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Recent questions and answers in Calculus
+2
votes
5
answers
1
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
1 day
ago
in
Calculus
by
severustux
(
121
points)

385
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tifr2019
engineeringmathematics
calculus
limits
+19
votes
5
answers
2
GATE201436
If $\int \limits_0^{2 \pi} x \: \sin x dx=k\pi$, then the value of $k$ is equal to ______.
answered
5 days
ago
in
Calculus
by
Lakshman Patel RJIT
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54.7k
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2.8k
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gate20143
calculus
integration
limits
numericalanswers
easy
+17
votes
4
answers
3
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
5 days
ago
in
Calculus
by
Lakshman Patel RJIT
Veteran
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54.7k
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1.9k
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gate20143
calculus
limits
integration
normal
+4
votes
3
answers
4
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
in
Calculus
by
Lakshman Patel RJIT
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54.7k
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1.9k
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isro2013
maximaminima
+1
vote
1
answer
5
TIFR2011MathsA19
The derivative of the function $\int_{0}^{\sqrt{x}} e^{t^{2}}dt$ at $x = 1$ is $e^{1}$ .
answered
Nov 25
in
Calculus
by
seetal samal
(
37
points)

151
views
tifrmaths2011
calculus
differentiability
+1
vote
1
answer
6
TIFR2011MathsA21
Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.
answered
Nov 25
in
Calculus
by
seetal samal
(
37
points)

105
views
tifrmaths2011
continuity
+2
votes
2
answers
7
Integration
Solve the following $\int_{0}^{\infty}e^{x^2}x^4dx$
answered
Nov 22
in
Calculus
by
Lakshman Patel RJIT
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54.7k
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185
views
engineeringmathematics
integration
calculus
+2
votes
1
answer
8
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
in
Calculus
by
`JEET
Boss
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12.9k
points)

34
views
isi2016mmamma
calculus
differentiability
+35
votes
4
answers
9
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
in
Calculus
by
Obafgkme
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61
points)

4k
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gate2012
calculus
maximaminima
normal
nielit
0
votes
1
answer
10
MadeEasy Workbook: Calculus  Maxima Minima
answered
Oct 31
in
Calculus
by
Kushagra गुप्ता
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1.9k
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77
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engineeringmathematics
calculus
maximaminima
madeeasybooklet
+2
votes
2
answers
11
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
in
Calculus
by
techbd123
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(
3.1k
points)

35
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isi2015mma
calculus
limits
summation
series
nongate
0
votes
1
answer
12
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
in
Calculus
by
`JEET
Boss
(
12.9k
points)

35
views
isi2017dcg
calculus
functions
+1
vote
1
answer
13
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
in
Calculus
by
chirudeepnamini
Active
(
3.1k
points)

8
views
isi2015mma
calculus
functions
nongate
+12
votes
3
answers
14
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
in
Calculus
by
neeraj2681
(
149
points)

1.4k
views
gate1996
calculus
differentiability
normal
+1
vote
1
answer
15
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
in
Calculus
by
joshi_nitish
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31.2k
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38
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isi2014dcg
calculus
function
limitcontinuity
differentiable
+2
votes
3
answers
16
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
in
Calculus
by
Abhishek Kumar 40
(
253
points)

39
views
isi2014dcg
calculus
functions
range
0
votes
1
answer
17
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
in
Calculus
by
techbd123
Active
(
3.1k
points)

28
views
isi2014dcg
calculus
limits
continuitydifferentiability
+1
vote
1
answer
18
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
in
Calculus
by
Nirmal Gaur
Active
(
2.1k
points)

48
views
+1
vote
1
answer
19
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
in
Calculus
by
techbd123
Active
(
3.1k
points)

15
views
isi2014dcg
calculus
functions
limits
continuity
0
votes
1
answer
20
ISI2015MMA78
The value of $\underset{n \to \infty}{\lim} \bigg[ (n+1) \int_0^1 x^n \text{ln}(1+x) dx \bigg]$ is $0$ $\text{ln }2$ $\text{ln }3$ $\infty$
answered
Oct 4
in
Calculus
by
`JEET
Boss
(
12.9k
points)

8
views
isi2015mma
calculus
limits
definiteintegration
nongate
0
votes
1
answer
21
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
in
Calculus
by
`JEET
Boss
(
12.9k
points)

10
views
isi2015mma
calculus
limits
nongate
0
votes
1
answer
22
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
in
Calculus
by
`JEET
Boss
(
12.9k
points)

23
views
isi2015mma
calculus
limits
nongate
+2
votes
1
answer
23
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
in
Calculus
by
`JEET
Boss
(
12.9k
points)

32
views
isi2016dcg
limits
+1
vote
1
answer
24
ISI2014DCG5
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1\} \:\:\:\:\:\:\: B=\{(x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is nonempty none of the above
answered
Oct 1
in
Calculus
by
Sourajit25
Active
(
1.3k
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79
views
isi2014dcg
calculus
functions
sets
+1
vote
1
answer
25
ISI2015MMA25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \bigg( \frac{3x1}{3x+1} \bigg) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
answered
Oct 1
in
Calculus
by
`JEET
Boss
(
12.9k
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24
views
isi2015mma
calculus
limits
nongate
+1
vote
1
answer
26
ISI2015MMA20
The limit $\underset{n \to \infty}{\lim} \bigg( 1 \frac{1}{n^2} \bigg) ^n$ equals $e^{1}$ $e^{1/2}$ $e^{2}$ $1$
answered
Oct 1
in
Calculus
by
`JEET
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(
12.9k
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11
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isi2015mma
calculus
limits
nongate
+1
vote
1
answer
27
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
in
Calculus
by
`JEET
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(
12.9k
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39
views
isi2018dcg
calculus
functions
differentiation
0
votes
1
answer
28
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
answered
Sep 30
in
Calculus
by
`JEET
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12.9k
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12
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isi2018dcg
calculus
differentiation
polynomials
+3
votes
2
answers
29
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 $\infty$ $0$ $\log_e 2$ $1$
answered
Sep 30
in
Calculus
by
techbd123
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3.1k
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92
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isi2014dcg
calculus
limits
summation
series
+3
votes
4
answers
30
ISI2014DCG3
$\underset{x \to \infty}{\lim} \bigg( \frac{3x1}{3x+1} \bigg) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
answered
Sep 29
in
Calculus
by
techbd123
Active
(
3.1k
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127
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isi2014dcg
calculus
limits
+2
votes
1
answer
31
ISI2015DCG52
$\underset{x \to 1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$
answered
Sep 28
in
Calculus
by
`JEET
Boss
(
12.9k
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22
views
isi2015dcg
calculus
limits
0
votes
1
answer
32
ISI2016DCG49
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}1}{x^{\frac{1}{4}}1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these
answered
Sep 28
in
Calculus
by
`JEET
Boss
(
12.9k
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12
views
isi2016dcg
calculus
limits
+1
vote
1
answer
33
ISI2016DCG50
The domain of the function $\ln(3x^{2}4x+5)$ is set of positive real numbers set of real numbers set of negative real numbers set of real numbers larger than $5$
answered
Sep 28
in
Calculus
by
`JEET
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(
12.9k
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8
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isi2016dcg
calculus
functions
domain
logarithms
0
votes
1
answer
34
ISI2016DCG53
$\underset{x\rightarrow1}{\lim}\dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$
answered
Sep 28
in
Calculus
by
`JEET
Boss
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12.9k
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10
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isi2016dcg
calculus
limits
0
votes
1
answer
35
ISI2017DCG27
The limit of the sequence $\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \dots$ is $1$ $2$ $2\sqrt{2}$ $\infty$
answered
Sep 27
in
Calculus
by
lokendra14
(
339
points)

12
views
isi2017dcg
calculus
limits
0
votes
1
answer
36
ISI2014DCG51
The function $f(x)$ defined as $f(x)=x^36x^2+24x$, where $x$ is real, is strictly increasing strictly decreasing increasing in $( \infty, 0)$ and decreasing in $(0, \infty)$ decreasing in $( \infty, 0)$ and increasing in $(0, \infty)$
answered
Sep 27
in
Calculus
by
`JEET
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12.9k
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19
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isi2014dcg
calculus
maximaminima
+2
votes
1
answer
37
ISI2014DCG2
Let $a_n=\bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1 – \frac{1}{\sqrt{n+1}} \bigg), \: n \geq 1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$
answered
Sep 27
in
Calculus
by
`JEET
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12.9k
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91
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isi2014dcg
calculus
limits
+2
votes
1
answer
38
ISI2014DCG19
It is given that $e^a+e^b=10$ where $a$ and $b$ are real. Then the maximum value of $(e^a+e^b+e^{a+b}+1)$ is $36$ $\infty$ $25$ $21$
answered
Sep 27
in
Calculus
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`JEET
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12.9k
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25
views
isi2014dcg
calculus
maximaminima
maximumvalues
0
votes
1
answer
39
ISI2017DCG3
If $2f(x)3f(\frac{1}{x})=x^2 \: (x \neq0)$, then $f(2)$ is $\frac{2}{3}$ $ – \frac{3}{2}$ $ – \frac{7}{4}$ $\frac{5}{4}$
answered
Sep 26
in
Calculus
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`JEET
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12.9k
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12
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isi2017dcg
calculus
functions
0
votes
1
answer
40
ISI2014DCG44
The function $f(x)=\sin x(1+ \cos x)$ which is defined for all real values of $x$ has a maximum at $x= \pi /3$ has a maximum at $x= \pi$ has a minimum at $x= \pi /3$ has neither a maximum nor a minimum at $x=\pi/3$
answered
Sep 26
in
Calculus
by
`JEET
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12.9k
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8
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isi2014dcg
calculus
maximaminima
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