Calculus · Engineering Mathematics · GATE ME

Let $f(.)$ be a twice differentiable function from $ \mathbb{R}^{2} \rightarrow \mathbb{R}$. If $P, \mathbf{x}_{0} \in \mathbb{R}^{2}$ where $\vert \v...

The figure shows the plot of a function over the interval [-4, 4]. Which one of the options given CORRECTLY identifies the function? ...

Given $\int^{\infty}_{-\infty}e^{-x^2}dx=\sqrt{\pi}$ If a and b are positive integers, the value of $\int^{\infty}_{-\infty}e^{-a(x+b)^2}dx...

A polynomial ψ(s) = ansn + an-1sn-1 + ......+ a1s + a0 of degree n > 3 with constant real coefficients an, an-1, ... a0&nb...

The limit $\rm p = \displaystyle\lim_{x \rightarrow \pi} \left( \frac{x^2 + α x + 2 \pi^2}{x - \pi + 2 \sin x} \right)$ has a finite value for a rea...

Define [x] as the greatest integer less than or equal to x, for each x ϵ (-∞, ∞). If y = [x], then area under y for x ϵ [1,4] is

The value of $$\mathop {\lim }\limits_{x \to 0} \left( {{{{x^3} - \sin \left( x \right)} \over x}} \right)$$ is

The values of $$x$$ for which the function $$f\left( x \right) = {{{x^2} - 3x - 4} \over {{x^2} + 3x - 4}}$$ is NOT continuous are

$$\mathop {Lt}\limits_{x \to 0} {{{{\log }_e}\left( {1 + 4x} \right)} \over {{e^{3x}} - 1}}$$ is equal to

The value of $$\mathop {Lim}\limits_{x \to 0} \left( {{{ - \sin x} \over {2\sin x + x\cos x}}} \right)\,\,\,$$ is __________.

At $$x=0,$$ the function $$f\left( x \right) = \left| x \right|$$ has

The value of $$\mathop {Lim}\limits_{x \to 0} \,{{1 - \cos \left( {{x^2}} \right)} \over {2{x^4}}}$$ is

The value of the integral $$\int\limits_0^2 {{{{{\left( {x - 1} \right)}^2}\sin \left( {x - 1} \right)} \over {{{\left( {x - 1} \right)}^2} + \cos \le...

$$\mathop {Lt}\limits_{x \to 0} \left( {{{{e^{2x}} - 1} \over {\sin \left( {4x} \right)}}} \right)\,\,$$ is equal to

If a function is continuous at a point,

$$\mathop {Lt}\limits_{x \to 0} {{x - \sin x} \over {1 - \cos x}}$$ is

The value of the definite integral $$\int_1^e {\sqrt x \ln \left( x \right)dx} $$ is

The area enclosed between the straight line $$y=x$$ and the parabola $$y = {x^2}$$ in the $$x-y$$ plane is

Consider the function $$f\left( x \right) = \left| x \right|$$ in the interval $$\,\, - 1 \le x \le 1.\,\,\,$$ At the point $$x=0, f(x)$$ is

$$\,\mathop {Lim}\limits_{x \to 0} \left( {{{1 - \cos x} \over {{x^2}}}} \right)$$ is

At $$x=0,$$ the function $$f\left( x \right) = {x^3} + 1$$ has

If $$f(x)$$ is even function and a is a positive real number , then $$\int\limits_{ - a}^a {f\left( x \right)dx\,\,} $$ equals ________.

A series expansion for the function $$\sin \theta $$ is _______.

What is $$\mathop {Lim}\limits_{\theta \to 0} {{\sin \theta } \over \theta }\,\,$$ equal to ?

The parabolic are $$y = \sqrt x ,1 \le x \le 2$$ is revolved around the $$x$$-axis. The volume of the solid of revolution is

The function $$y = \left| {2 - 3x} \right|$$

The value of the integral $$\int\limits_{ - a}^a {{{dx} \over {1 + {x^2}}}} $$

The area enclosed between the curves $${y^2} = 4x\,\,$$ and $${{x^2} = 4y}$$ is

The distance between the origin and the point nearest to it on the surface $$\,\,{z^2} = 1 + xy\,\,$$ is

In the Taylor series expansion of $${e^x}$$ about $$x=2,$$ the coefficient of $$\,\,{\left( {x - 2} \right)^4}\,\,$$ is

The value of $$\,\,\mathop {Lim}\limits_{x \to 8} {{{x^{1/3}} - 2} \over {x - 8}}\,\,$$ is

The minimum value of function $$\,\,y = {x^2}\,\,$$ in the interval $$\,\,\left[ {1,5} \right]\,\,$$ is

$$\int\limits_{ - a}^a {\left[ {{{\sin }^6}\,x + {{\sin }^7}\,x} \right]dx} $$ is equal to

Changing the order of integration in the double integral $${\rm I} = \int\limits_0^8 {\int\limits_{{\raise0.5ex\hbox{$\scriptstyle x$} \kern-0.1em/\k...

If $$\,\,\,x = a\left( {\theta + Sin\theta } \right)$$ and $$y = a\left( {1 - Cos\theta } \right)$$ then $$\,\,{{dy} \over {dx}} = \,\_\_\_\_\_.$$

Value of the function $$\mathop {Lim}\limits_{x \to a} \,{\left( {x - a} \right)^{x - a}}$$ is _______.

Area bounded by the curve $$y = {x^2}$$ and the lines $$x=4$$ and $$y=0$$ is given by

If a function is continuous at a point its first derivative

The area bounded by the parabola $$2y = {x^2}$$ and the lines $$x=y-4$$ is equal to _________.

If $$H(x, y)$$ is homogeneous function of degree $$n$$ then $$x{{\partial H} \over {\partial x}} + y{{\partial H} \over {\partial y}} = nH$$

The value of $$\int\limits_0^\infty {{e^{ - {y^3}}}.{y^{1/2}}} $$ dy is _________.

$$\mathop {Lim}\limits_{x \to 0} {{x\left( {{e^x} - 1} \right) + 2\left( {\cos x - 1} \right)} \over {x\left( {1 - \cos x} \right)}} = \_\_\_\_\_\_.$$...

The function $$f\left( {x,y} \right) = {x^2}y - 3xy + 2y + x$$ has

If the value of the double integral$\int_{x=3}^{4} \int_{y=1}^{2} \frac{dydx}{(x + y)^2}$is $\log_e(\frac{a}{24})$, then $a$ is __________ (answer in ...

A parametric curve defined by $$x = \cos \left( {{{\pi u} \over 2}} \right),y = \sin \left( {{{\pi u} \over 2}} \right)\,\,$$ in the range $$0 \le u \...

$$\mathop {Lt}\limits_{x \to \infty } \left( {\sqrt {{x^2} + x - 1} - x} \right)$$ is

Consider the function $$f\left( x \right) = 2{x^3} - 3{x^2}\,\,$$ in the domain $$\,\left[ { - 1,2} \right].$$ The global minimum of $$f(x)$$ is _____...

Consider an ant crawling along the curve $$\,{\left( {x - 2} \right)^2} + {y^2} = 4,$$ where $$x$$ and $$y$$ are in meters. The ant starts at the poin...

Consider a spatial curve in three -dimensional space given in parametric form by $$\,\,x\left( t \right)\,\, = \,\,\cos t,\,\,\,y\left( t \right)\,\, ...

The value of the integral $$\,\int\limits_0^2 {\int\limits_0^x {{e^{x + y}}\,\,dy} } $$ $$dx$$ is

A political party orders an arch for the entrance to the ground in which the annual convention is being held. The profile of the arch follows the equa...

The infinite series $${\,f\left( x \right) = x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - {{{x^7}} \over {7!}} + - - - \,\,}$$ Converges to

Let $$\,\,f = {y^x}.$$ What is $$\,\,{{{\partial ^2}f} \over {\partial x\partial y}}\,\,$$ at $$x=2,$$ $$y=1$$?

The length of the curve $$\,y = {2 \over 3}{x^{3/2}}$$ between $$x=0$$ & $$x=1$$ is

Which of the following integrals is unbounded?

Consider the shaded triangular region $$P$$ shown in the figure. What is $$\int\!\!\!\int\limits_p {xy\,dx\,dy\,?} $$ ...

If $$\,\,\,y = x + \sqrt {x + \sqrt {x + \sqrt {x + .....\alpha } } } \,\,\,$$ then $$y(2)=$$ __________.

$$\mathop {Lim}\limits_{x \to 0} {{{e^x} - \left( {1 + x + {{{x^2}} \over 2}} \right)} \over {{x^3}}} = $$

By a change of variables $$x(u, v) = uv,$$ $$\,\,y\left( {u,v} \right) = {v \over u}$$ in a double integral, the integral $$f(x, y)$$ changes to $$\,...

The volume of an object expressed in spherical co-ordinates is given by $$V = \int\limits_0^{2\pi } {\int\limits_0^{{\raise0.5ex\hbox{$\scriptstyle \...