For the function $$f(x,y) = \frac{{xy}}{{{x^2} + {y^2}}},$$ the limit of $$f(x,y)$$ as $$(x,y) \to (0,0)$$ along the line $$y = - x$$ is

• A) 0
• B) 1
• C) $$- \frac{1}{2}$$
• D) $$\frac{1}{2}$$

$$\begin{gathered} {\text{Suppose in }}{\mathbb{R}^2}{\text{, }}f,{\text{ }}{f_x},{\text{ }}{f_y}{\text{ and }}{f_{xy}}{\text{ exist on neighborhood }}N{\text{ of }}\left( {{x_0},{y_0}} \right){\text{. Then }}{f_{yx}}\left( {{x_0},{y_0}} \right){\text{ exists, and}} \hfill \\\ {f_{yx}}\left( {{x_0},{y_0}} \right) = {f_{xy}}\left( {{x_0},{y_0}} \right){\text{ because if }}{f_{xy}}{\text{ is - - - - }}{\text{.}} \hfill \\\\ \end{gathered}$$

• A) continuous
• B) differntiable
• C) bounded
• D) partially differentiable

$${\text{If }}f{\text{ is continuous on a compact set }}S{\text{ in }}{\mathbb{R}^n},\,\,{\text{then }}f - - - - {\text{ on }}S.$$

• A) is also uniformly continuous
• B) All above are equally valid
• C) attains all its bounds
• D) is also defined on all the limit points of “S”

$$\begin{gathered} {\text{Let }}u{\text{ and }}v{\text{ be functions of two variables with continuous second - order partial derivatives in a region }}S{\text{. }} \hfill \\ {\text{Suppose that }}{u_x} = {v_y}{\text{ and }}{u_y} = - {v_x}{\text{ in }}S{\text{. Then, }}{u_{xx}} - {u_{yy}} - - - - . \hfill \\\ \end{gathered}$$

• A) =0
• B) =1
• C) none of these.
• D) >0

Let f be continuous on a region S in ${\mathbb{R}^n}$. Suppose that A and B are in S and _____________. Then $f\left( C \right) = u$ for some C in S.

• A) $$f(A){\rm{ }} < {\rm{ }}u{\rm{ }} < {\rm{ }}f(B)$$
• B) $$f(A){\rm{ }} \le {\rm{ }}u{\rm{ }} \le {\rm{ }}f(B)$$
• C) $$f(A){\rm{ }} > {\rm{ }}u{\rm{ }} > f(B)$$
• D) $$f(A){\rm{ }} \ge {\rm{ }}u{\rm{ }} \ge f(B)$$

If $$f(x,y)$$ is continuous at $$({x_0},{y_0})$$ and $${f_{yx}}({x_0},{y_0})$$ exists. Then,

• A) $${f_x}({x_0},{y_0}) = {f_y}({x_0},{y_0})$$
• B) $${f_{xx}}({x_0},{y_0}) = {f_{yy}}({x_0},{y_0})$$
• C) None of these
• D) $${f_{yx}}({x_0},{y_0}) = {f_{xy}}({x_0},{y_0})$$

$${\text{In }}{\mathbb{R}^n},{\text{ }}f\left( X \right) = \frac{1}{{\left| {X - {X_{_0}}} \right|}},and\mathop {\lim }\limits_{X \to {X_0}} f\left( X \right) = \infty ,{\text{ then }}\,f\left( X \right) > M > 0\,\, \Rightarrow {\text{ }}0 < \left| {X - {X_{_0}}} \right| < \delta = - - - - .$$

• A) $$\frac{1}{{\sqrt M }}$$
• B) $$\sqrt M$$
• C) $$M$$
• D) $$\frac{1}{M}\,$$

$${\text{In }}{\mathbb{R}^n}{\text{, }}\mathop {\lim }\limits_{X \to {X_0}} f\left( X \right) = - \infty ,\,\,{\text{if}}$$

• A) $$\mathop {\lim }\limits_{ - X \to {X_0}} f\left( X \right) = \infty$$
• B) $$\mathop {\lim }\limits_{X \to - {X_0}} f\left( X \right) = \infty$$
• C) $$\mathop {\lim }\limits_{X \to {X_0}} f\left( { - X} \right) = \infty$$
• D) $$\mathop {\lim }\limits_{X \to {X_0}} \left( { - f} \right)\left( X \right) = \infty$$

If $f\left( {x,y,z} \right) = \cos \left( {\frac{1}{{{x^2} + 2{y^2} + {z^2}}}} \right)$ then $\mathop {\lim }\limits_{\left| X \right| \to \infty } f\left( X \right) = \_\_\_\_\_$.

• A) infinite
• B) $1$
• C) $0$
• D) $-1$

$$\begin{gathered} {\text{In }}{\mathbb{R}^n},{\text{ monotonicity, limits inferior and superior of sequences, and }} \hfill \\ {\text{divergence to }} \pm \infty \,\,{\text{are undefined for }}n{\text{ }} > {\text{ }}1\,\,{\text{because }}{\mathbb{R}^n}{\text{ is - - - - - - - - }}{\text{.}} \hfill \\\ \end{gathered}$$

• A) $${\text{complete}}$$
• B) $${\text{a Field}}$$
• C) $${\text{not compact for }}n > 1$$
• D) $${\text{not ordered for }}n > 1$$

$$\begin{array}{*{20}{l}} {{\text{Let }}f{\text{ be defined on an interval }}I{\text{ in }}{\mathbb{R}^2}.\,{\text{Suppose that }}{x_1}{\text{ and }}{x_{\text{2}}}{\text{ are in }}I{\text{ and}}\,f\left( {{x_1}} \right){\text{ }} < {\text{ }}y{\text{ }} < {\text{ }}f\left( {{x_2}} \right){\text{. }}} \\\ {{\text{Then }}f\left( x \right) = y{\text{ for some }}x{\text{ in }}I.} \\\ {} \end{array}$$

• A) False
• B) True
• C)
• D)

Let the function $f(s) = \left\{ \begin{array}{l}\frac{{\sin s}}{s},s \ne 0\\1,s = 0\end{array} \right.$ is continuous over domain _____________.

• A) (c) $$(0,\infty )$$
• B) (b) $$( - \infty ,\infty )$$
• C) Both a & b
• D) (a) $$\mathbb{R}$$

If $f(s) = \sqrt s$ and $g(x,y) = 1 - {x^2} - {y^2}$ then domain of f and g are ______________________________.

• A) $${D_f} = [0,\infty ],{D_g} = {\mathbb{R}^2}$$
• B) $${D_f} = [0,\infty ),{D_g} = {\mathbb{R}^2}$$
• C) $${D_g} = ( - \infty ,\infty ),{D_f} = {\mathbb{R}^2}$$
• D) $${D_f} = [0,\infty ],{D_g} = \mathbb{R}$$

$${\text{If a function }}f{\text{ is continuous on a Compact set }}S{\text{ in }}{\mathbb{R}^n}{\text{, then f is - - - - - - - - on }}S{\text{.}}$$

• A) bounded above
• B) unbounded
• C) bounded
• D) bounded below

$$\begin{gathered} {\text{Let }}u{\text{ and }}v{\text{ be functions of two variables with continuous second - order partial derivatives in a region }}S{\text{. }} \hfill \\ {\text{Suppose that }}{u_x} = {v_y}{\text{ and }}{u_y} = - {v_x}{\text{ in }}S{\text{. Then, }}{v_{xx}} + {v_{yy}} - - - - . \hfill \\ \hfill \\\ \end{gathered}$$

• A) =1
• B) >0
• C) >0 but <1
• D) =0

$$\begin{gathered} {\text{In }}{\mathbb{R}^n},{\text{ if the first order partial derivative of a function }}f\left( X \right){\text{ exits at }}{X_0}{\text{, }} \hfill \\ {\text{then it is essentially continuous at }}{X_0}. \hfill \\\ \end{gathered}$$

• A) True
• B) False
• C)
• D)

$${\text{The function }}d{x_i}{\text{ gives the value at a point in }}{\mathbb{R}^n}{\text{as;}}$$

• A) $$d{x_i}\left( X \right) = {x_{i - 1}} + {x_{i + 1}}$$
• B) $$d{x_i}\left( X \right) = {x_i}$$
• C) $$d{x_i}\left( X \right) = {x_{i - 1}}$$
• D) $$d{x_i}\left( X \right) = {x_{i + 1}}$$

$${\text{In}}\,{\mathbb{R}^2}{\text{, }}f\left( X \right) = {\left| X \right|^2}{\text{ and }}\Phi {\text{ = }}\left( {\frac{1}{{\sqrt 2 }},\frac{1}{{\sqrt 2 }}} \right),{\text{ then }}\frac{{\partial f\left( X \right)}}{{\partial \Phi }} =$$

• A) $$\sqrt 2 \left( {{x_1} + {x_2}} \right)$$
• B) $$\frac{{{x_1} + {x_2}}}{2}$$
• C) $$\frac{{{x_1} + {x_2}}}{{\sqrt 2 }}$$
• D) $${x_1} + {x_2}$$

$${\text{How many }}third{\text{ order partial derivatives of }}g\left( {x,y} \right) = xy + {x^2}{y^3}{\text{ exist in }}{\mathbb{R}^3}?$$

• A) 3
• B) 8
• C) 6
• D) 9

A function f is ______________ on a subset S of its domain in ${\mathbb{R}^n}$ if for every $\varepsilon > 0$ there is a $\delta > 0$ such that $\left| {f(X) - f(X')} \right| < \varepsilon$ whenever $\left| {X - X'} \right| < \delta$ and $X,X' \in S$.

• A) Compact
• B) All of these
• C) Uniformly continuous
• D) Differentiable