Recent questions tagged calculus

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1

GATE CSE 1995 | Question: 7(B)
Compute without using power series expansion $\displaystyle \lim_{x \to 0} \frac{\sin x}{x}.$

Lakshman Patel RJIT asked in Calculus Apr 25

by Lakshman Patel RJIT

310 views

3 votes

2 answers

2

GATE CSE 2021 Set 2 | Question: 25
Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function on the interval $[-3, 3]$ and a differentiable function in the interval $(-3,3)$ such that for every $x$ in the interval, $f’(x) \leq 2$. If $f(-3)=7$, then $f(3)$ is at most __________

Arjun asked in Calculus Feb 18

by Arjun

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2 answers

3

GATE CSE 2021 Set 1 | Question: 20
Consider the following expression.$\displaystyle \lim_{x\rightarrow-3}\frac{\sqrt{2x+22}-4}{x+3}$The value of the above expression (rounded to 2 decimal places) is ___________.

Arjun asked in Calculus Feb 18

by Arjun

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2 votes

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4

NIELIT 2016 MAR Scientist C - Section B: 2
The function $f(x)=x^{5}-5x^{4}+5x^{3}-1$ has one minima and two maxima two minima and one maxima two minima and two maxima one minima and one maxima

Lakshman Patel RJIT asked in Calculus Apr 2, 2020

by Lakshman Patel RJIT

244 views

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1 answer

5

NIELIT 2016 MAR Scientist C - Section B: 10
$\underset{x \rightarrow 0}{\lim} \dfrac{x^{3}+x^{2}-5x-2}{2x^{3}-7x^{2}+4x+4}=?$ $-0.5$ $(0.5)$ $\infty$ None of the above

Lakshman Patel RJIT asked in Calculus Apr 2, 2020

by Lakshman Patel RJIT

154 views

0 votes

1 answer

6

NIELIT 2016 MAR Scientist C - Section B: 11
$\displaystyle \int_{0}^{\dfrac{\pi}{2}} \sin^{7}\theta \cos ^{4} \theta d\theta=?$ $16/1155$ $16/385$ $16\pi/385$ $8\pi/385$

Lakshman Patel RJIT asked in Calculus Apr 2, 2020

by Lakshman Patel RJIT

116 views

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7

NIELIT 2016 MAR Scientist C - Section B: 12
$\displaystyle \lim_{x \rightarrow a}\frac{1}{x^{2}-a^{2}} \displaystyle \int_{a}^{x}\sin (t^{2})dt=$? $2a \sin (a^{2})$ $2a$ $\sin (a^{2})$ None of the above

Lakshman Patel RJIT asked in Calculus Apr 2, 2020

by Lakshman Patel RJIT

93 views

0 votes

1 answer

8

NIELIT 2016 MAR Scientist C - Section B: 13
$\displaystyle \lim_{x \rightarrow 0}\frac{1}{x^{6}} \displaystyle \int_{0}^{x^{2}}\frac{t^{2}dt}{t^{6}+1}=$? $1/4$ $1/3$ $1/2$ $1$

Lakshman Patel RJIT asked in Calculus Apr 2, 2020

by Lakshman Patel RJIT

117 views

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9

NIELIT 2016 MAR Scientist C - Section B: 17
A ladder $13$ feet long rests against the side of a house. The bottom of the ladder slides away from the house at a rate of $0.5$ ft/s. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is $5$ ... $-\dfrac{5}{24} \text {ft/s} \\$ $-\dfrac{5}{12} \text{ ft/s}$

Lakshman Patel RJIT asked in Calculus Apr 2, 2020

by Lakshman Patel RJIT

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1 answer

10

NIELIT 2017 OCT Scientific Assistant A (CS) - Section B: 18
What is the maximum value of the function $f(x) = 2x^{2} – 2x + 6$ in the interval $[0,2]?$ $6$ $10$ $12$ $5,5$

Lakshman Patel RJIT asked in Calculus Apr 1, 2020

by Lakshman Patel RJIT

209 views

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11

NIELIT 2017 OCT Scientific Assistant A (CS) - Section B: 19
The value of the Integral $I = \displaystyle{}\int_{0}^{\pi/2} x^{2}\sin x dx$ is $(x+2)/2$ $2/(\pi-2)$ $\pi – 2$ $\pi + 2$

Lakshman Patel RJIT asked in Calculus Apr 1, 2020

by Lakshman Patel RJIT

150 views

0 votes

1 answer

12

NIELIT 2017 DEC Scientific Assistant A - Section B: 10
The function $f\left ( x \right )=\dfrac{x^{2}-1}{x-1}$ at $x=1$ is : Continuous and differentiable Continuous but not differentiable Differentiable but not continuous Neither continuous nor differentiable

Lakshman Patel RJIT asked in Calculus Mar 31, 2020

by Lakshman Patel RJIT

357 views

0 votes

1 answer

13

NIELIT 2016 MAR Scientist B - Section B: 5
The greatest and the least value of $f(x)=x^4-8x^3+22x^2-24x+1$ in $[0,2]$ are $0,8$ $0,-8$ $1,8$ $1,-8$

Lakshman Patel RJIT asked in Calculus Mar 31, 2020

by Lakshman Patel RJIT

187 views

0 votes

1 answer

14

NIELIT 2016 MAR Scientist B - Section B: 9
The value of improper integral $\displaystyle\int_{0}^{1} x\ln x =?$ $1/4$ $0$ $-1/4$ $1$

Lakshman Patel RJIT asked in Calculus Mar 31, 2020

by Lakshman Patel RJIT

196 views

0 votes

1 answer

15

NIELIT 2016 MAR Scientist B - Section B: 10
Maxima and minimum of the function $f(x)=2x^3-15x^2+36x+10$ occur; respectively at $x=3$ and $x=2$ $x=1$ and $x=3$ $x=2$ and $x=3$ $x=3$ and $x=4$

Lakshman Patel RJIT asked in Calculus Mar 31, 2020

by Lakshman Patel RJIT

200 views

0 votes

1 answer

16

NIELIT 2016 MAR Scientist B - Section B: 11
What is the derivative w.r.t $x$ of the function given by $\large \Phi(x)= \displaystyle \int_{0}^{x^2}\sqrt t\:dt$, $2x^2$ $\sqrt x$ $0$ $1$

Lakshman Patel RJIT asked in Calculus Mar 31, 2020

by Lakshman Patel RJIT

180 views

0 votes

1 answer

17

NIELIT 2016 MAR Scientist B - Section B: 13
$\underset{x\to 0}{\lim} \dfrac{(1-\cos x)}{2}$ is equal to $0$ $1$ $1/3$ $1/2$

Lakshman Patel RJIT asked in Calculus Mar 31, 2020

by Lakshman Patel RJIT

216 views

0 votes

1 answer

18

NIELIT 2016 MAR Scientist B - Section B: 14
The minimum value of $\mid x^2-5x+2\mid$ is $-5$ $0$ $-1$ $-2$

Lakshman Patel RJIT asked in Calculus Mar 31, 2020

by Lakshman Patel RJIT

193 views

0 votes

1 answer

19

NIELIT 2016 DEC Scientist B (CS) - Section B: 26
Consider the function $f(x)=\sin(x)$ in the interval $\bigg [\dfrac{ \pi}{4},\dfrac{7\pi}{4}\bigg ]$. The number and location(s) of the 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 $\dfrac{3\pi}{2}$

Lakshman Patel RJIT asked in Calculus Mar 31, 2020

by Lakshman Patel RJIT

209 views

13 votes

3 answers

20

GATE CSE 2020 | Question: 1
Consider the functions $e^{-x}$ $x^{2}-\sin x$ $\sqrt{x^{3}+1}$ Which of the above functions is/are increasing everywhere in $[ 0,1]$? Ⅲ only Ⅱ only Ⅱ and Ⅲ only Ⅰ and Ⅲ only

Arjun asked in Calculus Feb 12, 2020

by Arjun

5.8k views

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1 answer

21

TIFR2020-A-8
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 one global minimum inside $(0,1)$. What can you say about the behaviour of the first derivative $f'$ ... is zero at at least one point $f'$ is zero at at least two points, $f''$ is zero at at least two points

Lakshman Patel RJIT asked in Calculus Feb 10, 2020

by Lakshman Patel RJIT

351 views

2 votes

1 answer

22

ISI2014-DCG-2
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 $0$

Arjun asked in Calculus Sep 23, 2019

by Arjun

347 views

4 votes

4 answers

23

ISI2014-DCG-3
$\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$

Arjun asked in Calculus Sep 23, 2019

by Arjun

565 views

3 votes

2 answers

24

ISI2014-DCG-4
$\underset{n \to \infty}{\lim} \dfrac{1}{n} \bigg( \dfrac{n}{n+1} + \dfrac{n}{n+2} + \cdots + \dfrac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$

Arjun asked in Calculus Sep 23, 2019

by Arjun

395 views

2 votes

2 answers

25

ISI2014-DCG-6
If $f(x)$ is a real valued function such that $2f(x)+3f(-x)=15-4x$, for every $x \in \mathbb{R}$, then $f(2)$ is $-15$ $22$ $11$ $0$

Arjun asked in Calculus Sep 23, 2019

by Arjun

259 views

2 votes

3 answers

26

ISI2014-DCG-7
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

Arjun asked in Calculus Sep 23, 2019

by Arjun

221 views

2 votes

1 answer

27

ISI2014-DCG-12
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

Arjun asked in Calculus Sep 23, 2019

by Arjun

333 views

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