Recent questions tagged calculus

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1

GATE2020-CE-1-2
The value of $\lim_{x\to\infty}\dfrac{x^2-5x+4}{4x^2+2x}$ is $0 \\$ $\dfrac{1}{4} \\$ $\dfrac{1}{2} \\$ $1$

asked Feb 28, 2020 in Engineering Mathematics by jothee (2.7k points)

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2

GATE2020-CE-1-27
A continuous function $f(x)$ is defined. If the third derivative at $x_i$ is to be computed by using the fourth order central finite-divided-difference scheme (with step length $=h$ ...

asked Feb 28, 2020 in Calculus by jothee (2.7k points)

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GATE2020-CE-1-39
If $C$ represents a line segment between $(0,0,0)$ and $(1,1,1)$ in Cartesian coordinate system, the value (expressed as integer) of the line integral $\int_C [(y+z)dx+(x+z)dy+(x+y)dz] $ is ______

asked Feb 28, 2020 in Calculus by jothee (2.7k points)

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4

GATE2020 CE-2-3
The integral $\int\limits_{0}^{1} (5x^3 + 4x^2 + 3x + 2) dx$ is estimated numerically using three alternative methods namely the rectangular,trapezoidal and Simpson's rules with a common step size. In this context, which one of the following ... NON-zero error. Only the rectangular rule of estimation will give zero error. Only Simpson's rule of estimation will give zero error.

asked Feb 13, 2020 in Calculus by jothee (2.7k points)

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GATE2020 CE-2-24
Velocity distribution in a boundary layer is given by $\dfrac{u}{U_\infty} = \sin\large \left( \dfrac{\pi}{2}\dfrac{y}{\delta} \right)$, where $u$ is the velocity at vertical coordinate $y,\: U_\infty$ is the free stream velocity and $\delta$ is the boundary layer ... $\ s^{-1}$, round off to two decimal places) at $y = 0$, is ________.

asked Feb 13, 2020 in Calculus by jothee (2.7k points)

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6

GATE2017 CE-2-19
The divergence of the vector field $V=x^2 i + 2y^3 j + z^4 k$ at $x=1, \: y=2, \: z=3$ is ________

asked Aug 7, 2019 in Calculus by gatecse (3.9k points)

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GATE2017 CE-2-27
Consider the following definite integral: $I= \int_0^1 \dfrac{(\sin ^{-1}x)^2}{\sqrt{1-x^2}} dx$. The value of the integral is $\dfrac{\pi ^3}{24} \\$ $\dfrac{\pi ^3}{12} \\$ $\dfrac{\pi ^3}{48} \\$ $\dfrac{\pi ^3}{64}$

asked Aug 7, 2019 in Calculus by gatecse (3.9k points)

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8

GATE2019 CE-1: 1
Which one of the following is correct? $\lim_{x\rightarrow 0} ( \dfrac{\sin4x}{\sin2x})=2 $ and $\lim_{x\rightarrow 0} ( \dfrac{\tan x}{x})=1 \\$ $\lim_{x\rightarrow 0} ( \dfrac{\sin4x}{\sin2x})=1$ and $\lim_{x\rightarrow 0} ( \dfrac{\tan x}{x})=1 \\$ ... $\lim_{x\rightarrow 0} ( \dfrac{\sin4x}{\sin2x})=2$ and $\lim_{x\rightarrow 0} ( \dfrac{\tan x}{x})= \infty$

asked Feb 14, 2019 in Calculus by Arjun (2.8k points)

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GATE2019 CE-1: 4
For a small value of $h$, the Taylor series expansion for $f(x+h)$ is $f(x)+h{f}' (x) + \dfrac{h^2}{2!}{f}''(x) + \dfrac{h^3}{3!}{f}'''(x)+\dots \infty \\$ $f(x)-h{f}' (x) + \dfrac{h^2}{2!}{f}''(x) - \dfrac{h^3}{3!}{f}'''(x)+ \dots \infty \\$ ... $f(x)-h{f}' (x) + \dfrac{h^2}{2}{f}''(x) - \dfrac{h^3}{3}{f}'''(x)+ \dots \infty $

asked Feb 14, 2019 in Calculus by Arjun (2.8k points)

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GATE2019 CE-1: 26
Which one of the following is NOT a correct statement? The function $\sqrt[x]{x}, \: (x>0)$, has the global maxima at $x=e$ The function $\sqrt[x]{x}, \: (x>0)$, has the global minima at $x=e$ The function $x^3$ has neither global minima nor global maxima The function $\mid x \mid$ has the global minima at $x=0$

asked Feb 14, 2019 in Calculus by Arjun (2.8k points)

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11

GATE2019 CE-1: 30
Consider two functions: $x=\psi \text{ ln } \phi$ and $y= \phi \text{ ln } \psi$. Which one of the following is the correct expression for $\frac{\partial \psi}{\partial x}$? $\dfrac{x \: \text{ln } \psi}{\text{ln } \phi \text{ ln } \psi -1} \\$ ... $\dfrac{\: \text{ln } \psi}{\text{ln } \phi \text{ ln } \psi -1}$

asked Feb 14, 2019 in Calculus by Arjun (2.8k points)

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12

GATE2019 CE-2: 3
The following inequality is true for all $x$ close to $0$. $2-\dfrac{x^2}{3} < \dfrac{x \sin x}{1- \cos x} <2$ What is the value of $\underset{x \to 0}{\lim} \dfrac{x \sin x}{1 – \cos x}$? $0$ $1/2$ $1$ $2$

asked Feb 12, 2019 in Calculus by Arjun (2.8k points)

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13

GATE2019 CE-2: 4
What is curl of the vector field $2x^2y \textbf{i} + 5z^2 \textbf{j} – 4yz \textbf{k}$? $6z \textbf{i} + 4x \textbf{j} – 2x^2 \textbf{k}$ $6z \textbf{i} - 8xy \textbf{j} + 2x^2 y\textbf{k}$ $- 14 z \textbf{i} + 6y \textbf{j} + 2x^2 \textbf{k}$ $-14z \textbf{i} – 2x^2 \textbf{k}$

asked Feb 12, 2019 in Calculus by Arjun (2.8k points)

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14

GATE2016-2-2
The optimum value of the function $f(x)=x^2-4x+2$ is $2$ (maximum) $2$ (minimum) $ – 2$ (maximum) $ – 2$ (minimum)

asked Mar 28, 2018 in Calculus by Milicevic3306 (11.9k points)

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15

GATE2016-2-5
What is the value of $\underset{x \rightarrow 0 \\ y \rightarrow 0}{\lim} \dfrac{xy}{x^2+y^2}$? $1$ $-1$ $0$ Limit does not exist

asked Mar 28, 2018 in Calculus by Milicevic3306 (11.9k points)

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16

GATE2016-2-29
The area between the parabola $x^2=8y$ and the straight line $y=8$ is _______.

asked Mar 28, 2018 in Calculus by Milicevic3306 (11.9k points)

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17

GATE2016-1-29
The magnitudes of vectors $\textbf{P, Q}$ and $\textbf{R}$ are $100$ kN, $250$ kN and $150$ kN, respectively as shown in the figure. The respective values of the magnitude (in kN) and the direction (with respect to the x-axis) of the resultant vector are $290.9$ and $96.0^{\circ}$ $368.1$ and $94.7^{\circ}$ $330.4$ and $118.9^{\circ}$ $400.1$ and $113.5^{\circ}$

asked Mar 28, 2018 in Calculus by Milicevic3306 (11.9k points)

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18

GATE2017 CE-1: 2
The number of parameters in the univariate exponential and Gaussian distributions, repsectively, are $2$ and $2$ $1$ and $2$ $2$ and $1$ $1$ and $1$

asked Mar 26, 2018 in Engineering Mathematics by Milicevic3306 (11.9k points)

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19

GATE2017 CE-1: 3
Let $x$ be a continuous variable defined over the interval $(- \infty, \infty)$, and $f(x) = e^{-x-e^{-x}}$. The integral $g(x)= \int f(x) \: dx$ is equal to $e^{e^{-x}}$ $e^{-e^{-x}}$ $e^{-e^{x}}$ $e^{-x}$