Calculus · Engineering Mathematics · GATE CE

The function $f(x) = x^3 - 27x + 4$, $1 \leq x \leq 6$ has

The smallest positive root of the equation $$x^5 - 5 x^4 - 10 x^3 + 50 x^2 + 9 x - 45 = 0$$ lies in the range

For the integral $\rm I=\displaystyle\int^1_{-1}\frac{1}{x^2}dx$ which of the following statements is TRUE?  ...

The following function is defined over the interval [-L, L]: f(x) = px4 + qx5. If it is expressed as a Fourier series,  $\rm f(x)=a_0 +\d...

Consider the polynomial f(x) = x3 $$-$$ 6x2 + 11x $$-$$ 6 on the domain S, given by 1 $$\le$$ x $$\le$$ 3. The first and second derivatives are f'(x) ...

$$\int {\left( {x - {{{x^2}} \over 2} + {{{x^3}} \over 3} - {{{x^4}} \over 4} + ....} \right)dx} $$ is equal to :

Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statements is/are TRUE about the function f(x) = max{3 $$-$$ x, ...

A set of observations of independent variable (x) and the corresponding dependent variable (y) is given below. .tg {border-collapse:collapse;border-...

Let $$\,\,W = f\left( {x,y} \right),\,\,$$ where $$x$$ and $$y$$ are functions of $$t.$$ Then, according to the chain rule, $${{dw} \over {dt}}$$ is e...

$$\mathop {Lim}\limits_{x \to 0} \left( {{{\tan x} \over {{x^2} - x}}} \right)$$ is equal to _________.

Let $$x$$ be a continuous variable defined over the interval $$\left( { - \infty ,\infty } \right)$$, and $$f\left( x \right) = {e^{ - x - {e^{ - x}}}...

$$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {1 \over x}} \right)^{2x}}\,\,$$ is equal to

Given $$i = \sqrt { - 1} ,$$ the value of the definite integral, $$\,{\rm I} = \int\limits_0^{\pi /2} {{{\cos x + \sin x} \over {\cos x - i\,\sin x}}d...

$$\,\,\mathop {Lim}\limits_{x \to \infty } \left( {{{x + \sin x} \over x}} \right)\,\,$$ equal to

With reference to the conventional Cartesian $$(x,y)$$ coordinate system, the vertices of a triangle have the following coordinates: $$\,\left( {{x_1...

The solution $$\int\limits_0^{\pi /4} {{{\cos }^4}3\theta {{\sin }^3}\,6\theta d\theta \,\,} $$ is :

The infinite series $$1 + x + {{{x^2}} \over {2!}} + {{{x^3}} \over {3!}} + {{{x^4}} \over {4!}} + ........$$ corresponds to

What should be the value of $$\lambda $$ such that the function defined below is continuous at $$x = {\pi \over 2}$$? $$f\left( x \right) = \left\{ {...

The $$\mathop {Lim}\limits_{x \to 0} {{\sin \left( {{2 \over 3}x} \right)} \over x}\,\,\,$$ is

The value of the function, $$f\left( x \right) = \mathop {Lim}\limits_{x \to 0} {{{x^3} + {x^2}} \over {2{x^3} - 7{x^2}}}\,\,\,$$ is

The following function has local minima at which value of $$x,$$ $$f\left( x \right) = x\sqrt {5 - {x^2}} $$

The value of the following definite integral in $$\int\limits_{{\raise0.5ex\hbox{$\scriptstyle { - \pi }$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{...

Limit of the following series as $$x$$ approaches $${\pi \over 2}$$ is $$f\left( x \right) = x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - {{{x^...

Consider the following integral $$\mathop {Lim}\limits_{x \to 0} \int\limits_1^a {{x^{ - 4}}} dx$$ ________.

The function $$f\left( x \right) = {e^x}$$ is _________.

Number of inflection points for the curve $$\,\,\,y = x + 2{x^4}\,\,\,\,$$ is_______.

Limit of the function, $$\mathop {Lim}\limits_{n \to \infty } {n \over {\sqrt {{n^2} + n} }}$$ is _______.

The continuous function $$f(x, y)$$ is said to have saddle point at $$(a, b)$$ if

A discontinuous real function can be expressed as

The Taylor's series expansion of sin $$x$$ is ______.

If $$\varphi \left( x \right) = \int\limits_0^{{x^2}} {\sqrt t \,dt\,} $$ then $${{d\varphi } \over {dx}} = \_\_\_\_\_\_\_.$$

If $$y = \left| x \right|$$ for $$x < 0$$ and $$y=x$$ for $$x \ge 0$$ then

The function $$f\left( x \right) = \left| {x + 1} \right|$$ on the interval $$\left[ { - 2,0} \right]$$ is __________.

The function $$f\left( x \right) = {x^3} - 6{x^2} + 9x + 25$$ has

The value of $$\varepsilon $$ in the mean value theoram of $$f\left( b \right) - f\left( a \right) = \left( {b - a} \right)\,\,f'\left( \varepsilon \...

The expression for computing the effective interest rate $(i_{eff})$ using continuous compounding for a nominal interest rate of 5% is $i_{eff} = \lim...

For the function f(x) = ex |sin x|; x ∈ ℝ, which of the following statements is/are TRUE?...

Consider the following definite integral $$${\rm I} = \int\limits_0^1 {{{{{\left( {{{\sin }^{ - 1}}x} \right)}^2}} \over {\sqrt {1 - {x^2}} }}dx} $$$...

The tangent to the curve represented by $$y=x$$ $$ln$$ $$x$$ is required to have $${45^ \circ }$$ inclination with the $$x-$$axis. The coordinates of ...

The expression $$\mathop {Lim}\limits_{a \to 0} \,{{{x^a} - 1} \over a}\,\,$$ is equal to

What is the value of the definite integral? $$\,\,\int\limits_0^a {{{\sqrt x } \over {\sqrt x + \sqrt {a - x} }}dx\,\,} $$?

A parabolic cable is held between two supports at the same level. The horizontal span between the supports is $$L.$$ The sag at the mid-span is $$h.$$...

Given a function $$f\left( {x,y} \right) = 4{x^2} + 6{y^2} - 8x - 4y + 8,$$ the optimal values of $$f(x,y)$$ is

The function $$f\left( x \right) = 2{x^3} - 3{x^2} - 36x + 2\,\,\,$$ has its maxima at

Limit of the following sequence as $$n \to \infty $$ $$\,\,\,$$ is $$\,\,\,$$ $${x_n} = {n^{{1 \over n}}}$$

The value of the following improper integral is $$\,\int\limits_0^1 {x\,\log \,x\,dx} = \_\_\_\_\_.$$

Limit of the function $$f\left( x \right) = {{1 - {a^4}} \over {{x^4}}}\,\,as\,\,x \to \infty $$ is given by

The Taylor series expansion of sin $$x$$ about $$x = {\pi \over 6}$$ is given by

If $$f\left( {x,y,z} \right) = $$ $${\left( {{x^2} + {y^2} + {z^2}} \right)^{{\raise0.5ex\hbox{$\scriptstyle { - 1}$} \kern-0.1em/\kern-0.15em \lower...