Recent questions and answers in Calculus

+1 vote

2 answers

1

TIFR2014-A-18
We are given a collection of real numbers where a real number $a_{i}\neq 0$ occurs $n_{i}$ times. Let the collection be enumerated as $\left\{x_{1}, x_{2},...x_{n}\right\}$ so that $x_{1}=x_{2}=...=x_{n_{1}}=a_{1}$ and so on, and $n=\sum _{i}n_{i}$ is finite. What is ... $\min_{i} |a_{i}|$ $\min_{i} \left(n_{i}|a_{i}|\right)$ $\max_{i} |a_{i}|$ None of the above.

answered 6 days ago in Calculus by Arjun Veteran (408k points) | 167 views

+2 votes

2 answers

2

TIFR2019-A-13
Consider the integral $\int^{1}_{0} \frac{x^{300}}{1+x^2+x^3} dx$ What is the value of this integral correct up to two decimal places? $0.00$ $0.02$ $0.10$ $0.33$ $1.00$

answered Jun 9 in Calculus by Arjun Veteran (408k points) | 209 views

+19 votes

3 answers

3

GATE2014-1-6
Let the function ... There exists $\theta \in (\frac{\pi}{6},\frac{\pi}{3})$ such that $f'(\theta)\neq 0$ I only II only Both I and II Neither I Nor II

answered Jun 8 in Calculus by 97apoorva singh (77 points) | 3.5k views

0 votes

1 answer

4

Differentiability
$\varphi \left ( x \right )=x^{2}\cos \frac{1}{x}$ when $x\neq 0$ $=0$ when $x=0$ Is it differentiable at $x=0$? Is it continuous ?

answered Jun 7 in Calculus by ankitgupta.1729 Boss (13.3k points) | 51 views

0 votes

1 answer

5

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

answered May 27 in Calculus by Mk Utkarsh Boss (34.5k points) | 96 views

+12 votes

5 answers

6

GATE2015-1-4
$\lim_{x\rightarrow \infty } x^{ \tfrac{1}{x}}$ is $\infty $ 0 1 Not defined

answered May 26 in Calculus by SALIL KUMAR (11 points) | 2k views

+3 votes

1 answer

7

Limits

answered May 16 in Calculus by Satbir Boss (13.2k points) | 93 views

+3 votes

1 answer

8

MadeEasy Test Series 2018: Calculus - Limits
The value of $\lim_{x\rightarrow \infty }\left ( \frac{4^{x+2} + 3^{x}}{4^{x-2}} \right )$ is ____________

answered May 16 in Calculus by Satbir Boss (13.2k points) | 139 views

+2 votes

6 answers

9

GATE2019-13
Compute $\lim_{x \rightarrow 3} \frac{x^4-81}{2x^2-5x-3}$ $1$ $53/12$ $108/7$ Limit does not exist

answered May 13 in Calculus by Bikki_gupta (77 points) | 1.6k views

0 votes

1 answer

10

ISI2019-MMA-21
A function $f:\mathbb{R^2} \rightarrow \mathbb{R}$ is called degenerate on $x_i$, if $f(x_1,x_2)$ remains constant when $x_i$ varies $(i=1,2)$. Define $f(x_1,x_2) = \mid 2^{\pi _i/x_1} \mid ^{x_2} \text{ for } x_1 \neq 0$, where $i = \sqrt {-1}$. ... $x_1$ but not on $x_2$ $f$ is degenerate on $x_2$ but not on $x_1$ $f$ is neither degenerate on $x_1$ nor on $x_2$

answered May 13 in Calculus by pratekag Active (2k points) | 387 views

0 votes

0 answers

11

ISI2018-MMA-28
Consider the following functions $f(x)=\left\{\begin{matrix} 1 &, if\ |x| \leq 1 \\ 0 & ,if\ |x|>1 \end{matrix}\right.$ ... at $ 1$ $h_2$ is continuous everywhere and $h_1$ has discontinuity at $ 2$ $h_1$ has discontinuity at $ 2$ and $h_2$ has discontinuity at $ 1$.

asked May 11 in Calculus by akash.dinkar12 Boss (40.5k points) | 28 views

0 votes

0 answers

12

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.

asked May 11 in Calculus by akash.dinkar12 Boss (40.5k points) | 17 views

0 votes

0 answers

13

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

asked May 11 in Calculus by akash.dinkar12 Boss (40.5k points) | 41 views

0 votes

1 answer

14

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 $i = 1, 2, . . . , n.$ ... $a_n \rightarrow p, b_n \rightarrow p,c_n \rightarrow 1-2p$ $a_n \rightarrow1/2, b_n \rightarrow1/2,c_n \rightarrow0$ $a_n \rightarrow0, b_n \rightarrow0,c_n \rightarrow1$

answered May 11 in Calculus by srestha Veteran (111k points) | 25 views

+1 vote

1 answer

15

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 May 7 in Calculus by pratekag Active (2k points) | 227 views

+1 vote

1 answer

16

ISI2019-MMA-30
Consider the function $h$ defined on $\{0,1,…….10\}$ with $h(0)=0, \: h(10)=10 $ and $2[h(i)-h(i-1)] = h(i+1) – h(i) \: \text{ for } i = 1,2, \dots ,9.$ Then the value of $h(1)$ is $\frac{1}{2^9-1}\\$ $\frac{10}{2^9+1}\\$ $\frac{10}{2^{10}-1}\\$ $\frac{1}{2^{10}+1}$

answered May 7 in Calculus by pratekag Active (2k points) | 338 views

0 votes

1 answer

17

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 $1/4$ $1/6$ $1/8$ $1/24$

answered May 7 in Calculus by pratekag Active (2k points) | 637 views

0 votes

1 answer

18

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 $f(0)$ $f’(0)$ $f’’(0)$ $f(100)$

answered May 7 in Calculus by pratekag Active (2k points) | 374 views

0 votes

1 answer

19

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 $(0,2)$

answered May 7 in Calculus by pratekag Active (2k points) | 216 views

0 votes

1 answer

20

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 $-5$ $-3$ $3$ $5$

answered May 6 in Calculus by pratekag Active (2k points) | 151 views

0 votes

0 answers

21

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) $f(x) = Maximum (x,-x)$

asked May 4 in Calculus by balchandar reddy san Active (3k points) | 52 views

+1 vote

1 answer

22

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.

answered Mar 18 in Calculus by Kushagra Chatterjee Loyal (9.5k points) | 109 views

0 votes

1 answer

23

UPPCL AE 2018:81

answered Mar 11 in Calculus by Ram Swaroop Active (3k points) | 52 views

+1 vote

1 answer

24

Continuity and Differentiability
If the function f(x) =[(x-2)3 /a] sin(x-2) + acos(x-2), [.] denotes greatest integer function, is continuous & differentiable in (4,6) then find ‘a’ range: (A) a ϵ (-∞,∞) (B) a ϵ [64, ∞) (C) a ϵ [128, ∞) (D) Not defined

answered Mar 10 in Calculus by jinal99 (37 points) | 310 views

0 votes

2 answers

25

ISRO2012-ECE Engineering Mathematics
a) 1-x b) (1-x)/x c)1/x d)x/(1-x)

answered Mar 10 in Calculus by abhishekmehta4u Boss (33.9k points) | 71 views

0 votes

1 answer

26

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 $0$

answered Feb 21 in Calculus by venkatesh pagadala (495 points) | 133 views

+1 vote

1 answer

27

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.

answered Feb 21 in Calculus by Kushagra Chatterjee Loyal (9.5k points) | 125 views

+2 votes

3 answers

28

VALUE OF LIMIT
What is value of $\lim_{x\rightarrow 0, Y\rightarrow 0}\frac{xY}{x^2+Y^2}$ 1 -1 0 Does not exist

answered Jan 28 in Calculus by Sai Shravan (169 points) | 291 views

+5 votes

3 answers

29

TIFR2011-A-11
$\int_{0}^{1} \ln x\, \mathrm{d}x=$ $1$ $-1$ $\infty $ $-\infty $ None of the above.

answered Jan 26 in Calculus by Lakshman Patel RJIT Boss (40k points) | 469 views

0 votes

0 answers

30

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

asked Jan 20 in Calculus by `JEET Active (3.3k points) | 88 views

0 votes

0 answers

31

MadeEasy Workbook: Calculus - Maxima Minima

asked Jan 19 in Calculus by chanchala3993 (5 points) | 52 views

0 votes

0 answers

32

Calculating average mean
The aggregate monthly expenditure of a family was $ 6240 during first 3 months, $ 6780 during next 4 months, and $7236 during last 5 months of a year. If total saving during the year is $ 7080. Find average monthly income of family?

asked Jan 13 in Calculus by Alina (9 points) | 35 views

0 votes

0 answers

33

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.

asked Jan 12 in Calculus by jhaanuj2108 (185 points) | 73 views

0 votes

1 answer

34

How to solve such question with different variable in the limit.
$\int_{a}^{x} \frac{sint}{t} dt$ (x > 0), then f(x) has _________________. (A) a maximum at x = $n\pi$ where n is even (B) a minimum at x = $n\pi$ where n is odd (C) a maximum at x = $n\pi$ where n is odd (D) a minimum at x = $\frac{n\pi}{2}$ where n is odd.

answered Jan 12 in Calculus by Pranavapp (21 points) | 65 views

0 votes

1 answer

35

Integration
How to solve it?

answered Jan 11 in Calculus by adeemajain (193 points) | 61 views

0 votes

1 answer

36

ME TEST SERIES

answered Jan 10 in Calculus by Kunal Kadian Active (2.7k points) | 52 views

+1 vote

1 answer

37

UPPCL AE 2018:85

answered Jan 5 in Calculus by Lakshman Patel RJIT Boss (40k points) | 94 views

0 votes

1 answer

38

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

answered Jan 5 in Calculus by Nandkishor3939 Active (1k points) | 50 views

0 votes

1 answer

39

calculus question
Question Number 4?

answered Jan 5 in Calculus by GATE2021 (11 points) | 46 views

0 votes

1 answer

40

GATE 2013 MA Calculus

answered Jan 3 in Calculus by pradeepchaudhary Active (1.1k points) | 29 views

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