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Recent questions and answers in Calculus
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1
GATE ECE 2020 | Question: 3
The partial derivative of the function $f(x, y, z) = e^{1-x\cos y} + xze^{-1/(1+y^{2})}$ with respect to $x$ at the point $(1,0,e)$ is $-1$ $0$ $1 \\$ $\dfrac{1}{e}$
go_editor
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in
Calculus
Feb 13, 2020
by
go_editor
1.9k
points
180
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gate2020-ec
calculus
derivatives
partial-derivatives
0
votes
0
answers
2
GATE ECE 2020 | Question: 51
For the solid $S$ shown below, the value of $\underset{S}{\iiint} xdxdydz$ (rounded off to two decimal places) is _______________.
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asked
in
Calculus
Feb 13, 2020
by
go_editor
1.9k
points
130
views
gate2020-ec
numerical-answers
calculus
definite-integrals
0
votes
0
answers
3
GATE ECE 2019 | Question: 16
The value of the contour integral $\frac{1}{2\pi j} \oint\left(z+\frac{1}{z}\right)^{2}dz$ evaluated over the unit circle $\mid z \mid=1$ is_______.
Arjun
asked
in
Calculus
Feb 12, 2019
by
Arjun
6.0k
points
88
views
gate2019-ec
numerical-answers
calculus
integrals
0
votes
0
answers
4
GATE ECE 2019 | Question: 19
The value of the integral $ \displaystyle{}\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.
Arjun
asked
in
Calculus
Feb 12, 2019
by
Arjun
6.0k
points
121
views
gate2019-ec
numerical-answers
calculus
definite-integrals
0
votes
0
answers
5
GATE ECE 2019 | Question: 26
Consider a differentiable function $f(x)$ on the set of real numbers, such that $f(-1)=0$ and $ \mid f’(x) \mid \leq 2.$ Given these conditions, which one of the following inequalities is necessarily true for all $x \in[-2,2]?$ $f(x)\leq \frac{1}{2} \mid x+1 \mid$ $f(x)\leq 2 \mid x+1 \mid $ $f(x)\leq \frac{1}{2} \mid x \mid$ $f(x)\leq 2 \mid x \mid$
Arjun
asked
in
Calculus
Feb 12, 2019
by
Arjun
6.0k
points
126
views
gate2019-ec
calculus
maxima-minima
0
votes
0
answers
6
GATE ECE 2019 | Question: 27
Consider the line integral $\int_{c} (xdy-ydx)$ the integral being taken in a counterclockwise direction over the closed curve $C$ that forms the boundary of the region $R$ shown in the figure below. The region $R$ is the area enclosed by the union of a $2 \times 3$ ... circle of radius $1$. The line integral evaluates to $6+ \dfrac{\pi}{2}$ $8+\pi$ $12+\pi$ $16+2\pi$
Arjun
asked
in
Calculus
Feb 12, 2019
by
Arjun
6.0k
points
239
views
gate2019-ec
integrals
calculus
0
votes
0
answers
7
GATE ECE 2016 Set 3 | Question: 4
The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1-x)}}$ is equal to _______
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in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
87
views
gate2016-ec-3
numerical-answers
calculus
definite-integrals
0
votes
0
answers
8
GATE ECE 2016 Set 2 | Question: 3
As $x$ varies from $-1$ to $+3$, which one of the following describes the behaviour of the function $f(x)=x^{3}-3x^{2}+1?$ $f(x)$ increases monotonically. $f(x)$ increases, then decreases and increases again. $f(x)$ decreases, then increases and decreases again. $f(x)$ increases and then decreases.
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Calculus
Mar 28, 2018
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Milicevic3306
15.8k
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50
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gate2016-ec-2
calculus
maxima-minima
0
votes
0
answers
9
GATE ECE 2016 Set 2 | Question: 4
How many distinct value of $x$ satisfy the equation $\sin(x)=x/2$, where $x$ is in radians? $1$ $2$ $3$ $4$ or more
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in
Calculus
Mar 28, 2018
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Milicevic3306
15.8k
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60
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gate2016-ec-2
calculus
functions
0
votes
0
answers
10
GATE ECE 2016 Set 1 | Question: 3
Given the following statements about a function $f: \Bbb R \rightarrow \Bbb R$, select the right option: P: If $f(x)$ is continuous at $x = x_0$ then it is also differentiable at $x = x_0$. Q: If $f(x)$ is continuous at $x = x_0$ then it may not be ... is false P is false, Q is true, R is true P is false, Q is true, R is false P is true, Q is false, R is true
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asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
49
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gate2016-ec-1
calculus
continuity-and-differentiability
0
votes
0
answers
11
GATE ECE 2016 Set 1 | Question: 26
The integral $\frac{1}{2\pi} \iint_D(x+y+10) \,dx\,dy$, where $D$ denotes the disc: $x^2+y^2\leq 4$,evaluates to _________
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in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
43
views
gate2016-ec-1
numerical-answers
calculus
definite-integrals
0
votes
0
answers
12
GATE ECE 2015 Set 3 | Question: 2
The contour on the $x-y$ plane, where the partial derivative of $x^{2} + y^{2}$ with respect to $y$ is equal to the partial derivative of $6y+4x$ with respect to $x$, is $y=2$ $x=2$ $x+y=4$ $x-y=0$
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in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
63
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gate2015-ec-3
calculus
derivatives
partial-derivatives
0
votes
0
answers
13
GATE ECE 2015 Set 3 | Question: 5
The value of $\displaystyle{}\sum_{n=0}^{\infty} n \left(\dfrac{1}{2}\right)^{n}$ is ________.
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asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
46
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gate2015-ec-3
numerical-answers
calculus
taylor-series
0
votes
0
answers
14
GATE ECE 2015 Set 2 | Question: 27
The value of the integral $\int_{-\infty}^{\infty} 12\cos(2\pi t) \dfrac{\sin(4\pi t)}{4 \pi t}dt$ is _________.
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asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
76
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gate2015-ec-2
numerical-answers
calculus
definite-integrals
0
votes
0
answers
15
GATE ECE 2015 Set 1 | Question: 2
A function $f(x)=1-x^2+x^3$ is defined in the closed interval $[-1,1]$. The value of $x$, in the open interval $(-1,1)$ for which the mean value theorem is satisfied, is $-1/2$ $-1/3$ $1/3$ $1/2$
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
41
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gate2015-ec-1
calculus
mean-value-theorem
0
votes
0
answers
16
GATE ECE 2015 Set 1 | Question: 28
Which one of the following graphs describes the function $f(x)=e^{-x}(x^2+x+1)$?
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
49
views
gate2015-ec-1
calculus
functions
0
votes
0
answers
17
GATE ECE 2015 Set 1 | Question: 29
The maximum area (in square units) of a rectangle whose vertices lie on the eclipse $x^2+4y^2=1$ is __________.
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asked
in
Calculus
Mar 28, 2018
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Milicevic3306
15.8k
points
49
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gate2015-ec-1
numerical-answers
calculus
definite-integrals
0
votes
0
answers
18
GATE ECE 2014 Set 4 | Question: 1
The series $\Sigma_{n=0}^{\infty} \frac{1}{n!}$ converges to $2 \text{ ln } 2$ $\sqrt{2}$ $2$ $e$
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in
Calculus
Mar 26, 2018
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Milicevic3306
15.8k
points
43
views
gate2014-ec-4
calculus
taylor-series
convergence
0
votes
0
answers
19
GATE ECE 2014 Set 4 | Question: 29
For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the triangle, the angle between the hypotenuse and the side is $12^{\circ}$ $36^{\circ}$ $60^{\circ}$ $45^{\circ}$
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Calculus
Mar 26, 2018
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Milicevic3306
15.8k
points
41
views
gate2014-ec-4
calculus
maxima-minima
0
votes
0
answers
20
GATE ECE 2014 Set 3 | Question: 1
The maximum value of the function $f(x) = \text{ln } (1+x) – x $ (where $x >-1$) occurs at $x=$_______.
Milicevic3306
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in
Calculus
Mar 26, 2018
by
Milicevic3306
15.8k
points
50
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gate2014-ec-3
calculus
maxima-minima
numerical-answers
0
votes
0
answers
21
GATE ECE 2014 Set 3 | Question: 26
The maximum value of $f(x)$= $2x^{3}$ – $9x^{2}$ + $12x – 3$ in the interval $0\leq x\leq 3$ is _______.
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Calculus
Mar 26, 2018
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Milicevic3306
15.8k
points
47
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gate2014-ec-3
calculus
maxima-minima
numerical-answers
0
votes
0
answers
22
GATE ECE 2014 Set 2 | Question: 3
For $0 \leq t < \infty ,$ the maximum value of the function $f(t)= e^{-t}-2e^{-2t}$ occurs at $t= log_{e}4$ $t= log_{e}2$ $t= 0$ $t= log_{e}8$
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Calculus
Mar 26, 2018
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Milicevic3306
15.8k
points
47
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gate2014-ec-2
calculus
maxima-minima
0
votes
0
answers
23
GATE ECE 2014 Set 2 | Question: 4
The value of $\lim_{x\rightarrow \infty }(1 +\tfrac{1}{x})^{x}$ is $\text{ln }2$ $1.0$ $e$ $\infty$
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Calculus
Mar 26, 2018
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Milicevic3306
15.8k
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43
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gate2014-ec-2
calculus
limits
0
votes
0
answers
24
GATE ECE 2014 Set 2 | Question: 45
The value of the integral $\int_{-\infty }^{\infty } \text{sinc}^{2}(5t) \: dt$ is _______.
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Calculus
Mar 26, 2018
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Milicevic3306
15.8k
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49
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gate2014-ec-2
numerical-answers
calculus
definite-integrals
0
votes
0
answers
25
GATE ECE 2014 Set 1 | Question: 26
The Taylor series expansion of $3\sin x + 2 \cos x$ is $2 + 3x-x^{2} – \frac{x^{3}}{2} + \dots$ $2 – 3x + x^{2} – \frac{x^{3}}{2} + \dots$ $2 + 3x + x^{2} + \frac{x^{3}}{2} + \dots$ $2 – 3x – x^{2} + \frac{x^{3}}{2} + \dots$
Milicevic3306
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Calculus
Mar 26, 2018
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Milicevic3306
15.8k
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46
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gate2014-ec-1
calculus
taylor-series
0
votes
0
answers
26
GATE ECE 2013 | Question: 20
A polynomial $f(x) = a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x - a_{0}$ with all coefficients positive has no real roots no negative real root odd number of real roots at least one positive and one negative real root
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Calculus
Mar 26, 2018
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Milicevic3306
15.8k
points
72
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gate2013-ec
calculus
polynomials
0
votes
0
answers
27
GATE ECE 2012 | Question: 46
The maximum value of $f(x)=x^3-9x^2+24x+5$ in the interval $[1,6]$ is $21$ $25$ $41$ $46$
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Calculus
Mar 25, 2018
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Milicevic3306
15.8k
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39
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gate2012-ec
calculus
maxima-minima
0
votes
0
answers
28
GATE ECE 2012 | Question: 25
If $x=\sqrt{-1}$, then the value of $x^x$ is $e^{\frac{-\pi}{2}}$ $e^{\frac{\pi}{2}}$ $x$ $1$
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Calculus
Mar 25, 2018
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Milicevic3306
15.8k
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42
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gate2012-ec
calculus
0
votes
0
answers
29
GATE ECE 2018 | Question: 55
Let $X\left[ k \right ] = k + 1,0\leq k\leq 7$ be $8$-point $\:\text{DFT}\:$ of a sequence $x[n]$. where $X\left [ k \right ]=\sum_{n=0}^{N-1}x \left [ n \right ]e^{-j2\pi nk/N}$. The value (correct to two decimal places) of $\sum_{n=0}^{3}x \left [ 2n \right ]$ is ________.
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gatecse
1.5k
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78
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gate2018-ec
numerical-answers
calculus
0
votes
0
answers
30
GATE ECE 2018 | Question: 52
Let $r=x^{2}+y-z$ and $z^{3}-xy+yz+y^{3}=1.$ Assume that $x$ and $y$ are independent variables. At $\left( x,y,z \right)=\left ( 2,-1,1 \right ),$ the value (correct to two decimal places) of $\dfrac{\partial r}{\partial x}$ is _________ .
gatecse
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Calculus
Feb 19, 2018
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gatecse
1.5k
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63
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gate2018-ec
numerical-answers
calculus
partial-derivatives
0
votes
0
answers
31
GATE ECE 2018 | Question: 24
Taylor series expansion of $f\left ( x \right )=\int ^{x}_{0}e^{-\left ( \frac{t^{2}}{2} \right )}dt$ around $x=0$ has the form $f\left ( x \right )={a}_{0}+a_{1}x+a_{2}x^{2}+...$ The coefficient $a_{2}$ (correct to two decimal places) is equal to ________.
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in
Calculus
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by
gatecse
1.5k
points
87
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gate2018-ec
numerical-answers
calculus
taylor-series
0
votes
0
answers
32
GATE ECE 2018 | Question: 6
Consider $p(s)=s^{3}+ a_{2}s^{2}+a_{1}s+a_{0}$ with all real coefficients. It is known that its derivatives ${p}'(s)$ has no real roots. The number of real roots of $p(s)$ is $0$ $1$ $2$ $3$
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Calculus
Feb 19, 2018
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gatecse
1.5k
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60
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gate2018-ec
calculus
derivatives
0
votes
0
answers
33
GATE ECE 2017 Set 2 | Question: 30
The minimum value of the function $f(x)=\frac{1}{3} x(x^2-3)$ in the interval $-100≤x≤100$ occurs at $x =$ ________.
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Calculus
Nov 23, 2017
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admin
36.1k
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83
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gate2017-ec-2
numerical-answers
calculus
maxima-minima
0
votes
0
answers
34
GATE ECE 2017 Set 2 | Question: 26
The values of the integrals $\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dy \right )dx$ and $\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dx \right )dy$ are same and equal to $0.5$ same and equal to $-0.5$ $0.5$ and $-0.5$, respectively $-0.5$ and $0.5$, respectively
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Calculus
Nov 23, 2017
by
admin
36.1k
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50
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gate2017-ec-2
calculus
definite-integrals
0
votes
0
answers
35
GATE ECE 2017 Set 1 | Question: 26
Let $f(x)=e^{x+x^{2}}$ for real $x$ . From among the following, choose the Taylor series approximation of $f(x)$ around $x=0$, which includes all powers of $x$ less than or equal to $3$. $1 + x + x^{2} + x^{3} $ $1 + x +\frac{3}{2} x^{2} + x^{3} $ $1 + x +\frac{3}{2} x^{2} + \frac{7}{6}x^{3} $ $1 + x +3 x^{2} + 7x^{3} $
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Calculus
Nov 17, 2017
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admin
36.1k
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64
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gate2017-ec-1
calculus
taylor-series
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Recent questions and answers in Calculus