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Recent questions and answers in Calculus
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1
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ISI2016DCG69
Consider the differential equation $(x^{2}y^{2})\frac{\mathrm{d} y}{\mathrm{d} x}=2xy.$ Assuming $y=10$ for $x=0,$ its solution is $x^{2}+(y5)^{2}=25$ $x^{2}+y^{2}=100$ $(x5)^{2}+y^{2}=125$ $(x5)^{2}+(y5)^{2}=50$
answered
Mar 14
in
Calculus
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haralk10
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isi2016dcg
calculus
differentialequation
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0
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1
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2
ISI2016DCG68
The general solution of the differential equation $x+yx{y}'=0$ is (assuming $C$ as an arbitrary constant of integration) $y=x(\log x+C)$ $x=y(\log y+C)$ $y=x(\log y+C)$ $y=y(\log x+C)$
answered
Mar 14
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Calculus
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haralk10
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isi2016dcg
calculus
differentialequation
nongate
0
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1
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3
ISI2016DCG67
The general solution of the differential equation $2y{y}'x=0$ is (assuming $C$ as an arbitrary constant of integration) $x^{2}y^{2}=C$ $2x^{2}y^{2}=C$ $2y^{2}x^{2}=C$ $x^{2}+y^{2}=C$
answered
Mar 14
in
Calculus
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haralk10
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isi2016dcg
calculus
differentialequation
nongate
0
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1
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4
ISI2015DCG50
The piecewise linear function for the following graph is $f(x) = \begin{cases} = x, \: x \leq 2 \\ =4, \: 2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$ $f(x) = \begin{cases} = x2, \: x \leq 2 \\ =4, \: 2<x<3 \\ =x1, \: x \geq 3 \end{cases}$ ... $f(x) = \begin{cases} = 2x, \: x \leq 2 \\ =4, \: 2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$
answered
Mar 14
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Calculus
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haralk10
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28
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isi2015dcg
calculus
functions
0
votes
1
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5
ISI2014DCG48
If $x$ is real, the set of real values of $a$ for which the function $y=x^2ax+12a^2$ is always greater than zero is $ \frac{2}{3} < a \leq \frac{2}{3}$ $ \frac{2}{3} \leq a < \frac{2}{3}$ $ \frac{2}{3} < a < \frac{2}{3}$ None of these
answered
Mar 12
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Calculus
by
haralk10
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22
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isi2014dcg
calculus
functions
quadraticequations
0
votes
1
answer
6
TIFR2020A8
Consider a function $f:[0,1]\rightarrow [0,1]$ which is twice differentiable in $(0,1).$ Suppose it has exactly one global maximum and exactly global minimum inside $(0,1).$ What can you say about the behaviour of the first derivative $f'$ and and second derivative $f''$ on ... is zero at atleast one point $f'$ is zero at atleast two points, $f''$ is zero at atleast two points
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Mar 1
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Calculus
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ankitgupta.1729
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tifr2020
engineeringmathematics
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0
votes
1
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7
TIFR2020A13
What is the area of the largest rectangle that can be inscribed in a circle of radius $R$? $R^{2}/2$ $\pi \times R^{2}/2$ $R^{2}$ $2R^{2}$ None of the above
answered
Feb 19
in
Calculus
by
ankitgupta.1729
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tifr2020
+2
votes
2
answers
8
ISI2014DCG6
If $f(x)$ is a real valued function such that $2f(x)+3f(x)=154x$, for every $x \in \mathbb{R}$, then $f(2)$ is $15$ $22$ $11$ $0$
answered
Jan 29
in
Calculus
by
Akumar20
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35
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114
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isi2014dcg
calculus
functions
+1
vote
2
answers
9
MadeEasy Test Series: Calculus  Limits
How it solve this ?
answered
Jan 9
in
Calculus
by
Mayank Harbola
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45
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253
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madeeasytestseries
calculus
limits
engineeringmathematics
+1
vote
2
answers
10
GATE199302.6
The value of the double integral $\int^{1}_{0} \int_{0}^{\frac{1}{x}} \frac {x}{1+y^2} dxdy$ is_________.
answered
Dec 31, 2019
in
Calculus
by
Verma Ashish
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gate1993
calculus
integration
normal
+3
votes
5
answers
11
TIFR2019A15
Consider the matrix $A = \begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ What is $\lim_{n→\infty}$A^n$ ? $\begin{bmatrix} \ 0 & 0 & 0\\ 0& 0 ... $\text{The limit exists, but it is none of the above}$
answered
Dec 5, 2019
in
Calculus
by
severustux
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131
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519
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tifr2019
engineeringmathematics
calculus
limits
+22
votes
5
answers
12
GATE201436
If $\int \limits_0^{2 \pi} x \: \sin x dx=k\pi$, then the value of $k$ is equal to ______.
answered
Nov 30, 2019
in
Calculus
by
Lakshman Patel RJIT
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gate20143
calculus
integration
limits
numericalanswers
easy
+22
votes
4
answers
13
GATE2014347
The value of the integral given below is $\int \limits_0^{\pi} \: x^2 \: \cos x\:dx$ $2\pi$ $\pi$ $\pi$ $2\pi$
answered
Nov 30, 2019
in
Calculus
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Lakshman Patel RJIT
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gate20143
calculus
limits
integration
normal
+6
votes
3
answers
14
ISRO201349
What is the least value of the function $f(x) = 2x^{2}8x3$ in the interval $[0, 5]$? $15$ $7$ $11$ $3$
answered
Nov 28, 2019
in
Calculus
by
Lakshman Patel RJIT
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isro2013
maximaminima
+1
vote
1
answer
15
TIFR2011MathsA19
The derivative of the function $\int_{0}^{\sqrt{x}} e^{t^{2}}dt$ at $x = 1$ is $e^{1}$ .
answered
Nov 25, 2019
in
Calculus
by
seetal samal
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37
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188
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tifrmaths2011
calculus
differentiation
+1
vote
1
answer
16
TIFR2011MathsA21
Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.
answered
Nov 25, 2019
in
Calculus
by
seetal samal
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37
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120
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tifrmaths2011
continuity
+2
votes
2
answers
17
Integration
Solve the following $\int_{0}^{\infty}e^{x^2}x^4dx$
answered
Nov 22, 2019
in
Calculus
by
Lakshman Patel RJIT
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217
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engineeringmathematics
integration
calculus
+2
votes
1
answer
18
ISI2016MMA8
Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable with $g'(x^2)=x^3$ for all $x>0$ and $g(1) =1$. Then $g(4)$ equals $64/5$ $32/5$ $37/5$ $67/5$
answered
Nov 19, 2019
in
Calculus
by
`JEET
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19.7k
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61
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isi2016mmamma
calculus
differentiation
+1
vote
3
answers
19
ISI2017DCG23
A determinant is chosen at random from the set of all determinants of order $2$ with elements $0$ or $1$ only. The probability of choosing a nonzero determinant is $\frac{3}{16}$ $\frac{3}{8}$ $\frac{1}{4}$ none of these
answered
Nov 19, 2019
in
Calculus
by
chirudeepnamini
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5.3k
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69
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isi2017dcg
probability
determinants
+39
votes
4
answers
20
GATE20129
Consider the function $f(x) = \sin(x)$ in the interval $x =\left[\frac{\pi}{4},\frac{7\pi}{4}\right]$. The number and location(s) of the local minima of this function are One, at $\dfrac{\pi}{2}$ One, at $\dfrac{3\pi}{2}$ Two, at $\dfrac{\pi}{2}$ and $\dfrac{3\pi}{2}$ Two, at $\dfrac{\pi}{4}$ and $\dfrac{3\pi}{2}$
answered
Nov 13, 2019
in
Calculus
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Obafgkme
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67
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gate2012
calculus
maximaminima
normal
nielit
0
votes
1
answer
21
MadeEasy Workbook: Calculus  Maxima Minima
answered
Oct 31, 2019
in
Calculus
by
Kushagra गुप्ता
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engineeringmathematics
calculus
maximaminima
madeeasybooklet
+2
votes
2
answers
22
ISI2015MMA10
The value of the infinite product $P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^31}{n^3+1} \times \cdots \text{ is }$ $1$ $2/3$ $7/3$ none of the above
answered
Oct 25, 2019
in
Calculus
by
techbd123
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83
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isi2015mma
calculus
limits
0
votes
1
answer
23
ISI2017DCG6
Let $f(x) = \dfrac{x1}{x+1}, \: f^{k+1}(x)=f\left(f^k(x)\right)$ for all $k=1, 2, 3, \dots , 99$. Then $f^{100}(10)$ is $1$ $10$ $100$ $101$
answered
Oct 21, 2019
in
Calculus
by
`JEET
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56
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isi2017dcg
calculus
functions
+1
vote
1
answer
24
ISI2015MMA36
For nonnegative integers $m$, $n$ define a function as follows $f(m,n) = \begin{cases} n+1 & \text{ if } m=0 \\ f(m1, 1) & \text{ if } m \neq 0, n=0 \\ f(m1, f(m,n1)) & \text{ if } m \neq 0, n \neq 0 \end{cases}$ Then the value of $f(1,1)$ is $4$ $3$ $2$ $1$
answered
Oct 19, 2019
in
Calculus
by
chirudeepnamini
Loyal
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5.3k
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30
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isi2015mma
calculus
functions
nongate
+14
votes
3
answers
25
GATE19961.6
The formula used to compute an approximation for the second derivative of a function $f$ at a point $X_0$ is $\dfrac{f(x_0 +h) + f(x_0 – h)}{2}$ $\dfrac{f(x_0 +h)  f(x_0 – h)}{2h}$ $\dfrac{f(x_0 +h) + 2f(x_0) + f(x_0 – h)}{h^2}$ $\dfrac{f(x_0 +h)  2f(x_0) + f(x_0 – h)}{h^2}$
answered
Oct 18, 2019
in
Calculus
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neeraj_bhatt
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407
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1.7k
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gate1996
calculus
differentiation
normal
+1
vote
1
answer
26
ISI2014DCG13
Let the function $f(x)$ be defined as $f(x)=\mid x1 \mid + \mid x2 \:\mid$. Then which of the following statements is true? $f(x)$ is differentiable at $x=1$ $f(x)$ is differentiable at $x=2$ $f(x)$ is differentiable at $x=1$ but not at $x=2$ none of the above
answered
Oct 14, 2019
in
Calculus
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joshi_nitish
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isi2014dcg
calculus
differentiation
+2
votes
3
answers
27
ISI2014DCG7
If $f(x) = \dfrac{\sqrt{3} \sin x}{2+\cos x}$, then the range of $f(x)$ is the interval $[1 , \sqrt{3}{/2}]$ the interval $[\sqrt{3}{/2}, 1]$ the interval $[1, 1]$ none of these
answered
Oct 11, 2019
in
Calculus
by
Abhishek Kumar 40
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253
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74
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isi2014dcg
calculus
functions
range
0
votes
1
answer
28
ISI2014DCG29
If $f(x) = \sin \bigg( \dfrac{1}{x^2+1} \bigg),$ then $f(x)$ is continuous at $x=0$, but not differentiable at $x=0$ $f(x)$ is differentiable at $x=0$, and $f’(0) \neq 0$ $f(x)$ is differentiable at $x=0$, and $f’(0) = 0$ None of the above
answered
Oct 11, 2019
in
Calculus
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techbd123
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isi2014dcg
calculus
continuity
differentiation
+1
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1
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29
Mean Value Theorem
f(x) is a differentiable function that satisfies 5 ≤ f′(x) ≤ 14 for all x. Let a and b be the maximum and minimum values, respectively, that f(11)−f(3) can possibly have, then what is the value of a+b?
answered
Oct 10, 2019
in
Calculus
by
Nirmal Gaur
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1
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30
ISI2014DCG37
Let $f: \bigg( – \dfrac{\pi}{2}, \dfrac{\pi}{2} \bigg) \to \mathbb{R}$ be a continuous function, $f(x) \to +\infty$ as $x \to \dfrac{\pi^}{2}$ and $f(x) \to – \infty$ as $x \to \dfrac{\pi^+}{2}$. Which one of the following functions satisfies the above properties of $f(x)$? $\cos x$ $\tan x$ $\tan^{1} x$ $\sin x$
answered
Oct 8, 2019
in
Calculus
by
techbd123
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isi2014dcg
calculus
functions
limits
continuity
0
votes
1
answer
31
ISI2015MMA78
The value of $\displaystyle \lim_{n \to \infty} \left[ (n+1) \int_0^1 x^n \ln(1+x) dx \right]$ is $0$ $\ln 2$ $\ln 3$ $\infty$
answered
Oct 4, 2019
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Calculus
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`JEET
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isi2015mma
calculus
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definiteintegrals
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0
votes
1
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32
ISI2015MMA22
Let $a_n= \bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1 \frac{1}{\sqrt{n+1}} \bigg), \: \: n \geq1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$
answered
Oct 4, 2019
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Calculus
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`JEET
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29
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isi2015mma
calculus
limits
nongate
0
votes
1
answer
33
ISI2015MMA19
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k1) }{n}} \end{vmatrix}\:\:\:$ is $2$ $2e$ $2 \pi$ $2i$
answered
Oct 4, 2019
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Calculus
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`JEET
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isi2015mma
calculus
limits
nongate
+2
votes
1
answer
34
ISI2016DCG45
The value of $\underset{x \to 0}{\lim} \dfrac{\tan^{2}\:xx\:\tan\:x}{\sin\:x}$ is $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $0$ None of these
answered
Oct 2, 2019
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Calculus
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`JEET
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isi2016dcg
limits
+1
vote
1
answer
35
ISI2015MMA25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
answered
Oct 1, 2019
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Calculus
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`JEET
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isi2015mma
calculus
limits
nongate
+1
vote
1
answer
36
ISI2015MMA20
The limit $\underset{n \to \infty}{\lim} \left( 1 \frac{1}{n^2} \right) ^n$ equals $e^{1}$ $e^{1/2}$ $e^{2}$ $1$
answered
Oct 1, 2019
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Calculus
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`JEET
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isi2015mma
calculus
limits
nongate
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vote
1
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37
ISI2018DCG9
Let $f(x)=1+x+\dfrac{x^2}{2}+\dfrac{x^3}{3}...+\dfrac{x^{2018}}{2018}.$ Then $f’(1)$ is equal to $0$ $2017$ $2018$ $2019$
answered
Sep 30, 2019
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Calculus
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`JEET
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isi2018dcg
calculus
functions
differentiation
0
votes
1
answer
38
ISI2018DCG10
Let $f’(x)=4x^33x^2+2x+k,$ $f(0)=1$ and $f(1)=4.$ Then $f(x)$ is equal to $4x^43x^3+2x^2+x+1$ $x^4x^3+x^2+2x+1$ $x^4x^3+x^2+2(x+1)$ none of these
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Sep 30, 2019
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isi2018dcg
calculus
differentiation
polynomials
+3
votes
2
answers
39
ISI2014DCG4
$\underset{n \to \infty}{\lim} \dfrac{1}{n} \bigg( \dfrac{n}{n+1} + \dfrac{n}{n+2} + \cdots + \dfrac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
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Sep 30, 2019
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Calculus
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techbd123
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+4
votes
4
answers
40
ISI2014DCG3
$\underset{x \to \infty}{\lim} \left( \frac{3x1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
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Sep 29, 2019
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