Multivariable Differentiation and Optimization - Calculus - Mathematics Previous Year Questions

1

2008 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2008

For \((x, y) \in \mathbb{R}^2\), let \( f(x, y) = \begin{cases} \frac{2xy}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \) Then

A

\(f_x\) and \(f_y\) exist at \((0,0)\), and \(f\) is continuous at \((0,0)\)

B

\(f_x\) and \(f_y\) exist at \((0,0)\), and \(f\) is discontinuous at \((0,0)\)

C

\(f_x\) and \(f_y\) do not exist at \((0,0)\), and \(f\) is continuous at \((0,0)\)

D

\(f_x\) and \(f_y\) do not exist at \((0,0)\), and \(f\) is discontinuous at \((0,0)\)

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2

2008 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2008

Let \(q_1, q_2, \cdots, q_n\) be the generalized coordinates and \(\dot{q}_1, \dot{q}_2, \cdots, \dot{q}_n\) be the generalized velocities in a conservative force field. If under a transformation \(\varphi\), the new coordinate system has the generalized coordinates \(Q_1, Q_2, \cdots, Q_n\) and velocities \(\dot{Q}_1, \dot{Q}_2, \cdots, \dot{Q}_n\). Then the equation \(\frac{\partial L}{\partial q_i} = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right)\) takes the form

A

\(\frac{\partial L}{\partial Q_k} = \varphi \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{Q}_k}\right)\)

B

\(\varphi \frac{\partial L}{\partial Q_k} = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{Q}_k}\right)\)

C

\(\frac{\partial L}{\partial Q_k} = -\varphi \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{Q}_k}\right)\)

D

\(\frac{\partial L}{\partial Q_k} = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{Q}_k}\right)\)

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3

2008 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2008

Let \(E = \{(x, y) \in \mathbb{R}^2 : |x| \le 1, |y| \le 1\}\). Define \(f: E \to \mathbb{R}\) by \(f(x, y) = \frac{x + y}{1 + x^2 + y^2}\). Then the range of \(f\) is a

A

connected open set

B

connected closed set

C

bounded open set

D

closed and unbounded set

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4

2009 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2009

The value of \( f \) at a local minimum in the rectangular region \( R=\left\{(x, y) \in \mathbb{R}^{2}:|x|<\frac{3}{2},|y|<\frac{3}{2}\right\} \) is

A

\(-2 \)

B

\(-3 \)

C

\(-\frac{7}{8} \)

D

\(0 \)

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5

2010 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2010

Let \( f(x,y) = \begin{cases} \frac{xy}{(x^2+y^2)^{3/2}} [1-\cos(x^2+y^2)], & (x,y) \neq (0,0) \\ k, & (x,y) = (0,0) \end{cases} \) Then the value of \( k \) for which \( f(x,y) \) is continuous at \( (0,0) \) is

A

0

B

\(\frac{1}{2} \)

C

1

D

\(\frac{3}{2} \)

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6

2011 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2011

If \( x, y \) and \( z \) are positive real numbers, then the minimum value of \( x^2 + 8y^2 + 27z^2 \) where \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1 \) is

A

108

B

216

C

405

D

1048

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7

2012 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2012

The maximum value of the function \( f(x,y,z) = xyz \) subject to the constraint
\( xy + yz + zx - a = 0, a > 0 \) is

A

\(a^{3/2} \)

B

\((a/3)^{3/2} \)

C

\((3/a)^{3/2} \)

D

\((3a/2)^{3/2} \)

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8

2015 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2015

Let \(D = \{(x,y) \in \mathbb{R}^2: 1 \le x \le 1000, 1 \le y \le 1000\}\). Define
\(f(x,y) = \frac{x y}{2} + \frac{500}{x} + \frac{500}{y}\).
Then the minimum value of \(f\) on \(D\) is equal to ______

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9

2015 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2015

Suppose that among all continuously differentiable functions \(y(x), x \in \mathbb{R}\), with \(y(0) = 0\) and \(y(1) = \frac{1}{2}\), the function \(y_0(x)\) minimizes the functional
\(\int_0^1 \left( e^{-y' - x} + (1+y) y' \right) dx\).
Then \(y_0\left(\frac{1}{2}\right)\) is equal to

A

0

B

\(\frac{1}{8}\)

C

\(\frac{1}{4}\)

D

\(\frac{1}{2}\)

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10

2016 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2016

Maximum {x + y : (x,y) ∈ B̄(0,1)} is equal to __________

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11

2016 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2016

For any \((x,y) \in \mathbb{R}^2 \setminus \overline{B(0,1)}\), let
\[ f(x,y) = \text{distance}\left((x,y), \overline{B(0,1)}\right) \\ = \inf\left\{\sqrt{(x-x_1)^2 + (y-y_1)^2} : (x_1,y_1) \in \overline{B(0,1)}\right\}. \]
Then, \(\|\nabla f(3,4)\|\| is equal to __________

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12

2017 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2017

If \(u(x, y) = 1 + x + y + f(xy)\), where \(f : \mathbb{R}^2 \to \mathbb{R}\) is a differentiable function, then \(u\) satisfies

A

\(x \frac{\partial u}{\partial x} - y \frac{\partial u}{\partial y} = x^2 - y^2\)

B

\(x \frac{\partial u}{\partial x} - y \frac{\partial u}{\partial y} = 0\)

C

\(x \frac{\partial u}{\partial x} - y \frac{\partial u}{\partial y} = x - y\)

D

\(y \frac{\partial u}{\partial x} - x \frac{\partial u}{\partial y} = x - y\)

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13

2017 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2017

Let \(f:\mathbb{R}^2\to\mathbb{R}\) be defined by \(f(x,y)=\begin{cases}\sin\left(\frac{y^2}{x}\right)\sqrt{x^2+y^2}, & x\neq 0,\\ 0, & x=0.\end{cases}\) Then, at \((0,0)\),

A

\(f\) is continuous and the directional derivative of \(f\) does NOT exist in some direction

B

\(f\) is NOT continuous and the directional derivatives of \(f\) exist in all directions

C

\(f\) is NOT differentiable and the directional derivatives of \(f\) exist in all directions

D

\(f\) is differentiable

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14

2017 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2017

Let \( X_1, X_2, \ldots, X_n \) (\( n \ge 2 \)) be a random sample from a \( N(\theta, \theta) \) population, where \( \theta > 0 \), and let \( W = \frac{1}{n} \sum_{i=1}^{n} X_i^2 \). Then the maximum likelihood estimator of \( \theta \) is

A

\(\frac{1}{2} + \frac{1}{2} \sqrt{1-4W} \)

B

\(\frac{1}{2} + \frac{1}{2} \sqrt{1+4W} \)

C

\(\frac{-1}{2} + \frac{1}{2} \sqrt{1-4W} \)

D

\(\frac{-1}{2} + \frac{1}{2} \sqrt{1+4W} \)

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15

2018 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2018

Let \(u(x, y, z) = x^2 - 2y + 4z^2\) for \((x, y, z) \in \mathbb{R}^3\). Then the directional derivative of \(u\) in the direction \(\frac{3}{5}\hat{i} - \frac{4}{5}\hat{k}\) at the point \((5, 1, 0)\) is ________.

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16

2019 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2019

Let \(f : \mathbb{R}^2 \to \mathbb{R}\) be defined by \[f(x,y) = x^5 - 2x^3 y - x^2 y + 2y^3.\] ( \(\mathbb{R}\) is the set of all real numbers and \(\mathbb{R}^2 = \{(x,y) : x,y \in \mathbb{R}\}\) ) Which one of the following statements is TRUE?

A

\(f\) has a local maximum at origin

B

\(f\) has a local minimum at origin

C

\(f\) has a saddle point at origin

D

The origin is not a critical point of \(f\)

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17

2019 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2019

Let \(g : \mathbb{R}^2 \to \mathbb{R}^2\) be a function defined by \(g(x,y) = (e^x \cos y, e^x \sin y)\) and \((a,b) = g \left( 1, \frac{\pi}{3} \right)\). ( \(\mathbb{R}\) is the set of all real numbers and \(\mathbb{R}^2 = \{(x,y) : x,y \in \mathbb{R}\}\) ) Which one of the following statements is TRUE?

A

\(g\) is injective

B

If \(h\) is the continuous inverse of \(g\), defined in some neighbourhood of \((a,b) \in \mathbb{R}^2\), such that \(h(a,b) = \left( 1, \frac{\pi}{3} \right)\), then the Jacobian of \(h\) at \((a,b)\) is \(e^2\)

C

If \(h\) is the continuous inverse of \(g\), defined in some neighbourhood of \((a,b) \in \mathbb{R}^2\), such that \(h(a,b) = \left( 1, \frac{\pi}{3} \right)\), then the Jacobian of \(h\) at \((a,b)\) is \(e^{-2}\)

D

\(g\) is surjective

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18

2019 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2019

The temperature \(T: \mathbb{R}^3 \setminus \{(0,0,0)\} \to \mathbb{R}\) at any point \(P(x,y,z)\) is inversely proportional to the square of the distance of \(P\) from the origin. If the value of the temperature \(T\) at the point \(R(0,0,1)\) is \(\sqrt{3}\), then the rate of change of \(T\) at the point \(Q(1,1,2)\) in the direction of \(\overrightarrow{QR}\) is equal to ______ (round off to 2 places of decimal). (\(\mathbb{R}\) is the set of all real numbers, \(\mathbb{R}^3 = \{(x,y,z): x,y,z \in \mathbb{R}\}\) and \(\mathbb{R}^3 \setminus \{(0,0,0)\}\) denotes \(\mathbb{R}^3\) excluding the origin)

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19

2019 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2019

Let \(f: \mathbb{R}^2 \to \mathbb{R}\) be defined by \(f(x,y) = \begin{cases} (x^2+y^2)\sin\left(\frac{1}{x^2+y^2}\right), & \text{if } (x,y) \neq (0,0) \\ 0, & \text{if } (x,y) = (0,0). \end{cases}\) Consider the following statements: I. The partial derivatives \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\) exist at \((0,0)\) but are unbounded in any neighbourhood of \((0,0)\). II. \(f\) is continuous but not differentiable at \((0,0)\). III. \(f\) is not continuous at \((0,0)\). IV. \(f\) is differentiable at \((0,0)\). (\(\mathbb{R}\) is the set of all real numbers and \(\mathbb{R}^2 = \{(x,y): x,y \in \mathbb{R}\}\)) Which of the above statements is/are TRUE? (A) I and II only (B) I and IV only (C) IV only (D) III only

A

I and II only

B

I and IV only

C

IV only

D

III only

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20

2020 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2020

If \((4,0)\) and \(\left(0, -\frac{1}{2}\right)\) are critical points of the function \( f(x,y) = 5 - (\alpha + \beta)x^2 + \beta y^2 + (\alpha + 1)y^3 + x^3, \) where \(\alpha, \beta \in \mathbb{R}\), then

A

\(\left(4, -\frac{1}{2}\right)\) is a point of local maxima of \(f\)

B

\(\left(4, -\frac{1}{2}\right)\) is a saddle point of \(f\)

C

\(\alpha = 4, \beta = 2\)

D

\(\left(4, -\frac{1}{2}\right)\) is a point of local minima of \(f\)

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