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Recent questions tagged 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 $-1$ $0$ $1 \\$ $\dfrac{1}{e}$
jothee
asked
in
Calculus
Feb 13, 2020
by
jothee
1.9k
points
143
views
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 _______________.
jothee
asked
in
Calculus
Feb 13, 2020
by
jothee
1.9k
points
88
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
4.5k
points
69
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
4.5k
points
104
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
4.5k
points
103
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
4.5k
points
203
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 _______
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
68
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.
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
39
views
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
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
50
views
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
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
36
views
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 _________
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
28
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 $y=2$ $x=2$ $x+y=4$ $x-y=0$
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
47
views
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 ________.
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
32
views
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 _________.
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
54
views
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
28
views
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
36
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 __________.
Milicevic3306
asked
in
Calculus
Mar 28, 2018
by
Milicevic3306
15.8k
points
30
views
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 $2 \text{ ln } 2$ $\sqrt{2}$ $2$ $e$
Milicevic3306
asked
in
Calculus
Mar 26, 2018
by
Milicevic3306
15.8k
points
32
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 $12^{\circ}$ $36^{\circ}$ $60^{\circ}$ $45^{\circ}$
Milicevic3306
asked
in
Calculus
Mar 26, 2018
by
Milicevic3306
15.8k
points
28
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
asked
in
Calculus
Mar 26, 2018
by
Milicevic3306
15.8k
points
42
views
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 _______.
Milicevic3306
asked
in
Calculus
Mar 26, 2018
by
Milicevic3306
15.8k
points
34
views
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 $t= log_{e}4$ $t= log_{e}2$ $t= 0$ $t= log_{e}8$
Milicevic3306
asked
in
Calculus
Mar 26, 2018
by
Milicevic3306
15.8k
points
33
views
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 $\text{ln }2$ $1.0$ $e$ $\infty$
Milicevic3306
asked
in
Calculus
Mar 26, 2018
by
Milicevic3306
15.8k
points
28
views
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 _______.
Milicevic3306
asked
in
Calculus
Mar 26, 2018
by
Milicevic3306
15.8k
points
33
views
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
asked
in
Calculus
Mar 26, 2018
by
Milicevic3306
15.8k
points
34
views
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
Milicevic3306
asked
in
Calculus
Mar 26, 2018
by
Milicevic3306
15.8k
points
57
views
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 $21$ $25$ $41$ $46$
Milicevic3306
asked
in
Calculus
Mar 25, 2018
by
Milicevic3306
15.8k
points
27
views
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 $e^{\frac{-\pi}{2}}$ $e^{\frac{\pi}{2}}$ $x$ $1$
Milicevic3306
asked
in
Calculus
Mar 25, 2018
by
Milicevic3306
15.8k
points
36
views
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 ________.
gatecse
asked
in
Calculus
Feb 19, 2018
by
gatecse
1.5k
points
61
views
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
asked
in
Calculus
Feb 19, 2018
by
gatecse
1.5k
points
52
views
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 ________.
gatecse
asked
in
Calculus
Feb 19, 2018
by
gatecse
1.5k
points
65
views
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 $0$ $1$ $2$ $3$
gatecse
asked
in
Calculus
Feb 19, 2018
by
gatecse
1.5k
points
46
views
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 $x =$ ________.
admin
asked
in
Calculus
Nov 23, 2017
by
admin
2.8k
points
60
views
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
admin
asked
in
Calculus
Nov 23, 2017
by
admin
2.8k
points
38
views
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} $
admin
asked
in
Calculus
Nov 17, 2017
by
admin
2.8k
points
47
views
gate2017-ec-1
calculus
taylor-series
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