\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {a,b} \right)} \frac{{\sin \sqrt {1 - {x^2} - 2{y^2}} }}{{\sqrt {1 - {x^2} - 2{y^2}} }} = 1,{\text{ if - - - - - - }}{\text{.}}

• A) \left( {{a^2} + 2{b^2}} \right)\mathop > \limits_ < 1
• B) \left( {{a^2} + 2{b^2}} \right) = 1
• C) \left( {{a^2} + 2{b^2}} \right) < 1
• D) \left( {{a^2} + 2{b^2}} \right) > 1

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) \[f(A){\rm{ }} \ge {\rm{ }}u{\rm{ }} \ge f(B)\]
• D) \[f(A){\rm{ }} > {\rm{ }}u{\rm{ }} > f(B)\]

\[{\text{In }}{\mathbb{R}^2},{\text{ }}\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {2,2} \right)} \frac{{\sin \left( {x - y} \right)}}{{\sqrt {x - y} }} = - - - - .\]

• A) \[0\,\]
• B) \[\infty \]
• C) \[1\]
• D) \[\sqrt 2 \]

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) Differentiable
• D) Uniformly continuous

\[\begin{gathered} {\text{If }}{z_0} = f\left( {{x_0},{y_0}} \right){\text{ then }}z\left( t \right) = f\left( {x + \phi t,y + \phi t} \right){\text{ represents a curve through }}\left( {{x_0},{y_0},{z_0}} \right){\text{ in the plane }} \hfill \\ {\text{determined by the unit vectors }} - - - - {\text{.}} \hfill \\\ \end{gathered} \]

• A) \[\hat \Phi {\text{ and reciprocal vector of }}\hat k\]
• B) \[\hat \Phi {\text{ and }}\hat j\]
• C) \[\hat \Phi {\text{ and }}\hat i\]
• D) \[\hat \Phi {\text{ and }}\hat k\]

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

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

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

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

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

\[\begin{gathered} {\text{If a function }}f{\text{ is }}continuous{\text{ on a }}compact{\text{ set }}S{\text{ in }}{\mathbb{R}^n}{\text{, and }}\alpha = \mathop {\inf }\limits_{X \in S} f\left( X \right),\,\beta = \mathop {\sup }\limits_{X \in S} f\left( X \right)\,, \hfill \\ {\text{then }}f\left( {{X_1}} \right){\text{ = }}\alpha {\text{ and }}f\left( {{X_2}} \right){\text{ = }}\beta \, - - - - - {X_1}{\text{ and }}{X_2}{\text{ in }}S. \hfill \\\ \end{gathered} \]

• A) for all
• B) for some
• C)
• D)

\begin{gathered} {\text{If }}{z_0} = f\left( {{x_0},{y_0}} \right){\text{ then }}z\left( t \right) = f\left( {x + \phi t,y + \phi t} \right){\text{ represents a curve through }}\left( {{x_0},{y_0},{z_0}} \right){\text{ in the plane }} \hfill \\ {\text{determined by the unit vectors }} - - - - {\text{.}} \hfill \\\ \end{gathered}

• A) \hat \Phi {\text{ and reciprocal vector of }}\hat k
• B) \hat \Phi {\text{ and }}\hat j
• C) \hat \Phi {\text{ and }}\hat k
• D) \hat \Phi {\text{ and }}\hat i

{\text{If }}f\left( t \right) = \sqrt t {\text{, }}g\left( {x,y} \right) = 1 - {x^2} - 2{y^2}{\text{, then the domain of }}f \circ g = {\text{ - - - - - - }}{\text{.}}

• A) \left\{ {\left( {x,y} \right):{x^2} - 2{y^2} \leqslant 1} \right\}
• B) \left\{ {\left( {x,y} \right):{x^2} - 2{y^2} \geqslant 1} \right\}
• C) \mathbb{R}
• D) {\mathbb{R}^2}

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

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

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

Which statement(s) is(are) true about the function \[f(x,y) = \frac{{{x^2} + {y^2}}}{{x - y}}\],

• A) \[{f_y}(x,y) = \frac{y}{{x - y}} + \frac{{{x^2} + {y^2}}}{{{{(x - y)}^2}}}.\]
• B) \[{f_y}(x,y),{\text{ }}{f_x}(x,y)\] are continuous everywhere.
• C) \[{f_x}(x,y) = \frac{{2x}}{{x - y}} - \frac{{{x^2} + {y^2}}}{{{{(x + y)}^2}}}\]
• D) \[f\] is differentiable every where except at the points where \[y = x\]

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

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

\[\begin{array}{*{20}{l}} {{\text{Let }}f{\text{ be defined and continuous on a region }}S{\text{ in }}{\mathbb{R}^{\text{n}}}.\,{\text{Suppose that }}{X_1}{\text{ and }}{X_{\text{2}}}{\text{ are in }}S{\text{ and}}\,f\left( {{X_1}} \right){\text{ }} < {\text{ Y }} < {\text{ }}f\left( {{X_2}} \right){\text{. }}} \\\\ {{\text{Then }}f\left( X \right) = Y{\text{ - - - - - - - }}X{\text{ in }}S.} \\\\ {} \end{array}\]

• A) for all
• B) for some
• C)
• D)

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

• A) False
• B) True
• C)
• D)

\[\begin{gathered} {\text{Analogous to the derivative of a function of one variable in }}{\mathbb{R}^2}{\text{, the directional derivative of a function}} \hfill \\ f{\text{ at }}{X_0}{\text{ in }}{\mathbb{R}^n}{\text{ has - - - - - - value(s)}}{\text{. }} \hfill \\\ \end{gathered} \]

• A) integral
• B) infinite many
• C) unique
• D) multiple

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

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

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