Recent questions and answers in Calculus - GATE Overflow

+2 votes

5 answers

1

TIFR2019-A-15
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 views

+19 votes

5 answers

2

GATE2014-3-6
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 Veteran (54.7k points) | 2.8k views

+17 votes

4 answers

3

GATE2014-3-47
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 (54.7k points) | 1.9k views

+4 votes

3 answers

4

ISRO-2013-49
What is the least value of the function $f(x) = 2x^{2}-8x-3$ in the interval $[0, 5]$? $-15$ $7$ $-11$ $-3$

answered Nov 28 in Calculus by Lakshman Patel RJIT Veteran (54.7k points) | 1.9k views

+1 vote

1 answer

5

TIFR-2011-Maths-A-19
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

+1 vote

1 answer

6

TIFR-2011-Maths-A-21
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

+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 Veteran (54.7k points) | 185 views

+2 votes

1 answer

8

ISI2016-MMA-8
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 (12.9k points) | 34 views

+35 votes

4 answers

9

GATE2012-9
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 (61 points) | 4k views

0 votes

1 answer

10

MadeEasy Workbook: Calculus - Maxima Minima

answered Oct 31 in Calculus by Kushagra गुप्ता Active (1.9k points) | 77 views

+2 votes

2 answers

11

ISI2015-MMA-10
The value of the infinite product $P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^3-1}{n^3+1} \times \cdots \text{ is }$ $1$ $2/3$ $7/3$ none of the above

answered Oct 25 in Calculus by techbd123 Active (3.1k points) | 35 views

0 votes

1 answer

12

ISI2017-DCG-6
Let $f(x) = \dfrac{x-1}{x+1}, \: f^{k+1}(x)=f\left(f^k(x)\right)$ 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

+1 vote

1 answer

13

ISI2015-MMA-36
For non-negative integers $m$, $n$ define a function as follows $f(m,n) = \begin{cases} n+1 & \text{ if } m=0 \\ f(m-1, 1) & \text{ if } m \neq 0, n=0 \\ f(m-1, f(m,n-1)) & \text{ if } m \neq 0, n \neq 0 \end{cases}$ 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

+12 votes

3 answers

14

GATE1996-1.6
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

+1 vote

1 answer

15

ISI2014-DCG-13
Let the function $f(x)$ be defined as $f(x)=\mid x-1 \mid + \mid x-2 \:\mid$. 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 Boss (31.2k points) | 38 views

+2 votes

3 answers

16

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

answered Oct 11 in Calculus by Abhishek Kumar 40 (253 points) | 39 views

0 votes

1 answer

17

ISI2014-DCG-29
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

+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

ISI2014-DCG-37
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

0 votes

1 answer

20

ISI2015-MMA-78
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

0 votes

1 answer

21

ISI2015-MMA-22
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

0 votes

1 answer

22

ISI2015-MMA-19
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k-1) }{n}} \end{vmatrix}\:\:\:$ is $2$ $2e$ $2 \pi$ $2i$

answered Oct 4 in Calculus by `JEET Boss (12.9k points) | 23 views

+2 votes

1 answer

23

ISI2016-DCG-45
The value of $\underset{x \to 0}{\lim} \dfrac{\tan^{2}\:x-x\:\tan\:x}{\sin\:x}$ 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

+1 vote

1 answer

24

ISI2014-DCG-5
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 non-empty none of the above

answered Oct 1 in Calculus by Sourajit25 Active (1.3k points) | 79 views

+1 vote

1 answer

25

ISI2015-MMA-25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \bigg( \frac{3x-1}{3x+1} \bigg) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$

answered Oct 1 in Calculus by `JEET Boss (12.9k points) | 24 views

+1 vote

1 answer

26

ISI2015-MMA-20
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 Boss (12.9k points) | 11 views

+1 vote

1 answer

27

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 $0$ $2017$ $2018$ $2019$

answered Sep 30 in Calculus by `JEET Boss (12.9k points) | 39 views

0 votes

1 answer

28

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

answered Sep 30 in Calculus by `JEET Boss (12.9k points) | 12 views

+3 votes

2 answers

29

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 $\infty$ $0$ $\log_e 2$ $1$

answered Sep 30 in Calculus by techbd123 Active (3.1k points) | 92 views

+3 votes

4 answers

30

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

answered Sep 29 in Calculus by techbd123 Active (3.1k points) | 127 views

+2 votes

1 answer

31

ISI2015-DCG-52
$\underset{x \to -1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$

answered Sep 28 in Calculus by `JEET Boss (12.9k points) | 22 views

0 votes

1 answer

32

ISI2016-DCG-49
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these

answered Sep 28 in Calculus by `JEET Boss (12.9k points) | 12 views

+1 vote

1 answer

33

ISI2016-DCG-50
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 Boss (12.9k points) | 8 views

0 votes

1 answer

34

ISI2016-DCG-53
$\underset{x\rightarrow-1}{\lim}\dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$

answered Sep 28 in Calculus by `JEET Boss (12.9k points) | 10 views

0 votes

1 answer

35

ISI2017-DCG-27
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

0 votes

1 answer

36

ISI2014-DCG-51
The function $f(x)$ defined as $f(x)=x^3-6x^2+24x$, 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 Boss (12.9k points) | 19 views

+2 votes

1 answer

37

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 $0$

answered Sep 27 in Calculus by `JEET Boss (12.9k points) | 91 views

+2 votes

1 answer

38

ISI2014-DCG-19
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 by `JEET Boss (12.9k points) | 25 views

0 votes

1 answer

39

ISI2017-DCG-3
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 by `JEET Boss (12.9k points) | 12 views

0 votes

1 answer

40

ISI2014-DCG-44
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 Boss (12.9k points) | 8 views

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