Calculus · Engineering Mathematics · GATE ECE

Consider the two-dimensional vector field $$\overrightarrow F (x,y) - x\overrightarrow i + y\overrightarrow j $$, where $$\overrightarrow i $$ and $$...

The families of curves represented by the solution of the equation $${{dy} \over {dx}} = - {\left( {{x \over y}} \right)^n}$$ for n = –1 and n = 1 re...

Let $$f\left( {x,y} \right) = {{a{x^2} + b{y^2}} \over {xy}}$$, where $$a$$ and $$b$$ are constants. If $${{\partial f} \over {\partial x}} = {{\parti...

Taylor series expansion of $$f\left( x \right) = \int\limits_0^x {{e^{ - \left( {{{{t^2}} \over 2}} \right)}}} dt$$ around 𝑥 = 0 has the form f(x) = ...

Given the following statements about a function $$f:R \to R,$$ select the right option: $$P:$$ If $$f(x)$$ is continuous at $$x = {x_0},$$ then it is...

The integral $$\int\limits_0^1 {{{dx} \over {\sqrt {\left( {1 - x} \right)} }}} $$ is equal ________.

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

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

The value of $$\sum\limits_{n = 0}^\infty {n{{\left( {{1 \over 2}} \right)}^n}\,\,} $$ 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+4...

A function $$f\left( x \right) = 1 - {x^2} + {x^3}\,\,$$ is defined in the closed interval $$\left[ { - 1,1} \right].$$ The value of $$x,$$ in the ope...

The series $$\sum\limits_{n = 0}^\infty {{1 \over {n!}}\,} $$ converges to

The maximum value of the function $$\,f\left( x \right) = \ln \left( {1 + x} \right) - x$$ (where $$x > - 1$$ ) occurs at $$x=$$________.

If $$z=xy$$ $$ln(xy),$$ then

For $$0 \le t < \infty ,$$ the maximum value of the function $$f\left( t \right) = {e^{ - t}} - 2{e^{ - 2t}}\,$$ occurs at

The value of $$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {1 \over x}} \right)^x}\,\,$$ is

If $$\,{e^y} = {x^{1/x}}\,\,$$ then $$y$$ has a

Which of the following function would have only odd powers of $$x$$ in its Taylor series expansion about the point $$x=0$$ ?

For real values of $$x,$$ the minimum value of function $$f\left( x \right) = {e^x} + {e^{ - x}}\,\,$$ is

$$\mathop {Lim}\limits_{\theta \to 0} {{\sin \left( {\theta /2} \right)} \over \theta }\,\,\,$$ is

The following plot shows a function $$y$$ which varies linearly with $$x$$. The value of the integral $$\,\,{\rm I} = \int\limits_1^2 {y\,dx\,\,} $$ ...

For the function $${e^{ - x}},$$ the linear approximation around $$x=2$$ is

For $$\left| x \right| < < 1,\,\cot \,h\left( x \right)\,\,\,$$ can be approximated as

The value of the integral $$1 = {1 \over {\sqrt {2\pi } }}\,\,\int\limits_0^\infty {{e^{ - {\raise0.5ex\hbox{$\scriptstyle {{x^2}}$} \kern-0.1em/\ker...

The curve given by the equation $${x^2} + {y^2} = 3axy$$ is

By reversing the order of integration $$\int\limits_0^2 {\int\limits_{{x^2}}^{2x} {f\left( {x,y} \right)dy\,dx} } $$ may be represented as ______.

The third term in the taylor's series expansion of $${e^x}$$ about $$'a'$$ would be ________.

The function $$y = {x^2} + {{250} \over x}$$ at $$x=5$$ attains

Consider the Earth to be a perfect sphere of radius $R$. Then the surface area of the region, enclosed by the 60°N latitude circle, that contains the ...

The value of the line integral $$\int_P^Q {({z^2}dx + 3{y^2}dy + 2xz\,dz)} $$ along the straight line joining the points $$P(1,1,2)$$ and $$Q(2,3,1)$$...

The value of the integral $$\int\limits\!\!\!\int_R {xy\,dx\,dy} $$ over the regioin R, given in the figure, is _________ (rounded off to the nearest ...

The function f(x) = 8loge x $$-$$ x2 + 3 attains its minimum over the interval [1, e] at x = __________. (Here loge x is the natural logarithm of x.)...

The value of the integral $$\int\!\!\!\int\limits_D {3({x^2} + {y^2})dx\,dy} $$, where D is the shaded triangular region shown in the diagram, is ____...

Let r = x2 + y - z and z3 - xy + yz + y3 = 1. Assume that x and y are independent variables. At (x, y, z) = (2, -1, 1), the value (correct to two deci...

The minimum value of the function $$f\left( x \right) = {1 \over 3}x\left( {{x^2} - 3} \right)\,\,$$ in the interval $$ - 100 \le x \le $$ $$100$$ oc...

The values of the integrals $$\int\limits_0^1 {\left( {\int\limits_0^1 {{{x - y} \over {{{\left( {x + y} \right)}^3}}}dy} } \right)} dx\,\,$$ and $$\,...

Let $$\,\,f\left( x \right) = {e^{x + {x^2}}}\,\,$$ for real $$x.$$ From among the following. Choose the Taylor series approximation of $$f$$ $$(x)$$ ...

A three dimensional region $$R$$ of finite volume is described by $$\,\,{x^2} + {y^2} \le {z^3},\,\,\,0 \le z \le 1$$ Where $$x, y, z$$ are real. Th...

The integral $$\,\,{1 \over {2\pi }}\int {\int_D {\left( {x + y + 10} \right)dxdy\,\,} } $$ where $$D$$ denotes the disc: $${x^2} + {y^2} \le 4,$$ eva...

The region specified by $$\left\{ {\left( {\rho ,\varphi ,{\rm Z}} \right):3 \le \rho \le 5,\,\,{\pi \over 8} \le \phi \le {\pi \over 4},\,\,3 \l...

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=...

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

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

The maximum area (in square units) of a rectangle whose vertices lie on the ellipse $${x^2} + 4{y^2} = 1\,\,$$ is

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

The volume under the surface $$z\left( {x,y} \right) = x + y$$ and above the triangle in the $$xy$$ plane defined by $$\left\{ {0 \le y \le x} \righ...

For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, t...

The maximum value of $$f\left( x \right) = 2{x^3} - 9{x^2} + 12x - 3$$ in the interval $$\,0 \le x \le 3$$ is __________.

The Taylor series expansion of $$\,\,{{\sin x} \over {x - \pi }}\,\,$$ at $$x = \pi $$ is given by

In the Taylor series expansion of $${e^x} + \sin x$$ about the point $$x = \pi ,$$ the coefficient of $${\left( {x = \pi } \right)^2}$$ is

The value of the integral of the function $$\,\,g\left( {x,y} \right) = 4{x^3} + 10{y^4}\,\,$$ along the straight line segment from the point $$(0,0)...

Consider the function $$\,f\left( x \right) = {x^2} - x - 2.\,$$ The maximum value of $$f(x)$$ in the closed interval $$\left[ { - 4,4} \right]\,$$

$$\int\limits_0^{{\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}} {\int\limits_0^{{\raise0.5ex\hb...