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Recent questions tagged calculus
Webpage for Calculus:
0
votes
1
answer
1
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
asked
Feb 10
in
Calculus
by
Lakshman Patel RJIT
Veteran
(
61.4k
points)

47
views
tifr2020
engineeringmathematics
calculus
maximaminima
+2
votes
1
answer
2
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 $0$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

142
views
isi2014dcg
calculus
limits
+4
votes
4
answers
3
ISI2014DCG3
$\underset{x \to \infty}{\lim} \left( \frac{3x1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

175
views
isi2014dcg
calculus
limits
+3
votes
2
answers
4
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 $\infty$ $0$ $\log_e 2$ $1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

141
views
isi2014dcg
calculus
limits
+2
votes
2
answers
5
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 $15$ $22$ $11$ $0$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

114
views
isi2014dcg
calculus
functions
+2
votes
3
answers
6
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
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

74
views
isi2014dcg
calculus
functions
range
+2
votes
1
answer
7
ISI2014DCG12
The integral $\int _0^{\frac{\pi}{2}} \frac{\sin^{50} x}{\sin^{50}x +\cos^{50}x} dx$ equals $\frac{3 \pi}{4}$ $\frac{\pi}{3}$ $\frac{\pi}{4}$ none of these
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

103
views
isi2014dcg
calculus
definiteintegrals
integration
+1
vote
1
answer
8
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
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

83
views
isi2014dcg
calculus
differentiation
+2
votes
1
answer
9
ISI2014DCG17
$\underset{x \to 2}{\lim} \dfrac{1}{1+e^{\frac{1}{x2}}}$ is $0$ $1/2$ $1$ nonexistent
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

49
views
isi2014dcg
calculus
limits
+2
votes
1
answer
10
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 $36$ $\infty$ $25$ $21$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

67
views
isi2014dcg
calculus
maximaminima
0
votes
1
answer
11
ISI2014DCG20
If $A(t)$ is the area of the region bounded by the curve $y=e^{\mid x \mid}$ and the portion of the $x$axis between $t$ and $t$, then $\underset{t \to \infty}{\lim} A(t)$ equals $0$ $1$ $2$ $4$
asked
Sep 23, 2019
in
Geometry
by
Arjun
Veteran
(
436k
points)

44
views
isi2014dcg
calculus
definiteintegrals
area
+1
vote
0
answers
12
ISI2014DCG21
Suppose that the function $h(x)$ is defined as $h(x)=g(f(x))$ where $g(x)$ is monotone increasing, $f(x)$ is concave, and $g’’(x)$ and $f’’(x)$ exist for all $x$. Then $h(x)$ is always concave always convex not necessarily concave None of these
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

41
views
isi2014dcg
calculus
functions
maximaminima
convexconcave
0
votes
1
answer
13
ISI2014DCG24
Let $f(x) = \dfrac{2x}{x1}, \: x \neq 1$. State which of the following statements is true. For all real $y$, there exists $x$ such that $f(x)=y$ For all real $y \neq 1$, there exists $x$ such that $f(x)=y$ For all real $y \neq 2$, there exists $x$ such that $f(x)=y$ None of the above is true
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

29
views
isi2014dcg
calculus
functions
+1
vote
1
answer
14
ISI2014DCG28
The area enclosed by the curve $\mid\: x \mid + \mid y \mid =1$ is $1$ $2$ $\sqrt{2}$ $4$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

33
views
isi2014dcg
calculus
areaunderthecurve
0
votes
1
answer
15
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
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

59
views
isi2014dcg
calculus
continuity
differentiation
+2
votes
0
answers
16
ISI2014DCG31
For real $\alpha$, the value of $\int_{\alpha}^{\alpha+1} [x]dx$, where $[x]$ denotes the largest integer less than or equal to $x$, is $\alpha$ $[\alpha]$ $1$ $\dfrac{[\alpha] + [\alpha +1]}{2}$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

47
views
isi2014dcg
calculus
integration
definiteintegrals
0
votes
0
answers
17
ISI2014DCG33
Let $f(x)$ be a continuous function from $[0,1]$ to $[0,1]$ satisfying the following properties. $f(0)=0$, $f(1)=1$, and $f(x_1)<f(x_2)$ for $x_1 < x_2$ with $0 < x_1, \: x_2<1$. Then the number of such functions is $0$ $1$ $2$ $\infty$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

44
views
isi2014dcg
calculus
functions
limits
+1
vote
1
answer
18
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 $f(x)$? $\cos x$ $\tan x$ $\tan^{1} x$ $\sin x$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

43
views
isi2014dcg
calculus
functions
limits
continuity
+1
vote
1
answer
19
ISI2014DCG39
The function $f(x) = x^{1/x}, \: x \neq 0$ has a minimum at $x=e$; a maximum at $x=e$; neither a maximum nor a minimum at $x=e$; None of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

61
views
isi2014dcg
maximaminima
calculus
0
votes
0
answers
20
ISI2014DCG42
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then $f$ has no local minima $f$ has no local maxima $f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd integers $k$ and local maxima at $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for even integers $k$ None of the above
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

34
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
21
ISI2014DCG43
Let $f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid 1, & \text{ if } x>0. \end{cases}$ Then $\underset{x \to a}{\lim} f(x)$ exists if $a=0$ for all $a \in R$ for all $a \neq 0$ only if $a=1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

31
views
isi2014dcg
calculus
functions
limits
0
votes
1
answer
22
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 $x=\pi/3$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

27
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
23
ISI2014DCG45
Which of the following is true? $\log(1+x) < x \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$ $\log(1+x) > x \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$ $\log(1+x) > x \frac{x^2}{2} + \frac{x^3}{3} \text{ for some } x>0$ $\log(1+x) < x \frac{x^2}{2} + \frac{x^3}{3} \text{ for some } x>0$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

39
views
isi2014dcg
calculus
functions
logarithms
0
votes
1
answer
24
ISI2014DCG46
The maximum value of the real valued function $f(x)=\cos x + \sin x$ is $2$ $1$ $0$ $\sqrt{2}$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

36
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
25
ISI2014DCG47
The value of the definite integral $\int_0^{\pi} \mid \frac{1}{2} + \cos x \mid dx$ is $\frac{\pi}{6} + \sqrt{3}$ $\frac{\pi}{6}  \sqrt{3}$ $0$ $\frac{1}{2}$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

37
views
isi2014dcg
calculus
integration
definiteintegrals
0
votes
1
answer
26
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
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

22
views
isi2014dcg
calculus
functions
quadraticequations
0
votes
1
answer
27
ISI2014DCG49
Let $f(x) = \dfrac{x}{(x1)(2x+3)}$, where $x>1$. Then the $4^{th}$ derivative of $f, \: f^{(4)} (x)$ is equal to $ \frac{24}{5} \bigg[ \frac{1}{(x1)^5}  \frac{48}{(2x+3)^5} \bigg]$ ... $\frac{64}{5} \bigg[ \frac{1}{(x1)^5} + \frac{48}{(2x+3)^5} \bigg]$
asked
Sep 23, 2019
in
Others
by
Arjun
Veteran
(
436k
points)

52
views
isi2014dcg
calculus
differentiation
functions
0
votes
0
answers
28
ISI2014DCG50
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1 \cos tx}$ is equal to $0$ $1$ $\infty$ $2$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

35
views
isi2014dcg
calculus
limits
0
votes
1
answer
29
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 $(0, \infty)$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

36
views
isi2014dcg
calculus
maximaminima
0
votes
0
answers
30
ISI2014DCG53
The value of the integral $\displaystyle{}\int_{1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$ $\frac{1}{2}$ $ – \frac{1}{2}$ $1$
asked
Sep 23, 2019
in
Calculus
by
Arjun
Veteran
(
436k
points)

56
views
isi2014dcg
calculus
integration
definiteintegrals
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