Recent activity in Calculus

Let $I= c\int \int _{R} xy^{2} dxdy$, where $R$ is the region shown in the figure and $c= 6 \times 10^{-4}$. The value of $I$ equals _________. (Give the answer up to two...

Let $R$ be a region in the first quadrant of the $xy$ plane enclosed by a closed curve $C$ considered in counter-clockwise direction. Which of the following expressions d...

A quadratic function of two variables is given as $$ f\left(x_{1}, x_{2}\right)=x_{1}^{2}+2 x_{2}^{2}+3 x_{1}+3 x_{2}+x_{1} x_{2}+1$$The magnitude of the maximum rate of...

Let, $f\left ( x,y, z \right ) = 4x^{2} + 7xy + 3xz^{2}$. The direction in which the function $\text{f(x, y, z)}$ increases most rapidly at point $P = (1,0,2)$ is$20\hat{...

​​​​Let $f(t)$ be a real-valued function whose second derivative is positive for $-\infty < t < \infty.$ Which of the following statements is/are always true?$f(t)$ has a...

The direction of vector $\textbf{A}$ is radically outward from the origin, with $\mid \textbf{A} \mid k r ^n$ where $r^2=x^2+y^2+z^2$ and $k$ is a constant. The value of ...

The value of $\int_{-\infty}^{+\infty} e^{-t} \delta (2t-2){d}t$, where $\delta (t)$ is the Dirac delta function, isA. $\dfrac{1}{2e}$B. $\dfrac{2}{e}$C. $\dfrac{1}{e^{2}...

The value of the integral$$\oint _{c}\dfrac{2z+5}{\left ( z-\dfrac{1}{2} \right ) \left (z^{2} -4z+5 \right )}dz$$over the contour $\mid z \mid=1$, taken in the anti-cloc...

Suppose the circles $x^{2}+y^{2}=1$ and $\left ( x-1\right )^{2}+\left ( y-1 \right )^{2}=r^{2}$ intersect each other orthogonally at the point $\left ( u,v \right )$. Th...

In the open interval $\left ( 0,1 \right )$, the polynomial $p\left ( x \right) =x^{4}-4x^{3}+2$ hastwo real rootsone real rootthree real rootsno real roots

Let $\left ( -1 -j \right ), \left ( 3 -j \right ), \left ( 3 + j \right )$ and $\left ( -1+ j \right )$ be the vertices of a rectangle $C$ in the complex plane. Assuming...

Let $y^{2}-2y+1=x$ and $\sqrt{x}+y=5$. The value of $x+\sqrt{y}$ equals _________. (Give the answer up to three decimal places).

Let $x$ and $y$ be integers satisfying the following equations$2x^{2}+y^{2}=34$$x+2y=11$The value of $(x+y)$ is _______.

Which of the following is true for all possible non-zero choices of integers $m,n;m\neq n,$ or all possible non-zero choices of real numbers $p,q;p\neq q,$ as applicable?...

The vector function expressed by$$F=a_{x}\left ( 5y-k_{1} z\right )+a_{y}\left ( 3z+k_{2}x \right )+a_{z}\left ( k_{3} y-4x\right )$$represents a conservative field, wher...