Recent questions and answers in Calculus - GATE CSE Doubts

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GATE2014-1-47 Video Solution
A function $f(x)$ is continuous in the interval $[0,2]$. It is known that $f(0) = f(2) = -1$ and $f(1) = 1$. Which one of the following statements must be true? There exists a $y$ in the interval $(0,1)$ such that $f(y) = f(y+1)$ For every $y$ in ... of the function in the interval $(0,2)$ is $1$ There exists a $y$ in the interval $(0,1)$ such that $f(y)$ = $-f(2-y)$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2015-2-26 Video Solution
Let $f(x)=x^{-\left(\frac{1}{3}\right)}$ and $A$ denote the area of region bounded by $f(x)$ and the X-axis, when $x$ varies from $-1$ to $1$. Which of the following statements is/are TRUE? $f$ is continuous in $[-1, 1]$ $f$ is not bounded in $[-1, 1]$ $A$ is nonzero and finite II only III only II and III only I, II and III

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2018-16 Video Solution
The value of $\int^{\pi/4} _0 x \cos(x^2) dx$ correct to three decimal places (assuming that $\pi = 3.14$) is ____

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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GATE2012-9 Video Solution
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 $\dfrac{3\pi}{2}$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2014-1-6 Video Solution
Let the function ... 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

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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GATE2015-3-9 Video Solution
The value of $\lim_{x \rightarrow \infty} (1+x^2)^{e^{-x}}$ is $0$ $\frac{1}{2}$ $1$ $\infty$

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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GATE2016-2-02 Video Solution
Let $f(x)$ be a polynomial and $g(x)=f'(x)$ be its derivative. If the degree of $(f(x)+f(-x))$ is $10$, then the degree of $(g(x) - g(-x))$ is __________.

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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GATE2015-3-45 Video Solution
If for non-zero $x, \: af(x) + bf(\frac{1}{x}) = \frac{1}{x} - 25$ where a $a \neq b \text{ then } \int_1^2 f(x)dx$ is $\frac{1}{a^2 - b^2} \begin{bmatrix} a(\ln 2 - 25) + \frac{47b}{2} \end{bmatrix}$ ... $\frac{1}{a^2 - b^2} \begin{bmatrix} a(\ln 2 - 25) - \frac{47b}{2} \end{bmatrix}$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2017-2-10 Video Solution
If $f(x) = R \: \sin ( \frac{\pi x}{2}) + S, f’\left(\frac{1}{2}\right) = \sqrt{2}$ and $\int_0^1 f(x) dx = \frac{2R}{\pi}$, then the constants $R$ and $S$ are $\frac{2}{\pi}$ and $\frac{16}{\pi}$ $\frac{2}{\pi}$ and 0 $\frac{4}{\pi}$ and 0 $\frac{4}{\pi}$ and $\frac{16}{\pi}$

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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GATE2008-1 Video Solution
$\lim_{x \to \infty}\frac{x-\sin x}{x+\cos x}$ equals $1$ $-1$ $\infty$ $-\infty$

asked Apr 19 in Calculus by admin (3.6k points) | 3 views

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GATE2016-1-3 Video Solution
$\lim _{x\rightarrow 4}\frac{\sin(x-4)}{x-4}$=____.

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2011-31 Video Solution
Given $i = \sqrt{-1}$, what will be the evaluation of the definite integral $\int \limits_0^{\pi/2} \dfrac{\cos x +i \sin x} {\cos x - i \sin x} dx$ ? $0$ $2$ $-i$ $i$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2015-1-44 Video Solution
Compute the value of: $\large \int_{\frac{1}{\pi}}^{\frac{2}{\pi}}\frac{\cos(1/x)}{x^{2}}dx$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2017-1-28 Video Solution
The value of $\lim_{x\rightarrow 1} \frac{x^{7}-2x^{5}+1}{x^{3}-3x^{2}+2}$ is $0$ is $-1$ is $1$ does not exist

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2008-25 Video Solution
A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve $3x^4-16x^3+24x^2+37$ is $0$ $1$ $2$ $3$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2014-3-47 Video Solution
The value of the integral given below is $\int \limits_0^{\pi} \: x^2 \: \cos x\:dx$ $-2\pi$ $\pi$ $-\pi$ $2\pi$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2013-22 Video Solution
Which one of the following functions is continuous at $x = 3?$ $f(x) = \begin{cases} 2,&\text{if $x = 3$ } \\ x-1& \text{if $x > 3$}\\ \frac{x+3}{3}&\text{if $x < 3$ } \end{cases}$ $f(x) = \begin{cases} 4,&\text{if $ ... $} \end{cases}$ $f(x) = \begin{cases} \frac{1}{x^3-27}&\text{if $x \neq 3$ } \end{cases}$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2010-5 Video Solution
What is the value of $\lim_{n \to \infty}\left(1 - \frac{1}{n}\right)^{2n}$ ? 0 $e^{-2}$ $e^{-1/2}$ 1

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2008-IT-31 Video Solution
If $f(x)$ is defined as follows, what is the minimum value of $f(x)$ for $x \in (0, 2]$ ? $f(x) = \begin{cases} \frac{25}{8x} \text{ when } x \leq \frac{3}{2} \\ x+ \frac{1}{x} \text { otherwise}\end{cases}$ $2$ $2 \frac{1}{12}$ $2\frac{1}{6}$ $2\frac{1}{2}$

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2000-2.3 Video Solution
Let $S = \sum_{i=3}^{100} i \log_{2} i$, and $T = \int_{2}^{100} x \log_{2}x dx$. Which of the following statements is true? $S > T$ $S = T$ $S < T$ and $2S > T$ $2S ≤ T$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2015-2-GA-3 Video Solution
Consider a function $f(x) = 1- |x| \text{ on } -1 \leq x \leq 1$. The value of $x$ at which the function attains a maximum, and the maximum value of the function are: $0, -1$ $-1, 0$ $0, 1$ $-1, 2$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE1997-4.1 Video Solution
What is the maximum value of the function $f(x) = 2x^2 - 2x + 6$ in the interval $\left[0,2 \right]$? 6 10 12 5.5

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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GATE1996-1.6 Video Solution
The formula used to compute an approximation for the second derivative of a function $f$ at a point $X_0$ is $\dfrac{f(x_0 +h) + f(x_0 – h)}{2}$ $\dfrac{f(x_0 +h) - f(x_0 – h)}{2h}$ $\dfrac{f(x_0 +h) + 2f(x_0) + f(x_0 – h)}{h^2}$ $\dfrac{f(x_0 +h) - 2f(x_0) + f(x_0 – h)}{h^2}$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2005-IT-35 Video Solution
What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$ $-1$ $0$ $1$ $\pi$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2007-1 Video Solution
Consider the following two statements about the function $f(x)=\left\vert x\right\vert$: P. $f(x)$ is continuous for all real values of $x$. Q. $f(x)$ is differentiable for all real values of $x$ . Which of the following is TRUE? $P$ is true and $Q$ is false. $P$ is false and $Q$ is true. Both $P$ and $Q$ are true. Both $P$ and $Q$ are false.

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2009-25 Video Solution
$\int^{\pi/4}_0 (1-\tan x)/(1+\tan x)\,dx $ $0$ $1$ $ln 2$ $1/2 ln 2$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE2014-1-46 Video Solution
The function $f(x) =x \sin x$ satisfies the following equation: $f''(x) + f(x) +t \cos x = 0$. The value of $t$ is______.

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE1996-3 Video Solution
Let $f$ be a function defined by $f(x) = \begin{cases} x^2 &\text{ for }x \leq 1\\ ax^2+bx+c &\text{ for } 1 < x \leq 2 \\ x+d &\text{ for } x>2 \end{cases}$ Find the values for the constants $a$, $b$, $c$ and $d$ so that $f$ is continuous and differentiable everywhere on the real line.

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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GATE1995-1.21 Video Solution
In the interval $[0, \pi]$ the equation $x=\cos x$ has No solution Exactly one solution Exactly two solutions An infinite number of solutions

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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GATE2010-ME Video Solution
The function $y=|2 - 3x|$ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ except at $x=\frac{3}{2}$ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ except at $x=\frac{2}{3}$ is continuous $∀ x ∈ R$ except $x=3$ and differentiable $∀ x ∈ R$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE1998-1.4 Video Solution
Consider the function $y=|x|$ in the interval $[-1, 1]$. In this interval, the function is continuous and differentiable continuous but not differentiable differentiable but not continuous neither continuous nor differentiable

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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GATE1998-8 Video Solution
Find the points of local maxima and minima, if any, of the following function defined in $0\leq x\leq 6$. $x^3-6x^2+9x+15$ Integrate $\int_{-\pi}^{\pi} x \cos x dx$

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE1995-25a Video Solution
Find the minimum value of $3-4x+2x^2$.

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE1993-02.1 Video Solution
In questions $2.1$ to $2.10$ below, each blank (___) is to be suitably filled in. $\lim_{x \to 0} \frac{x(e^x - 1) + 2(\cos x -1)}{x(1 - \cos x)}$ is_____________

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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GATE1987-1-xxvi Video Solution
If $f(x_{i}).f(x_{i+1})< 0$ then There must be a root of $f(x)$ between $x_i$ and $x_{i+1}$ There need not be a root of $f(x)$ between $x_{i}$ and $x_{i+1}$. There fourth derivative of $f(x)$ with respect to $x$ vanishes at $x_{i}$. The fourth derivative of $f(x)$ with respect to $x$ vanishes at $x_{i+1}$.

asked Apr 19 in Calculus by admin (3.6k points) | 1 view

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GATE1993-01.5 Video Solution
Fourier series of the periodic function (period 2π) defined by ... $\frac{{\pi }^2 }{4}$ $\frac{{\pi }^2 }{6}$ $\frac{{\pi }^2 }{8}$ $\frac{{\pi }^2 }{12}$

asked Apr 19 in Calculus by admin (3.6k points) | 2 views

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