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Hot questions in Calculus
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TIFR ECE 2023 | Question: 10
Convolution between two functions $f(t)$ and $g(t)$ is defined as follows: $f(t) * g(t)=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau$ Let $u(t)$ be the unit-step function, i.e., $u(t)=1$ for $t \geq 0$ and $u(t)=0$ for $t<0$. What is $f(t) * g(t)$ ... $\frac{1}{2}(\exp (-t)+\sin (t)-2 \cos (t)) u(t)$ $\frac{1}{2}(\exp (-t)-\sin (t)+2 \cos (t)) u(t)$
Convolution between two functions $f(t)$ and $g(t)$ is defined as follows:$$f(t) * g(t)=\int_{-\infty}^{\infty} f(\tau) g(t-\tau) d \tau$$Let $u(t)$ be the unit-step func...
admin
46.4k
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131
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admin
asked
Mar 14, 2023
Calculus
tifrece2023
engineering-mathematics
calculus
+
–
1
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0
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2
TIFR ECE 2015 | Question: 2
Let $x[n]=a^{\lfloor n \mid}$, ( $a$ is real, $0<a<1$ ) and the discrete time Fourier transform $\text{(DTFT)}$ of $x[n]$ is given by $X(\omega)=\sum_{n=-\infty}^{\infty} x[n] e^{-j \omega n}$. Then the $\text{DTFT}$ ... zero only at one value of $\omega \in[-\pi, \pi]$ Its maximum value is larger than $1$ Its minimum value is less than $-1$ None of the above
Let $x[n]=a^{\lfloor n \mid}$, ( $a$ is real, $0<a<1$ ) and the discrete time Fourier transform $\text{(DTFT)}$ of $x[n]$ is given by $X(\omega)=\sum_{n=-\infty}^{\infty}...
admin
46.4k
points
99
views
admin
asked
Dec 15, 2022
Calculus
tifr2015
calculus
discrete-fourier-transform
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1
votes
0
answers
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TIFR ECE 2014 | Question: 6
Let $g:[0, \pi] \rightarrow \mathbb{R}$ be continuous and satisfy \[ \int_{0}^{\pi} g(x) \sin (n x) d x=0 \] for all integers $n \geq 2$. Then which of the following can you say about $g?$ $g$ must be identically zero. $g(\pi / 2)=1$. $g$ need not be identically zero. $g(\pi)=0$. None of the above.
Let $g:[0, \pi] \rightarrow \mathbb{R}$ be continuous and satisfy\[\int_{0}^{\pi} g(x) \sin (n x) d x=0\]for all integers $n \geq 2$. Then which of the following can you ...
admin
46.4k
points
95
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
definite-integrals
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1
votes
0
answers
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TIFR ECE 2014 | Question: 8
Consider a square pulse $g(t)$ of height $1$ and width $1$ centred at $1 / 2$. Define $f_{n}(t)=\frac{1}{n}\left(g(t) *^{n} g(t)\right),$ where $*^{n}$ stands for $n$-fold convolution. Let $f(t)=\lim _{n \rightarrow \infty} f_{n}(t)$. Then, which ... $\infty$. $f(t)$ has width $\infty$ and height $1$ . $f(t)$ has width $0$ and height $\infty$. None of the above.
Consider a square pulse $g(t)$ of height $1$ and width $1$ centred at $1 / 2$. Define $f_{n}(t)=\frac{1}{n}\left(g(t) *^{n} g(t)\right),$ where $*^{n}$ stands for $n$-fol...
admin
46.4k
points
93
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
limits
+
–
1
votes
0
answers
5
TIFR ECE 2014 | Question: 13
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y)$. Then which of the following is $\text{TRUE?}$ $f(x) \leq f(y)$ whenever ... $f$ and $g$ are both convex, then $\min \{f, g\}$ is also convex. For a random variable $X, E(f(X)) \geq f(E(X))$.
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y...
admin
46.4k
points
92
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
functions
+
–
1
votes
0
answers
6
TIFR ECE 2014 | Question: 2
Evaluate the limit \[ \lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} . \] $e$ $1$ $2^{\frac{1}{3}}$ $0$ None of the above
Evaluate the limit\[\lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} .\]$e$$1$$2^{\frac{1}{3}}$$0$None of the above
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46.4k
points
84
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
limits
+
–
1
votes
0
answers
7
TIFR ECE 2014 | Question: 20
What is \[ \lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ? \] $0$ $\pi / 2$ $1 / \sqrt{2}$ $2 / \pi$ None of the above.
What is\[\lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ?\]$0$$\pi / 2$$1 / \sqrt{2}$$2 / \pi$None of the above....
admin
46.4k
points
82
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
limits
+
–
1
votes
0
answers
8
TIFR ECE 2013 | Question: 20
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is convex if for $x, y \in \mathbb{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq \alpha f(x)+(1-\alpha) f(y)$. Which of the following is not convex: $x^{2}$ $x^{3}$ $x$ $x^{4}$ $\mathrm{e}^{x}$
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is convex if for $x, y \in \mathbb{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq \alpha f(x)+(1-\alpha) f(y)$.Which...
admin
46.4k
points
89
views
admin
asked
Dec 12, 2022
Calculus
tifr2013
calculus
functions
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–
1
votes
0
answers
9
TIFR ECE 2013 | Question: 19
Which of the following is true for polynomials defined over real numbers $\mathbb{R}$. Every odd degree polynomial has a real root. Every odd degree polynomial has at least one complex root. Every even degree polynomial has at least one complex root. Every even degree polynomial has a real root. None of the above
Which of the following is true for polynomials defined over real numbers $\mathbb{R}$.Every odd degree polynomial has a real root.Every odd degree polynomial has at least...
admin
46.4k
points
35
views
admin
asked
Dec 12, 2022
Calculus
tifr2013
calculus
polynomials
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–
1
votes
0
answers
10
TIFR ECE 2012 | Question: 1
The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is $\ln \left(1+e^{-1 / 4}\right)$ $\ln (5 / 3)$ $0$ $\ln \left(1+e^{2}\right)$ None of the above
The minimum value of $f(x)=\ln \left(1+\exp \left(x^{2}-3 x+2\right)\right)$ for $x \geq 0$, where $\ln (\cdot)$ denotes the natural logarithm, is$\ln \left(1+e^{-1 / 4}\...
admin
46.4k
points
114
views
admin
asked
Dec 8, 2022
Calculus
tifr2012
calculus
maxima-minima
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–
1
votes
0
answers
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TIFR ECE 2012 | Question: 2
Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then \[ \lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n} \] is $0$ $\infty$ $\alpha_{k}$ $\alpha_{1}$ $\max _{j}|\alpha_{j}|$
Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then\[\lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n}\]is$0$$\infty$$\...
admin
46.4k
points
86
views
admin
asked
Dec 8, 2022
Calculus
tifr2012
calculus
limits
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1
votes
0
answers
12
TIFR ECE 2011 | Question: 13
If $a_k$ is an increasing function of $k$, i.e. $a_1<a_2<\ldots<a_k \ldots$. Then which of the following is $\text{TRUE.}$ $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k}}=\infty$ ... . Either $(a)$ or $(b)$. $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{a_{k}}=0$. None of the above.
If $a_k$ is an increasing function of $k$, i.e. $a_1<a_2<\ldots<a_k \ldots$. Then which of the following is $\text{TRUE.}$$\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \fr...
admin
46.4k
points
105
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
limits
+
–
1
votes
0
answers
13
TIFR ECE 2011 | Question: 10
Let $f(x)=|x|$, for $x \in(-\infty, \infty)$. Then $f(x)$ is not continuous but differentiable. $f(x)$ is continuous and differentiable. $f(x)$ is continuous but not differentiable. $f(x)$ is neither continuous nor differentiable. None of the above.
Let $f(x)=|x|$, for $x \in(-\infty, \infty)$. Then$f(x)$ is not continuous but differentiable.$f(x)$ is continuous and differentiable.$f(x)$ is continuous but not differe...
admin
46.4k
points
98
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
continuity-and-differentiability
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–
1
votes
0
answers
14
TIFR ECE 2011 | Question: 8
Let $f(x, y)$ be a function in two variables $x, y$. Then which of the following is true $\max _{x} \min _{y} f(x, y) \leq \min _{y} \max _{x} f(x, y)$. $\max _{x} \min _{y} f(x, y) \geq \min _{y} \max _{x} f(x, y)$ ... $\max _{x} \min _{y} f(x, y)=\min _{y} \max _{x} f(x, y)+\min _{y} \min _{x} f(x, y)$. None of the above.
Let $f(x, y)$ be a function in two variables $x, y$. Then which of the following is true$\max _{x} \min _{y} f(x, y) \leq \min _{y} \max _{x} f(x, y)$.$\max _{x} \min _{y...
admin
46.4k
points
88
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
maxima-minima
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–
1
votes
0
answers
15
TIFR ECE 2011 | Question: 4
Let $\lim _{n \rightarrow \infty} x_{n}=x$. Then which of the following is $\text{TRUE.}$ There exists an $n_{0}$, such that for all $n>n_{0},\left|x_{n}-x\right|=0$. There exists an $n_{0}$ ... $n>n_{0},\left|\frac{x_{n}}{x}\right| \leq \epsilon$ for any $\epsilon>0$. None of the above.
Let $\lim _{n \rightarrow \infty} x_{n}=x$. Then which of the following is $\text{TRUE.}$There exists an $n_{0}$, such that for all $n>n_{0},\left|x_{n}-x\right|=0$.There...
admin
46.4k
points
70
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
limits
+
–
1
votes
0
answers
16
TIFR ECE 2010 | Question: 20
The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is $2 t$ $t^{2}$ $2 t+t e^{-t^{2} / 2}$ Does not have a simple closed form expression None of the above
The function $f(t)$ is a convolution of $t^{2}$ with $\exp \left(-t^{2} / 2\right) / \sqrt{2 \pi}$. Its derivative is$2 t$$t^{2}$$2 t+t e^{-t^{2} / 2}$Does not have a sim...
admin
46.4k
points
103
views
admin
asked
Nov 30, 2022
Calculus
tifr2010
calculus
derivatives
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–
1
votes
0
answers
17
TIFR ECE 2010 | Question: 2
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is $\alpha>1 /(2 \pi)$ $\alpha \geq 1 / 6$ $\alpha \leq 1 /(2 \pi)$ $\alpha=1 / 4$ None of the above
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is$\alpha>1 /(2 \pi)$$\alpha \geq 1 / 6$$\alpha \leq 1 /(2 \pi)$$\alpha=1 / 4$None of the above
admin
46.4k
points
93
views
admin
asked
Nov 30, 2022
Calculus
tifr2010
calculus
maxima-minima
+
–
1
votes
0
answers
18
TIFR ECE 2022 | Question: 13
Calculate the minimum value attained by the function \[\sin (\pi x)-\sqrt{2} \pi x^{2}\] for values of $x$ which lie in the interval $[0,1]$. $\frac{1}{\sqrt{2}}\left(1-\frac{\pi}{8}\right)$ $0$ $1-\frac{\pi}{2 \sqrt{2}}$ $-\frac{1}{\sqrt{2}}\left(1+\frac{9 \pi}{2}\right)$ $-\sqrt{2} \pi$
Calculate the minimum value attained by the function\[\sin (\pi x)-\sqrt{2} \pi x^{2}\]for values of $x$ which lie in the interval $[0,1]$.$\frac{1}{\sqrt{2}}\left(1-\fra...
admin
46.4k
points
98
views
admin
asked
Nov 30, 2022
Calculus
tifrece2022
calculus
maxima-minima
+
–
1
votes
0
answers
19
TIFR ECE 2010 | Question: 17
Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit \[ \lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n} \] is $0$ $\infty$ $a_{k}$ $a_{1}$ $\left(\sum_{i=1}^{k} a_{k}\right) / k$
Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit\[\lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n}\]is$0$$\infty$$a_{k}$$a_{1...
admin
46.4k
points
84
views
admin
asked
Nov 30, 2022
Calculus
tifr2010
calculus
limits
+
–
1
votes
0
answers
20
TIFR ECE 2022 | Question: 15
Consider the difference below for $m \geq 5$: \[\sum_{n=1}^{m-1} \frac{1}{(1+n)^{2}}-\int_{x=1}^{m} \frac{1}{(1+x)^{2}} d x .\] Which statement about the difference is $\text{TRUE}?$ It is positive for infinitely many $m \geq 5$ ... is positive for infinitely many $m$ It is positive for all $m \geq 5,$ and is decreasing as $m$ increases It is negative for all $m \geq 5$
Consider the difference below for $m \geq 5$:\[\sum_{n=1}^{m-1} \frac{1}{(1+n)^{2}}-\int_{x=1}^{m} \frac{1}{(1+x)^{2}} d x .\]Which statement about the difference is $\te...
admin
46.4k
points
96
views
admin
asked
Nov 30, 2022
Calculus
tifrece2022
calculus
definite-integrals
+
–
1
votes
0
answers
21
TIFR ECE 2021 | Question: 3
Consider the following statements: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=1$. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=1$. Which of the following is $\text{TRUE?}$ Only Statement $1$ ... $1$ and $3$ are correct. All of Statements $1, 2,$ and $3$ are correct. None of the three Statements $1,2,$ and $3$ are correct.
Consider the following statements:$\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$.$\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=1$.$\lim _{x \rightarrow 0} \frac{1-\cos x}...
admin
46.4k
points
94
views
admin
asked
Nov 30, 2022
Calculus
tifrece2021
calculus
limits
+
–
1
votes
0
answers
22
TIFR ECE 2021 | Question: 7
Consider the function \[f(y)=\int_{1}^{y} \frac{1}{1+x^{2}} d x-\log _{e}(1+y)\] where $\log _{e}(x)$ denotes the natural logarithm of $x$. Which of the following is true: The function $f(y)$ ... $y \geq 1$. The derivative of function $f(y)$ does not exist at $y=1$.
Consider the function\[f(y)=\int_{1}^{y} \frac{1}{1+x^{2}} d x-\log _{e}(1+y)\]where $\log _{e}(x)$ denotes the natural logarithm of $x$.Which of the following is true:Th...
admin
46.4k
points
93
views
admin
asked
Nov 30, 2022
Calculus
tifrece2021
calculus
definite-integrals
+
–
1
votes
0
answers
23
TIFR ECE 2017 | Question: 6
Let $a, b \in\{0,1\}$. Consider the following statements where $*$ is the $\text{AND}$ operator, $\oplus$ is $\text{EXCLUSIVE-OR,}$ and ${ }^{c}$ denotes the complement function. $\max \left\{a * b, b \oplus a^{\mathrm{c}}\right\}=1$ ... $\text{(iii)}$ only $\text{(iii)}$ and $\text{(iv)}$ only $\text{(iv)}$ and $\text{(i)}$ only None of the above
Let $a, b \in\{0,1\}$. Consider the following statements where $*$ is the $\text{AND}$ operator, $\oplus$ is $\text{EXCLUSIVE-OR,}$ and ${ }^{c}$ denotes the complement f...
admin
46.4k
points
127
views
admin
asked
Nov 29, 2022
Calculus
tifrece2017
calculus
functions
+
–
1
votes
0
answers
24
TIFR ECE 2022 | Question: 4
Evaluate the value of \[\max \left(x^{2}+(1-y)^{2}\right),\] where the maximisation above is over $x$ and $y$ such that $0 \leq x \leq y \leq 1$. $0$ $2$ $1 / 2$ $1 / 4$ $1$
Evaluate the value of\[\max \left(x^{2}+(1-y)^{2}\right),\]where the maximisation above is over $x$ and $y$ such that $0 \leq x \leq y \leq 1$.$0$$2$$1 / 2$$1 / 4$$1$
admin
46.4k
points
80
views
admin
asked
Nov 30, 2022
Calculus
tifrece2022
calculus
maxima-minima
+
–
1
votes
0
answers
25
TIFR ECE 2021 | Question: 10
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$. Let the real number $a_{1}^{*}$ be such that it solves the following optimization problem \[d_{1}=\min _{a_{1} \in \mathbb{R}}\left\|\vec{u}-a_{1} \vec{v}_{1}\right\|,\] where we denote the length ... $\left\|\vec{u}-\left(\vec{p}_{2}-\vec{p}_{1}\right)\right\|$ $0$
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$. Let the real number $a_{1}^{*}$ be such that it solves the following optimization problem\[d_{1}=\min _{a_...
admin
46.4k
points
79
views
admin
asked
Nov 30, 2022
Calculus
tifrece2021
vector-analysis
vector-in-planes
+
–
1
votes
0
answers
26
TIFR ECE 2022 | Question: 6
Consider a degree-$5$ polynomial function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$. If $f$ exhibits at least four local maxima, which of the following is necessarily true? (Note: A local maximum is a point where the function value is the maximum in a ... derivative of $f(x)$ is negative for some $x \in[0,100]$ $f$ has exactly $4$ local maxima None of the above
Consider a degree-$5$ polynomial function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$. If $f$ exhibits at least four local maxima, which of the following is necess...
admin
46.4k
points
77
views
admin
asked
Nov 30, 2022
Calculus
tifrece2022
calculus
maxima-minima
+
–
1
votes
0
answers
27
TIFR ECE 2018 | Question: 3
Let $\lim _{n \rightarrow \infty} f(n)=\infty$ and $\lim _{n \rightarrow \infty} g(n)=\infty$. Then which of the following is necessarily $\text{TRUE.}$ $\lim _{n \rightarrow \infty}|f(n)-g(n)|=\infty$ $\lim _{n \rightarrow \infty}|f(n)-g(n)|=0$ $\lim _{n \rightarrow \infty}|f(n) / g(n)|=\infty$ $\lim _{n \rightarrow \infty}|f(n) / g(n)|=1$ None of the above
Let $\lim _{n \rightarrow \infty} f(n)=\infty$ and $\lim _{n \rightarrow \infty} g(n)=\infty$. Then which of the following is necessarily $\text{TRUE.}$$\lim _{n \rightar...
admin
46.4k
points
119
views
admin
asked
Nov 29, 2022
Calculus
tifrece2018
calculus
limits
+
–
1
votes
0
answers
28
TIFR ECE 2018 | Question: 15
Consider real-valued continuous functions $f:[0,2] \rightarrow(-\infty, \infty)$ and let \[A=\int_{0}^{1}|f(x)| d x \quad \text { and } B=\int_{1}^{2}|f(x)| d x .\] Which of the following is $\text{TRUE}?$ There exists an $f$ so that \[A+B<\int_{0}^{2} f(x) ... such that $\int_{0}^{1} f(x) d x=3$ There does not exist an $f$ so that \[A+B \leq-\int_{0}^{2} f(x) d x\]
Consider real-valued continuous functions $f:[0,2] \rightarrow(-\infty, \infty)$ and let\[A=\int_{0}^{1}|f(x)| d x \quad \text { and } B=\int_{1}^{2}|f(x)| d x .\]Which o...
admin
46.4k
points
95
views
admin
asked
Nov 29, 2022
Calculus
tifrece2018
calculus
definite-integrals
+
–
1
votes
0
answers
29
TIFR ECE 2017 | Question: 14
Consider the positive integer sequence \[x_{n}=n^{50} e^{-(\log (n))^{3 / 2}}, \quad n=1,2,3, \ldots\] Which of the following statements is $\text{TRUE?}$ For every $M>0$, there exists an $n$ such that $x_{n}>M$ ... and then increases with $n \geq 1$ Sequence $\left\{x_{n}\right\}$ eventually converges to zero as $n \rightarrow \infty$ None of the above
Consider the positive integer sequence\[x_{n}=n^{50} e^{-(\log (n))^{3 / 2}}, \quad n=1,2,3, \ldots\]Which of the following statements is $\text{TRUE?}$For every $M>0$, t...
admin
46.4k
points
84
views
admin
asked
Nov 29, 2022
Calculus
tifrece2017
calculus
maxima-minima
+
–
1
votes
0
answers
30
TIFR ECE 2016 | Question: 1
Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible? $\int_{1}^{\infty} x f(x) \mathrm{d} x=\infty$ ... $\int_{1}^{\infty} \frac{f(x)}{1+\ln x} \mathrm{~d} x=\infty$ None of the above
Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible?$\int_{1}^{\infty} ...
admin
46.4k
points
82
views
admin
asked
Nov 29, 2022
Calculus
tifrece2016
calculus
definite-integrals
+
–
1
votes
0
answers
31
TIFR ECE 2020 | Question: 4
Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions that are continuous and differentiable. Consider the following statements: $\min \{f, g\}$ is continuous $\max \{f, g\}$ is continuous $\max \{f, g\}$ is differentiable Which ... is correct Only statement $2$ is correct Only statement $3$ is correct Only statements $1$ and $2$ are correct None of the above
Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions that are continuous and differentiable. Consider the following statements:$\min \{f, g\}$ is continuous$\ma...
admin
46.4k
points
36
views
admin
asked
Nov 30, 2022
Calculus
tifrece2020
calculus
continuity-and-differentiability
+
–
1
votes
0
answers
32
TIFR ECE 2018 | Question: 4
Consider \[f(x)=\frac{(x \log x+x)^{5}(1+2 / x)^{x}}{(x+1 / x)^{5}(\log x+1 / \log x)^{6}}\] What can we say about $\lim _{x \rightarrow \infty} f(x)$ ? The function $f(x)$ does not have a limit as $x \rightarrow \infty$ ... $\lim _{x \rightarrow \infty} f(x)=e^{1 / 2}$ $\lim _{x \rightarrow \infty} f(x)=0$ $\lim _{x \rightarrow \infty} f(x)=\infty$
Consider\[f(x)=\frac{(x \log x+x)^{5}(1+2 / x)^{x}}{(x+1 / x)^{5}(\log x+1 / \log x)^{6}}\]What can we say about $\lim _{x \rightarrow \infty} f(x)$ ?The function $f(x)$ ...
admin
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78
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Nov 29, 2022
Calculus
tifrece2018
calculus
limits
+
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1
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0
answers
33
TIFR ECE 2019 | Question: 5
Consider the function $f(x)=e^{x^{2}}-8 x^{2}$ for all $x$ on the real line. For how many distinct values of $x$ do we have $f(x)=0?$ $1$ $4$ $2$ $3$ $5$
Consider the function $f(x)=e^{x^{2}}-8 x^{2}$ for all $x$ on the real line. For how many distinct values of $x$ do we have $f(x)=0?$ $1$$4$$2$$3$$5$
admin
46.4k
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Nov 30, 2022
Calculus
tifrece2019
calculus
functions
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1
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0
answers
34
TIFR ECE 2019 | Question: 3
Consider a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f(x)=1$ if $x$ is rational, and $f(x)=1-\epsilon,$ where $0<\epsilon<1$, if $x$ is irrational. Which of the following is $\text{TRUE}?$ $\lim _{x \rightarrow \infty} f(x)=1$ ... $1-\epsilon$ $\max _{x \geq 1} f(x)=1$ None of the above
Consider a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f(x)=1$ if $x$ is rational, and $f(x)=1-\epsilon,$ where $0<\epsilon<1$, if $x$ is irrational. Which...
admin
46.4k
points
29
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admin
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Nov 30, 2022
Calculus
tifrece2019
calculus
limits
+
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1
votes
0
answers
35
TIFR ECE 2019 | Question: 4
Let $f(x)=\sqrt{x^{2}-4 x+4},$ for $x \in(-\infty, \infty)$. Here, $\sqrt{y}$ denotes the non-negative square root of $y$ when $y$ is non-negative. Then, which of the following is $\text{TRUE}?$ $f(x)$ is ... differentiable $f(x)$ is continuous and differentiable $f(x)$ is continuous but not differentiable $f(x)$ is neither continuous nor differentiable None of the above
Let $f(x)=\sqrt{x^{2}-4 x+4},$ for $x \in(-\infty, \infty)$. Here, $\sqrt{y}$ denotes the non-negative square root of $y$ when $y$ is non-negative. Then, which of the fol...
admin
46.4k
points
26
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admin
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Nov 30, 2022
Calculus
tifrece2019
calculus
continuity-and-differentiability
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1
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0
answers
36
GATE ECE 2010 | Question: 26
If $e^{y}=x^{\frac{1}{x}}$, then $y$ has a maximum at $x=e$ minimum at $x=e$ maximum at $x=e^{-1}$ minimum at $x=e^{-1}$
If $e^{y}=x^{\frac{1}{x}}$, then $y$ has amaximum at $x=e$minimum at $x=e$maximum at $x=e^{-1}$minimum at $x=e^{-1}$
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46.4k
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40
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Sep 15, 2022
Calculus
gate2010-ec
calculus
maxima-minima
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0
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0
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37
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 $-1$ $0$ $1 \\$ $\dfrac{1}{e}$
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
1.9k
points
297
views
go_editor
asked
Feb 13, 2020
Calculus
gate2020-ec
calculus
derivatives
partial-derivatives
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–
0
votes
0
answers
38
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 _______________.
For the solid $S$ shown below, the value of $\underset{S}{\iiint} xdxdydz$ (rounded off to two decimal places) is _______________.
go_editor
1.9k
points
257
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Feb 13, 2020
Calculus
gate2020-ec
numerical-answers
calculus
definite-integrals
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0
votes
0
answers
39
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 $6+ \dfrac{\pi}{2}$ $8+\pi$ $12+\pi$ $16+2\pi$
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 $...
Arjun
6.5k
points
354
views
Arjun
asked
Feb 12, 2019
Calculus
gate2019-ec
integrals
calculus
+
–
0
votes
0
answers
40
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 $ $f(x)\leq \frac{1}{2} \mid x \mid$ $f(x)\leq 2 \mid x \mid$
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 follow...
Arjun
6.5k
points
207
views
Arjun
asked
Feb 12, 2019
Calculus
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calculus
maxima-minima
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