all directions at.(0,0)

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Question:

Using only the definitions, find $f_{xy}(0,0)\text{and}f_{yx}(0,0),$ if they exists, for the function

$f(x,y)=\begin{cases}\frac{x^2 y}{\sqrt{x^2+y^2}},&(x,y)\neq(0,0)\\0,&\text{otherwise}\end{cases}$

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Question:

Find the third Taylor polynomial of the function $f(x,y)=1+5xy+3^2 y\text{at}(1,2).$

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Question:

Find the following limits:

(i) $\lim_{x\to+\infty}\left(\frac{x^2}{8x^2-3}\right)^{1/3}$

(ii) $\lim_{x\to 0^+}(\sin x)^{\sin x}$

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Question:

State whether the following statements are true or false. Give reasons for your answers.

(i) $\lim_{x\to 0}\frac{x^2\sin(\frac{1}{x})}{\sin x}=1$

(ii) A real-valued function of three variables which is continuous everywhere is differentiable.

(iii) The function $F:\mathbb{R}^2\to\mathbb{R}^2,\text{defined by}F(x,y)=(y+2,x+y),$ is locally invertible at any $\left(x,y\right)\epsilon R^{2}$

(iv) $f:[-1,1]\times[-2,2]\to\mathbb{R},\text{defined by}$

$f(x,y)=\begin{cases}x,&\text{if}y\text{is rational}\\0,&\text{if}y\text{is not rational}\end{cases}$ is integrable.

(v) The function $f:\mathbb{R}^2\to\mathbb{R},\text{defined by}f(x,y)=1-y^2+x^2,$ (.0,0)

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Question:

What are the recent innovations, trends and Issues in International Banking?

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Question:

List the purpose for which EEFC funds are utilized. Also, discuss the objectives of External Commercial Borrowing in Developing Nations.

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Question:

What are the various kinds of risk in Banking? Discuss with an example in detail.

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