Recent questions and answers in Calculus - GATE Overflow

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1 answer

1

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

answered Jun 26 in Calculus by Sherrinford03 | 39 views

+1 vote

2 answers

2

ISI2019-MMA-24
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim _{n\rightarrow \infty} f''(x)$ exists for every $x \in \mathbb{R}$, where $f''(x) = f \circ f^{n-1}(x)$ for $n \geq 2$ ... $S \subset T$ $T \subset S$ $S = T$ None of the above

answered Jun 11 in Calculus by Amartya | 463 views

0 votes

1 answer

3

ISI2015-MMA-30
Suppose that a function $f$ defined on $\mathbb{R} ^2$ satisfies the following conditions: $\begin{array} &f(x+t,y) & = & f(x,y)+ty, \\ f(x,t+y) & = & f(x,y)+ tx \text{ and } \\ f(0,0) & = & K, \text{ a constant.} \end{array}$ Then for all $x,y \in \mathbb{R}, \:f(x,y)$ is equal to $K(x+y)$ $K-xy$ $K+xy$ none of the above

answered May 29 in Calculus by Amartya | 69 views

0 votes

1 answer

4

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

answered May 25 in Calculus by Mohit Kumar 6 | 40 views

0 votes

1 answer

5

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.

answered May 24 in Calculus by Amartya | 152 views

0 votes

1 answer

6

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 $-2$ $-1$ $1$ $2$

answered May 24 in Calculus by Amartya | 178 views

0 votes

2 answers

7

ISI2015-MMA-78
The value of $\displaystyle \lim_{n \to \infty} \left[ (n+1) \int_0^1 x^n \ln(1+x) dx \right]$ is $0$ $\ln 2$ $\ln 3$ $\infty$

answered May 23 in Calculus by Amartya | 66 views

0 votes

1 answer

8

ISI2015-MMA-69
Consider the function $f(x) = \begin{cases} \int_0^x \{5+ \mid 1-y \mid \} dy & \text{ if } x>2 \\ 5x+2 & \text{ if } x \leq 2 \end{cases}$ Then $f$ is not continuous at $x=2$ $f$ is continuous and differentiable everywhere $f$ is continuous everywhere but not differentiable at $x=1$ $f$ is continuous everywhere but not differentiable at $x=2$

answered May 23 in Calculus by Amartya | 72 views

0 votes

1 answer

9

ISI2015-MMA-57
Suppose $a>0$. Consider the sequence $a_n = n \{ \sqrt[n]{ea} – \sqrt[n]{a}, \:\:\:\:\: n \geq 1$. Then $\underset{n \to \infty}{\lim} a_n$ does not exist $\underset{n \to \infty}{\lim} a_n=e$ $\underset{n \to \infty}{\lim} a_n=0$ none of the above

answered May 22 in Calculus by Amartya | 59 views

+9 votes

5 answers

10

TIFR2014-A-9
Solve min $x^{2}+y^{2}$ subject to $\begin {align*} x + y &\geq 10,\\ 2x + 3y &\geq 20,\\ x &\geq 4,\\ y &\geq 4. \end{align*}$ $32$ $50$ $52$ $100$ None of the above

answered May 8 in Calculus by Joyoshish Saha | 598 views

0 votes

1 answer

11

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

answered Apr 1 in Calculus by aman_0709 | 73 views

0 votes

1 answer

12

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

answered Apr 1 in Calculus by haralk10 | 41 views

0 votes

1 answer

13

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 $x=4$

answered Apr 1 in Calculus by haralk10 | 46 views

0 votes

1 answer

14

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

answered Apr 1 in Calculus by haralk10 | 46 views

0 votes

0 answers

15

NIELIT 2016 MAR Scientist B - Section B: 8
If $\Delta f(x)= f(x+h)-f(x)$, then a constant $k,\Delta k$ $1$ $0$ $f(k)-f(0)$ $f(x+k)-f(x)$

asked Mar 31 in Calculus by Lakshman Patel RJIT | 49 views

0 votes

1 answer

16

ISI2016-DCG-69
Consider the differential equation $(x^{2}-y^{2})\frac{\mathrm{d} y}{\mathrm{d} x}=2xy.$ Assuming $y=10$ for $x=0,$ its solution is $x^{2}+(y-5)^{2}=25$ $x^{2}+y^{2}=100$ $(x-5)^{2}+y^{2}=125$ $(x-5)^{2}+(y-5)^{2}=50$

answered Mar 15 in Calculus by haralk10 | 42 views

0 votes

1 answer

17

ISI2016-DCG-68
The general solution of the differential equation $x+y-x{y}'=0$ is (assuming $C$ as an arbitrary constant of integration) $y=x(\log x+C)$ $x=y(\log y+C)$ $y=x(\log y+C)$ $y=y(\log x+C)$

answered Mar 14 in Calculus by haralk10 | 41 views

0 votes

1 answer

18

ISI2016-DCG-67
The general solution of the differential equation $2y{y}'-x=0$ is (assuming $C$ as an arbitrary constant of integration) $x^{2}-y^{2}=C$ $2x^{2}-y^{2}=C$ $2y^{2}-x^{2}=C$ $x^{2}+y^{2}=C$

answered Mar 14 in Calculus by haralk10 | 46 views

0 votes

1 answer

19

ISI2015-DCG-50
The piecewise linear function for the following graph is $f(x) = \begin{cases} = x, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$ $f(x) = \begin{cases} = x-2, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x-1, \: x \geq 3 \end{cases}$ ... $f(x) = \begin{cases} = 2-x, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$

answered Mar 14 in Calculus by haralk10 | 42 views

0 votes

1 answer

20

ISI2014-DCG-48
If $x$ is real, the set of real values of $a$ for which the function $y=x^2-ax+1-2a^2$ 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

answered Mar 13 in Calculus by haralk10 | 36 views

0 votes

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

answered Mar 1 in Calculus by ankitgupta.1729 | 82 views

0 votes

1 answer

22

TIFR2020-A-13
What is the area of the largest rectangle that can be inscribed in a circle of radius $R$? $R^{2}/2$ $\pi \times R^{2}/2$ $R^{2}$ $2R^{2}$ None of the above

answered Feb 19 in Calculus by ankitgupta.1729 | 50 views

+2 votes

2 answers

23

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 $-15$ $22$ $11$ $0$

answered Jan 30 in Calculus by Akumar20 | 139 views

+1 vote

2 answers

24

MadeEasy Test Series: Calculus - Limits
How it solve this ?

answered Jan 9 in Calculus by Mayank Harbola | 277 views

+1 vote

2 answers

25

GATE1993-02.6
The value of the double integral $\int^{1}_{0} \int_{0}^{\frac{1}{x}} \frac {x}{1+y^2} dxdy$ is_________.

answered Dec 31, 2019 in Calculus by Verma Ashish | 881 views

+3 votes

5 answers

26

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 Dec 5, 2019 in Calculus by severustux | 596 views

+22 votes

5 answers

27

GATE2014-3-6
If $\int \limits_0^{2 \pi} |x \: \sin x| dx=k\pi$, then the value of $k$ is equal to ______.

answered Dec 1, 2019 in Calculus by Lakshman Patel RJIT | 3.7k views

+22 votes

4 answers

28

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 Dec 1, 2019 in Calculus by Lakshman Patel RJIT | 2.5k views

+6 votes

3 answers

29

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, 2019 in Calculus by Lakshman Patel RJIT | 2.2k views

+1 vote

1 answer

30

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, 2019 in Calculus by seetal samal | 199 views

+1 vote

1 answer

31

TIFR-2011-Maths-A-21
Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.

answered Nov 25, 2019 in Calculus by seetal samal | 132 views

+2 votes

2 answers

32

Integration
Solve the following $\int_{0}^{\infty}e^{-x^2}x^4dx$

answered Nov 22, 2019 in Calculus by Lakshman Patel RJIT | 240 views

+2 votes

1 answer

33

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, 2019 in Calculus by `JEET | 88 views

+1 vote

3 answers

34

ISI2017-DCG-23
A determinant is chosen at random from the set of all determinants of order $2$ with elements $0$ or $1$ only. The probability of choosing a non-zero determinant is $\frac{3}{16}$ $\frac{3}{8}$ $\frac{1}{4}$ none of these

answered Nov 19, 2019 in Calculus by chirudeepnamini | 89 views

+41 votes

4 answers

35

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, 2019 in Calculus by Obafgkme | 5.2k views

0 votes

1 answer

36

MadeEasy Workbook: Calculus - Maxima Minima

answered Nov 1, 2019 in Calculus by Kushagra गुप्ता | 150 views

+2 votes

2 answers

37

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, 2019 in Calculus by techbd123 | 130 views

0 votes

1 answer

38

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, 2019 in Calculus by `JEET | 73 views

+1 vote

1 answer

39

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, 2019 in Calculus by chirudeepnamini | 64 views

+14 votes

3 answers

40

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, 2019 in Calculus by neeraj_bhatt | 2k views

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