# Recent questions tagged calculus

For a two-dimensional, incompressible flow having velocity components $u$ and $v$ in the $x$ and $y$ directions, respectively, the expression $\frac{\partial \left ( u^{2} \right )}{\partial x}+\frac{\partial \left ( uv \right )}{\partial y}$ ... $u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}$
A two dimensional flow has velocities in $x$ and $y$ directions given by $u = 2xyt$ and $v = -y^{2}t$, where $\text{t}$ denotes time. The equation for streamline passing through $x=1,\:y=1$ is $x^{2}y=1$ $xy^{2}=1$ $x^{2}y^{2}=1$ $x/y^{2}=1$
Value of $\left ( 1+i \right )^{8}$, where $i=\sqrt{-1}$, is equal to $4$ $16$ $4i$ $16i$
The value of $\int_{0}^{^{\pi }/_{2}}\int_{0}^{\cos\theta }r\sin\theta \:dr\:d\theta$ is $0$ $\frac{1}{6}$ $\frac{4}{3}$ $\pi$
The value of $\displaystyle{} \lim_{x \rightarrow 0} \left( \frac{1 – \cos x}{x^{2}}\right)$ is $\frac{1}{4}$ $\frac{1}{3}$ $\frac{1}{2}$ $1$
Let $\text{C}$ represent the unit circle centered at origin in the complex plane, and complex variable, $z=x+iy$. The value of the contour integral $\oint _{C}\dfrac{\cosh \:3z}{2z}\:dz$ (where integration is taken counter clockwise) is $0$ $2$ $\pi i$ $2 \pi i$
Let $f\left ( x \right )=x^{2}-2x+2$ be a continuous function defined on $x \in \left [ 1,3 \right ]$. The point $x$ at which the tangent of $f\left ( x \right )$ becomes parallel to the straight line joining $f\left ( 1 \right )$ and $f\left ( 3 \right )$ is $0$ $1$ $2$ $3$
Let $I=\displaystyle \int_{x=0}^1 \int_{y=0}^{x^2} xy^2 dy \: dx$. Then, $I$ may also be expressed as $\displaystyle \int_{y=0}^1 \int_{x=0}^{\sqrt{y}} xy^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=\sqrt{y}}^1 yx^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=\sqrt{y}}^1 xy^2 dx \: dy$ $\displaystyle \int_{y=0}^1 \int_{x=0}^{\sqrt{y}} yx^2 dx \: dy$
The directional derivative of $f(x,y,z) = xyz$ at point $(-1,1,3)$ in the direction of vector $\hat{i} – 2 \hat{j} +2 \hat{k}$ is $3\hat{i} – 3 \hat{j} - \hat{k} \\$ $- \dfrac{7}{3} \\$ $\dfrac{7}{3} \\$ $7$
The function $f(z)$ of complex variable $z=x+iy$, where $i=\sqrt{-1}$, is given as $f(z)=(x^3-3xy^2)+i \: v(x,y)$. For this function to be analytic, $v(x,y)$ should be $(3xy^2-y^3) +$ constant $(3x^2y^2-y^3) +$ constant $(x^3-3x^2 y) +$ constant $(3x^2y-y^3) +$ constant
The value of $\displaystyle{}\lim_{x \to \infty}\left ( \dfrac{1 -e^{-c\left ( 1-x \right )}}{1-x\:e^{-c\left ( 1-x \right )}} \right )$ is $\text{c} \\$ $\text{c + 1} \\$ $\dfrac{c}{c+1} \\$ $\dfrac{c+1}{c}$
Which of the following function $f(z)$, of the complex variable $z,$ is NOT analytic at all the points of the complex plane? $f\left ( z \right )=z^{2}$ $f\left ( z \right )=e^{z}$ $f\left ( z \right )=\sin z$ $f\left ( z \right )=\log z$
A vector field is defined as ... spherical shell formed by two concentric spheres with origin as the center, and internal and external radii of $1$ and $2$, respectively, is $0$ $2\pi$ $4\pi$ $8\pi$
An analytic function of a complex variable $z=x + iy \left ( i=\sqrt{-1} \right )$ is defined as $f\left ( z \right )=x^{2}-y^{2}+i\psi \left ( x,y \right ),$ where $\psi \left ( x,y \right )$ is a real function. The value of the imaginary part of $f(z)$ at $z=\left ( 1+i \right )$ is __________ (round off to $2$ decimal places).
The directional derivative of the function $f(x,y)=x^2+y^2$ along a line directed from $(0,0)$ to $(1,1)$, evaluated at the point $x=1, y=1$ is $\sqrt{2}$ $2$ $2 \sqrt{2}$ $4 \sqrt{2}$
An analytic function $f(z)$ of complex variable $z=x+iy$ may be written as $f(z)=u(x,y)+iv(x,y)$. Then $u(x,y)$ and $v(x,y)$ ...
Given a vector $\overrightarrow{u} = \dfrac{1}{3} \big(-y^3 \hat{i} + x^3 \hat{j} + z^3 \hat{k} \big)$ and $\hat{n}$ as the unit normal vector to the surface of the hemipshere $(x^2+y^2+z^2=1; \: z \geq 0)$ ... $S$ is $- \dfrac{\pi}{2} \\$ $\dfrac{\pi}{3} \\$ $\dfrac{\pi}{2} \\$ $\pi$
The derivative of $f(x)= \cos x$ can be estimated using the approximation $f'(x)=\dfrac{f(x+h)-f(x-h)}{2h}$. The percentage error is calculated as $\bigg( \dfrac{\text{Exact value - Approximate value}}{\text{Exact value}} \bigg) \times 100$. The percentage error in the derivative of $f(x)$ at $x=\pi /6$ ... $<0.1 \%$ $> 0.1 \% \text{ and } <1 \%$ $> 1 \% \text{ and } <5 \%$ $>5 \%$
A parabola $x=y^2$ with $0 \leq x \leq 1$ is shown in the figure. The volume of the solid of rotation obtained by rotating the shaded area by $360^{\circ}$ around x-axis is $\dfrac{\pi}{4} \\$ $\dfrac{\pi}{2} \\$ ${\pi} \\$ $2 \pi$
The value of the following definite integral is __________ (round off to three decimal places) $\int_1^e (x \: \ln \: x) dx$
1 vote
For a particular project, eight activities are to be carried out. Their relationships with other activities and expected durations are mentioned in the table below. ... $a-b-e-g-h$ $a-c-g-h$ $a-d-f-h$ $a-b-c-f-h$
For a particular project, eight activities are to be carried out. Their relationships with other activities and expected durations are mentioned in the table below. ... duration to complete the project remain the same critical path changes but the total duration to complete the project changes to $17$ days
For the spherical surface $x^2+y^2+z^2=1$, the unit outward normal vector at th point $\left( \dfrac{1}{\sqrt{2}}, \dfrac{1}{\sqrt{2}}, 0\right)$ is given by $\dfrac{1}{\sqrt{2}} \hat{i} +\dfrac{1}{\sqrt{2}} \hat{j} \\$ $\dfrac{1}{\sqrt{2}} \hat{i} -\dfrac{1}{\sqrt{2}} \hat{j} \\$ $\hat{k} \\$ $\dfrac{1}{\sqrt{3}} \hat{i} +\dfrac{1}{\sqrt{3}} \hat{j} +\dfrac{1}{\sqrt{3}} \hat{k}$
At $x=0$, the function $f(x)=x^3+1$ has a maximum value a minimum value a singularity a point of inflection
Consider the function $f(x) = \mid x \mid$ in the interval $-1 \leq x \leq 1$. At the point $x=0, \: f(x)$ is continuous and differentiable non-continuous and differentiable continuous and non-differentiable neither continuous nor differentiable
$\underset{x \rightarrow 0}{\lim} \bigg( \dfrac{1- \cos x}{x^2} \bigg)$ is $1/4$ $1/2$ $1$ $2$
The area enclosed between the straight line $y=x$ and the parabola $y=x^2$ in the $x-y$ plane is $1/6$ $1/4$ $1/3$ $1/2$
For a position vector $\overrightarrow{r} = x \hat{i}+y \hat{j} + z\hat{k}$ the norm of the vector can be defined as $\mid \overrightarrow{r} \mid = \sqrt{x^2+y^2+z^2}$. Given a function $\phi =\text{ln} \mid \overrightarrow{r} \mid$ ... $\dfrac{\overrightarrow{r}}{\overrightarrow{r} \cdot \overrightarrow{r} } \\$ $\dfrac{\overrightarrow{r}}{\mid \overrightarrow{r} \mid^3}$
Let $z$ be a complex variable. For a counter-clockwise integration around a unit circle $C$, centered at origin, $\oint_C \frac{1}{5z-4} dz=A \pi i$, the value of $A$ is $2/5$ $1/2$ $2$ $4/5$
The divergence of the vector field $\overrightarrow{u}=e^x(\cos \: y\hat{i}+\sin \: y \hat{j})$ is $0$ $e^x \cos y + e^x \sin y$ $2e^x \cos y$ $2e^x \sin y$
The Fourier cosine series for an even function $f(x)$ is given by $f(x)=a_0 + \Sigma_{n=1}^\infty a_n \cos (nx).$ The value of the coefficient $a_2$ for the function $f(x)=\cos ^2 (x)$ in $[0, \pi]$ is $-0.5$ $0.0$ $0.5$ $1.0$
The value of the integral over the closed surface $S$ bounding a volume $V$, where $\overrightarrow{r} = x \hat{i} + y \hat{j}+z \hat{k}$ is the position vector and $\overrightarrow{n}$ is the normal to the surface $S$, is $V$ $2V$ $3V$ $4V$
$F(z)$ is a function of the complex variable $z=x+iy$ given by $F(z)+ i \: z + k \: Re(z) + i \: Im(z)$. For what value of $k$ will $F(z)$ satisfy the Cauchy-Riemann equations? $0$ $1$ $-1$ $y$
According to the Mean Value Theorem, for a continuous function $f(x)$ in the interval $[a,b]$, there exists a value $\xi$ in this interval such that $\int_a^b f(x) dx =$ $f(\xi)(b-a)$ $f(b)(\xi-a)$ $f(a)(b-\xi)$ $0$
If $f(z)=(x^{2}+ay^{2})+i bxy$ is a complex analytic function of $z=x+iy$, where $i=\sqrt{-1}$, then $a=-1, b=-1$ $a=-1, b=2$ $a=1, b= 2$ $a=2, b=2$
The surface integral $\int \int _{s} F.n$ dS over the surface $S$ of the sphere $x^{2}+y^{2}+z^{2}=9$, where $F=(x+y) i+(x+z) j+(y+z)k$ and $n$ is the unit outward surface normal, yields ________.
The divergence of the vector $-yi+xj$ is ________.
A parametric curve defined by $x= \cos \left ( \dfrac{\Pi u}{2} \right ), y= \sin \left ( \dfrac{\Pi u}{2} \right )$ in the range $0 \leq u \leq 1$ is rotated about the $X$-axis by $360$ degrees. Area of the surface generated is $\dfrac{\Pi }{2} \\$ $\pi \\$ $2 \pi \\$ $4 \pi$
For the vector $\vec{V}=2yz\hat{i}+3xz \hat{j}+4xy \hat{k}$, the value of $\bigtriangledown$. $(\bigtriangledown \times \vec{V})$ is ________.
The value of $\displaystyle{}\lim_{x \rightarrow 0}\dfrac{x^{3}-\sin(x)}{x}$ is $0$ $3$ $1$ $-1$