Recent questions tagged calculus / GATE Overflow for GATE CSE

1 votes

2 answers

241

ISI2017-DCG-30
If $f(x)=e^{5x}$ and $h(x)=f’’(x)+2f’(x)+f(x)+2$ then $h(0)$ equals $38$ $8$ $4$ $0$

If $f(x)=e^{5x}$ and $h(x)=f’’(x)+2f’(x)+f(x)+2$ then $h(0)$ equals$38$$8$$4$$0$

gatecse

337 views

gatecse asked Sep 18, 2019

2 votes

1 answer

242

ISI2018-DCG-9
Let $f(x)=1+x+\dfrac{x^2}{2}+\dfrac{x^3}{3}...+\dfrac{x^{2018}}{2018}.$ Then $f’(1)$ is equal to $0$ $2017$ $2018$ $2019$

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$

gatecse

653 views

gatecse asked Sep 18, 2019

1 votes

1 answer

243

ISI2018-DCG-10
Let $f’(x)=4x^3-3x^2+2x+k,$ $f(0)=1$ and $f(1)=4.$ Then $f(x)$ is equal to $4x^4-3x^3+2x^2+x+1$ $x^4-x^3+x^2+2x+1$ $x^4-x^3+x^2+2(x+1)$ none of these

Let $f’(x)=4x^3-3x^2+2x+k,$ $f(0)=1$ and $f(1)=4.$ Then $f(x)$ is equal to$4x^4-3x^3+2x^2+x+1$$x^4-x^3+x^2+2x+1$$x^4-x^3+x^2+2(x+1)$none of these

gatecse

382 views

gatecse asked Sep 18, 2019

0 votes

1 answer

244

ISI2018-DCG-24
Let $[x]$ denote the largest integer less than or equal to $x.$ The number of points in the open interval $(1,3)$ in which the function $f(x)=a^{[x^2]},a\gt1$ is not differentiable, is $0$ $3$ $5$ $7$

Let $[x]$ denote the largest integer less than or equal to $x.$ The number of points in the open interval $(1,3)$ in which the function $f(x)=a^{[x^2]},a\gt1$ is not diff...

gatecse

406 views

gatecse asked Sep 18, 2019

0 votes

1 answer

245

ISI2018-DCG-28
Let $f(x)=e^{-\big( \frac{1}{x^2-3x+2} \big) };x\in \mathbb{R} \: \: \& x \notin \{1,2\}$. Let $a=\underset{n \to 1^+}{\lim}f(x)$ and $b=\underset{x \to 1^-}{\lim} f(x)$. Then $a=\infty, \: b=0$ $a=0, \: b=\infty$ $a=0, \: b=0$ $a=\infty, \: b=\infty$

Let $f(x)=e^{-\big( \frac{1}{x^2-3x+2} \big) };x\in \mathbb{R} \: \: \& x \notin \{1,2\}$. Let $a=\underset{n \to 1^+}{\lim}f(x)$ and $b=\underset{x \to 1^-}{\lim} f(x)$....

gatecse

306 views

gatecse asked Sep 18, 2019

0 votes

0 answers

246

ISI2018-DCG-29
Let $f(x)=(x-1)(x-2)(x-3)g(x); \: x\in \mathbb{R}$ where $g$ is twice differentiable function. Then there exists $y\in(1,3)$ such that $f’’(y)=0.$ there exists $y\in(1,2)$ such that $f’’(y)=0.$ there exists $y\in(2,3)$ such that $f’’(y)=0.$ none of the above is true.

Let $f(x)=(x-1)(x-2)(x-3)g(x); \: x\in \mathbb{R}$ where $g$ is twice differentiable function. Thenthere exists $y\in(1,3)$ such that $f’’(y)=0.$there exists $y\in(1,...

gatecse

350 views

gatecse asked Sep 18, 2019

0 votes

1 answer

247

Maths: Limits
$\LARGE \lim_{n \rightarrow \infty} \frac{n^{\frac{3}{4}}}{log^9 n}$

$\LARGE \lim_{n \rightarrow \infty} \frac{n^{\frac{3}{4}}}{log^9 n}$

Mk Utkarsh

592 views

Mk Utkarsh asked May 26, 2019

1 votes

4 answers

248

ISI2018-MMA-28
Consider the following functions $f(x)=\begin{cases} 1, & \text{if } \mid x \mid \leq 1 \\ 0, & \text{if } \mid x \mid >1 \end{cases}.$ ... at $\pm1$ $h_2$ is continuous everywhere and $h_1$ has discontinuity at $\pm2$ $h_1$ has discontinuity at $\pm 2$ and $h_2$ has discontinuity at $\pm1$.

Consider the following functions$f(x)=\begin{cases} 1, & \text{if } \mid x \mid \leq 1 \\ 0, & \text{if } \mid x \mid >1 \end{cases}.$ and $g(x)=\begin{cases} 1, & \te...

akash.dinkar12

1.2k views

akash.dinkar12 asked May 11, 2019

0 votes

1 answer

249

ISI2018-MMA-30
Consider the function $f(x)=\bigg(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots+\frac{x^n}{n!}\bigg)e^{-x}$, where $n\geq4$ is a positive integer. Which of the following statements is correct? $f$ has no local maximum For every $n$, $f$ has a local maximum at $x = 0$ ... at $x = 0$ when $n$ is even $f$ has no local extremum if $n$ is even and has a local maximum at $x = 0$ when $n$ is odd.

Consider the function$f(x)=\bigg(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots+\frac{x^n}{n!}\bigg)e^{-x}$,where $n\geq4$ is a positive integer. Which of the following statemen...

akash.dinkar12

1.1k views

akash.dinkar12 asked May 11, 2019

2 votes

1 answer

250

ISI2018-MMA-29
Let $f$ be a continuous function with $f(1) = 1$. Define $F(t)=\int_{t}^{t^2}f(x)dx$. The value of $F’(1)$ is $-2$ $-1$ $1$ $2$

Let $f$ be a continuous function with $f(1) = 1$. Define $$F(t)=\int_{t}^{t^2}f(x)dx$$.The value of $F’(1)$ is$-2$$-1$$1$$2$

akash.dinkar12

1.1k views

akash.dinkar12 asked May 11, 2019

0 votes

1 answer

251

ISI2018-MMA-19
Let $X_1,X_2, . . . ,X_n$ be independent and identically distributed with $P(X_i = 1) = P(X_i = −1) = p\ $and$ P(X_i = 0) = 1 − 2p$ for all $i = 1, 2, . . . , n.$ ... $a_n \rightarrow p, b_n \rightarrow p,c_n \rightarrow 1-2p$ $a_n \rightarrow1/2, b_n \rightarrow1/2,c_n \rightarrow0$ $a_n \rightarrow0, b_n \rightarrow0,c_n \rightarrow1$

Let $X_1,X_2, . . . ,X_n$ be independent and identically distributed with $P(X_i = 1) = P(X_i = −1) = p\ $and$ P(X_i = 0) = 1 − 2p$ for all $i = 1, 2, . . . , n.$ Def...

akash.dinkar12

712 views

akash.dinkar12 asked May 11, 2019

2 votes

1 answer

252

ISI2019-MMA-29
Let $\psi : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with $\psi(y) =0$ for all $y \notin [0,1]$ and $\int_{0}^{1} \psi(y) dy=1$. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function. Then the value of $\lim _{n\rightarrow \infty}n \int_{0}^{100} f(x)\psi(nx)dx$ is $f(0)$ $f’(0)$ $f’’(0)$ $f(100)$

Let $\psi : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with $\psi(y) =0$ for all $y \notin [0,1]$ and $\int_{0}^{1} \psi(y) dy=1$. Let $f:\mathbb{R} \rig...

Sayan Bose

2.0k views

Sayan Bose asked May 7, 2019

1 votes

1 answer

253

ISI2019-MMA-28
Consider the functions $f,g:[0,1] \rightarrow [0,1]$ given by $f(x)=\frac{1}{2}x(x+1) \text{ and } g(x)=\frac{1}{2}x^2(x+1).$ Then the area enclosed between the graphs of $f^{-1}$ and $g^{-1}$ is $1/4$ $1/6$ $1/8$ $1/24$

Consider the functions $f,g:[0,1] \rightarrow [0,1]$ given by$$f(x)=\frac{1}{2}x(x+1) \text{ and } g(x)=\frac{1}{2}x^2(x+1).$$Then the area enclosed between the graphs of...

Sayan Bose

2.1k views

Sayan Bose asked May 7, 2019

0 votes

1 answer

254

ISI2019-MMA-25
Let $a,b,c$ be non-zero real numbers such that $\int_{0}^{1} (1 + \cos^8x)(ax^2 + bx +c)dx = \int_{0}^{2}(1+ \cos^8x)(ax^2 + bx + c) dx =0$ Then the quadratic equation $ax^2 + bx +c=0$ has no roots in $(0,2)$ one root in $(0,2)$ and one root outside this interval one repeated root in $(0,2)$ two distinct real roots in $(0,2)$

Let $a,b,c$ be non-zero real numbers such that $\int_{0}^{1} (1 + \cos^8x)(ax^2 + bx +c)dx = \int_{0}^{2}(1+ \cos^8x)(ax^2 + bx + c) dx =0$Then the quadratic equation $ax...

Sayan Bose

1.2k views

Sayan Bose asked May 7, 2019

1 votes

2 answers

255

ISI2019-MMA-24
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim _{n\rightarrow \infty} f^n(x)$ exists for every $x \in \mathbb{R}$, where $f^n(x) = f \circ f^{n-1}(x)$ for $n \geq 2$ ... $S \subset T$ $T \subset S$ $S = T$ None of the above

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim _{n\rightarrow \infty} f^n(x)$ exists for every $x \in \mathbb{R}$, where $f^n(x) = f ...

Sayan Bose

1.6k views

Sayan Bose asked May 7, 2019

1 votes

1 answer

256

ISI2019-MMA-18
For the differential equation $\frac{dy}{dx} + xe^{-y}+2x=0$ It is given that $y=0$ when $x=0$. When $x=1$, $\:y$ is given by $\text{ln} \bigg(\frac{3}{2e} – \frac{1}{2} \bigg)$ $\text{ln} \bigg(\frac{3e}{2} – \frac{1}{4} \bigg)$ $\text{ln} \bigg(\frac{3}{e} – \frac{1}{2} \bigg)$ $\text{ln} \bigg(\frac{3}{2e} – \frac{1}{4} \bigg)$

For the differential equation $$\frac{dy}{dx} + xe^{-y}+2x=0$$It is given that $y=0$ when $x=0$. When $x=1$, $\:y$ is given by$\text{ln} \bigg(\frac{3}{2e} – \frac{1}{...

Sayan Bose

4.5k views

Sayan Bose asked May 6, 2019

0 votes

1 answer

257

ISI2019-MMA-6
The solution of the differential equation $\frac{dy}{dx} = \frac{2xy}{x^2-y^2}$ is $x^2 + y^2 = cy$, where $c$ is a constant $x^2 + y^2 = cx$, where $c$ is a constant $x^2 – y^2 = cy$ , where $c$ is a constant $x^2 - y^2 = cx$, where $c$ is a constant

The solution of the differential equation $$\frac{dy}{dx} = \frac{2xy}{x^2-y^2}$$is$x^2 + y^2 = cy$, where $c$ is a constant$x^2 + y^2 = cx$, where $c$ is a constant$x^2 ...

Sayan Bose

1.1k views

Sayan Bose asked May 6, 2019

0 votes

1 answer

258

ISI2019-MMA-5
If $f(a)=2, \: f’(a) = 1, \: g(a) =-1$ and $g’(a) =2$, then the value of $\lim _{x\rightarrow a}\frac{g(x) f(a) – f(x) g(a)}{x-a}$ is $-5$ $-3$ $3$ $5$

If $f(a)=2, \: f’(a) = 1, \: g(a) =-1$ and $g’(a) =2$, then the value of $$\lim _{x\rightarrow a}\frac{g(x) f(a) – f(x) g(a)}{x-...

Sayan Bose

778 views

Sayan Bose asked May 6, 2019

0 votes

1 answer

259

Gate 2002 - ME
Which of the following functions is not differentiable in the domain $[-1,1]$ ? (a) $f(x) = x^2$ (b) $f(x) = x-1$ (c) $f(x) = 2$ (d) $f(x) = Maximum (x,-x)$

Which of the following functions is not differentiable in the domain $[-1,1]$ ?(a) $f(x) = x^2$(b) $f(x) = x-1$(c) $f(x) = 2$(d) $f(x) = Maximum (x,-x)$

balchandar reddy san

2.7k views

balchandar reddy san asked May 4, 2019

2 votes

1 answer

260

ISI-MMA 2019 Sample Questions-23
For $n \geq1$, Let $a_{n} = \frac{1}{2^{2}} + \frac{2}{3^{2}} +.....+ \frac{n}{(n+1)^{2}}$ and $b_{n} = c_{0} + c_{1}r + c_{2}r^{2}+.....+c_{n}r^{n},$ where$|c_{k}| \leq M$ for all integers $k$ ... not a Cauchy sequence (C) $\{a_n\}$ is not a Cauchy sequence but $\{b_n\}$ is a Cauchy sequence (D) neither $\{a_n\}$ nor $\{b_n\}$ is a Cauchy sequence.

For $n \geq1$, Let$a_{n} = \frac{1}{2^{2}} + \frac{2}{3^{2}} +.....+ \frac{n}{(n+1)^{2}}$ and $b_{n} = c_{0} + c_{1}r + c_{2}r^{2}+.....+c_{n}r^{n},$where$|c_{k}| \leq M$...

ankitgupta.1729

1.2k views

ankitgupta.1729 asked Mar 17, 2019

1 votes

1 answer

261

ISI MMA-2015
Let, $a_{n} \;=\; \left ( 1-\frac{1}{\sqrt{2}} \right ) ... \left ( 1- \frac{1}{\sqrt{n+1}} \right )$ , $n \geq 1$. Then $\lim_{n\rightarrow \infty } a_{n}$ (A) equals $1$ (B) does not exist (C) equals $\frac{1}{\sqrt{\pi }}$ (D) equals $0$

Let, $a_{n} \;=\; \left ( 1-\frac{1}{\sqrt{2}} \right ) ... \left ( 1- \frac{1}{\sqrt{n+1}} \right )$ , $n \geq 1$. Then $\lim_{n\rightarrow \infty } a_{n}$(A) equals $1$...

ankitgupta.1729

1.3k views

ankitgupta.1729 asked Feb 21, 2019

2 votes

1 answer

262

ISI MMA-2015
If two real polynomials $f(x)$ and $g(x)$ of degrees $m\;(\geq2)$ and $n\;(\geq1)$ respectively, satisfy $f(x^{2}+1) = f(x)g(x)$ $,$ for every $x\in \mathbb{R}$ , then (A) $f$ has exactly one real root $x_{0}$ such that $f'(x_{0}) \neq 0$ (B) $f$ has exactly one real root $x_{0}$ such that $f'(x_{0}) = 0$ (C) $f$ has $m$ distinct real roots (D) $f$ has no real root.

If two real polynomials $f(x)$ and $g(x)$ of degrees $m\;(\geq2)$ and $n\;(\geq1)$ respectively, satisfy $f(x^{2}+1)...

ankitgupta.1729

1.2k views

ankitgupta.1729 asked Feb 20, 2019

14 votes

7 answers

263

GATE CSE 2019 | Question: 13
Compute $\displaystyle \lim_{x \rightarrow 3} \frac{x^4-81}{2x^2-5x-3}$ $1$ $53/12$ $108/7$ Limit does not exist

Compute $\displaystyle \lim_{x \rightarrow 3} \frac{x^4-81}{2x^2-5x-3}$$1$$53/12$$108/7$Limit does not exist

Arjun

6.4k views

Arjun asked Feb 7, 2019

0 votes

0 answers

264

How to solve such question.
$\frac{d}{dx}\int_{1}^{x^4} sect\space dt$

$$\frac{d}{dx}\int_{1}^{x^4} sect\space dt$$

`JEET

439 views

`JEET asked Jan 20, 2019

0 votes

1 answer

265

MadeEasy Workbook: Calculus - Maxima Minima

chanchala3993

619 views

chanchala3993 asked Jan 19, 2019

1 votes

1 answer

266

Applied Course | Mock GATE | Test 1 | Question: 11
The value of derivative of $f(x) = \mid x -1 \mid + \mid x -3 \mid \text{ at } x = 2$ is $-2$ $0$ $2$ Not defined

The value of derivative of $f(x) = \mid x -1 \mid + \mid x -3 \mid \text{ at } x = 2$ is$-2$$0$$2$Not defined

Applied Course

684 views

Applied Course asked Jan 16, 2019

0 votes

0 answers

267

Ace Test Series: Calculus - Integration
Can anyone help me with solving this type of problem? I want some resource from where I can learn to solve this type on integration, as according to solution it is a function of α, so I did not understand the solution.

Can anyone help me with solving this type of problem? I want some resource from where I can learn to solve this type on integration, as according to solution it is a func...

jhaanuj2108

472 views

jhaanuj2108 asked Jan 12, 2019

0 votes

1 answer

268

UPPCL AE 2018:81
The value of $\dfrac{|x|}{x}$ at $x= 0$ is: Infinity Not defined $1$ $0$

The value of $\dfrac{|x|}{x}$ at $x= 0$ is:InfinityNot defined$1$$0$

admin

316 views

admin asked Jan 5, 2019

0 votes

1 answer

269

self doubt
$\lim_{x\rightarrow \frac{\pi }{2}}cosx^{cosx}$ can we straight away say $0^{0}=0$ ?

$\lim_{x\rightarrow \frac{\pi }{2}}cosx^{cosx}$can we straight away say $0^{0}=0$ ?

manisha11

378 views

manisha11 asked Jan 5, 2019

0 votes

1 answer

270

calculus question
Question Number 4?

Question Number 4?

pradeepchaudhary

291 views

pradeepchaudhary asked Jan 4, 2019

tags	tag:apple
author	user:martin
title	title:apple
content	content:apple
exclude	-tag:apple
force match	+apple
views	views:100
score	score:10
answers	answers:2
is accepted	isaccepted:true
is closed	isclosed:true