\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{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 \\\ \hfill \\\\ \end{gathered}

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

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

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

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

\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{not compact for }}n > 1
• C) {\text{not ordered for }}n > 1
• D) {\text{complete}}

\begin{gathered} {\text{The domain of the real valued function }}f:{\mathbb{R}^n} \to \mathbb{R}{\text{ defined and given by;}} \hfill \\ f\left( X \right){\text{ = }}{\left( {1 - x_1^2 - x_2^2 - \cdots - x_2^n} \right)^{ - 1}}{\text{ is - - - - - }}{\text{.}} \hfill \\\ \end{gathered}

• A) \left\{ {X|\left| X \right| = 1} \right\}
• B) \left\{ {X|\left| X \right| \ne 1} \right\}
• C)
• D)

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

\begin{gathered} {\text{In }}{\mathbb{R}^2},f\left( {x,y} \right){\text{ = }}\left\{ \begin{gathered} \frac{{\sin \sqrt {1 - {x^2} - 2{y^2}} }}{{\sqrt {1 - {x^2} - 2{y^2}} }},\,\,{x^2} + 2{y^2} < 1 \hfill \\ 1,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x^2} + 2{y^2} = 1 \hfill \\\ \end{gathered} \right. \hfill \\ {\text{is - - - - - - - - - - on }}{x^2} + 2{y^2} = 1{\text{ }}{\text{.}} \hfill \\\ \end{gathered}

• A) continuous
• B) discontinuous
• C)
• D)

\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 some
• B) for all
• C)
• D)

\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) >0 but <1
• B) =0
• C) =1
• D) >0

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

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

\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}