GATE EC Calculus questions

Let $I=\int_{c}\left ( 2zdx+2ydy+2xdx \right )$ where $x,y,z$ are real, and let $C$ be the straight line segment from point $A:(0,2,1)$ to point $B:(4,1,-1)$.The value of $I$ is ____________.

Starting with $x=1$, the solution of the equation $x^{3}+x=1$, after two iterations of Newton-Raphson’s method (up to two decimal places) is__________.

The values of the integrals $$\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dy \right )dx$$ and $$\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dx \right )dy$$ are

The minimum value of the function $f(x)=\frac{1}{3} x(x^2-3)$ in the interval $-100≤x≤100$ occurs at $x =$ ________.

Given the following statements about a function $f: \Bbb R \rightarrow \Bbb R$, select the right option: P: If $f(x)$ is continuous at $x = x_0$ then it is also differentiable at $x = x_0$. Q: If $f(x)$ is continuous at $x = x_0$ then it may not be differentiable at $x = x_0$. R: If $f(x)$ is differentiable at $x = x_0$ then it is also …

The integral $\frac{1}{2\pi} \iint_D(x+y+10) \,dx\,dy$, where $D$ denotes the disc: $x^2+y^2\leq 4$,evaluates to _________

The region specified by $\{ (\rho,\varphi,z):3 \leq\rho\leq 5,\frac{\pi}{8}\leq\varphi\leq\frac{\pi}{4}, \: 3\leq z\leq4.5\}$ in cylindrical coordinates has volume of _________

As $x$ varies from $-1$ to $+3$, which one of the following describes the behaviour of the function $f(x)=x^{3}-3x^{2}+1?$

How many distinct value of $x$ satisfy the equation $\sin(x)=x/2$, where $x$ is in radians?

The integral $\int\limits_{0}^{1}\large\frac{dx}{\sqrt{(1-x)}}$ is equal to _______

A triangle in the $xy$-plane is bounded by the straight lines $2x=3y, \: y=0$ and $x=3$. The volume above the triangle and under the plane $x+y+z=6$ is _______

A function $f(x)=1-x^2+x^3$ is defined in the closed interval $[-1,1]$. The value of $x$, in the open interval $(-1,1)$ for which the mean value theorem is satisfied, is

Which one of the following graphs describes the function $f(x)=e^{-x}(x^2+x+1)$?

The maximum area (in square units) of a rectangle whose vertices lie on the eclipse $x^2+4y^2=1$ is __________.

The value of the integral $\int_{-\infty}^{\infty} 12\cos(2\pi t) \dfrac{\sin(4\pi t)}{4 \pi t}dt$ is _________.

The contour on the $x-y$ plane, where the partial derivative of $x^{2} + y^{2}$ with respect to $y$ is equal to the partial derivative of $6y+4x$ with respect to $x$, is

Consider the function $g(t) = e^{-t}\sin(2\pi t)u(t)$ where $u(t)$ is the unit step function. The area under $g(t)$ is ______.

The value of $\displaystyle{}\sum_{n=0}^{\infty} n \left(\dfrac{1}{2}\right)^{n}$ is ________.

The Newton-Raphson method is used to solve the equation $f(x) = x^{3} – 5x^{2} + 6x – 8 = 0.$ Taking the initial guess as $x = 5,$ the solution obtained at the end of the first iteration is __________.

The Taylor series expansion of $3\sin x + 2 \cos x$ is