calculus

The integral $\frac{1}{\pi} \int_0^{\infty} \frac{x^{2026}}{(1 + x^{2026})(1 + x^2)} dx$ evaluates to _______ (Round off to two decimal places)

Consider the set S of points (x, y) ∈ R² which minimize the real valued function f(x, y) = (x + y − 1)² + (x + y)² Which of the following statements is true about the set S?

Consider the set $S$ of points $(x, y) \in R^2$ which minimize the real valued function $$ f(x, y)=(x+y-1)^2+(x+y)^2 $$ Which of the following statements is true about the set $S$...

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

Consider the function f(t) = (max(0,t))² for -∞ < t < ∞, where max(a, b) denotes the maximum of a and b. Which of the following statements is/are true?

Let $f(t)$ be a real-valued function whose second derivative is positive for $- \infty

Consider the function $f(t) = (\text{max}(0,t))^2$ for $- \infty

In the following differential equation, the numerically obtained value of y(t), at t =1, is ________ (Round off to 2 decimal places). $\frac{dy}{dt} = \frac{e^{-at}}{2 + at}$, α =...

Consider the following equation in a 2-D real-space. |x1|^p + |x2|^p = 1 for p > 0 Which of the following statement(s) is/are true.

Consider the following equation in a 2-D real-space. $$|{x_1}{|^p} + |{x_2}{|^p} = 1$$ for $$p > 0$$ Which of the following statement(s) is/are true.

Let $f(x)$ be a real-valued function such that $f^{\prime}\left(x_0\right)=0$ for some $x_0 \in(0,1)$ and $f^{\prime \prime}\left(x_0\right)>0$ for all $x \in(0,1)$. Then $f(x)$ ha...

In the open interval $(0,1)$, the polynomial $p(x)=x^4-4 x^3+2$ has

Let $I = c \iint_R xy^2 \, dxdy$, 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 decimal place...

Let $f$ be a real-valued function of a real variable defined as $f(x)=x-[x]$, where $[x]$ denotes the largest integer less than or equal to $x$. The value of $\int_{0.25}^{1.25} f(...

A function $f(x)$ is defined as $f(x)=\begin{cases} e^x, & x<1 \ \ln x+ax^2+bx, & x\ge1 \end{cases}$, where $x \in \mathbb{R}$. Which one of the following statements is TRUE?

Let f(x) = 3x³ −7x² + 5x + 6. The maximum value of f(x) over the interval [0, 2] is ________ (up to 1 decimal place).

The number of roots of e^x + 0.5x² - 2 = 0 in the range [-5, 5] is

A function $$f(x)$$ is defined as $$f\left( x \right) = \left\{ {\matrix{ {{e^x},x < 1} \cr {\ln x + a{x^2} + bx,x \ge 1} \cr } \,\,,\,\,} \right.$$ where $$x \in R.$$ Which one of...

Let $$g\left( x \right) = \left\{ {\matrix{ { - x} & {x \le 1} \cr {x + 1} & {x \ge 1} \cr } } \right.$$ and $$f\left( x \right) = \left\{ {\matrix{ {1 - x,} & {x \le 0} \cr {{x^{2...

Let $$\,\,S = \sum\limits_{n = 0}^\infty {n{\alpha ^n}} \,\,$$ where $$\,\,\left| \alpha \right| < 1.\,\,$$ The value of $$\alpha $$ in the range $$\,\,0 < \alpha < 1,\,\,$$ such t...

The maximum value attained by the function $$f(x)=x(x-1) (x-2)$$ in the interval $$\left[ {1,2} \right]$$ is _________.

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

The volume enclosed by the surface $$f\left( {x,y} \right) = {e^x}$$ over the triangle bounded by the lines $$x=y;$$ $$x=0;$$ $$y=1$$ in the $$xy$$ plane is ________.

The Laplace transform of $$f\left( t \right) = 2\sqrt {t/\pi } $$$$\,\,\,\,\,$$ is$$\,\,\,\,\,$$ $${s^{ - 3/2}}.$$ The Laplace transform of $$g\left( t \right) = \sqrt {1/\pi t} $$...

If a continuous function $$f(x)$$ does not have a root in the interval $$\left[ {a,b} \right],\,\,$$ then which one of the following statements is TRUE?

Minimum of the real valued function $$f\left( x \right) = {\left( {x - 1} \right)^{2/3}}$$ occurs at $$x$$ equal to

The minimum value of the function $$f\left( x \right) = {x^3} - 3{x^2} - 24x + 100$$ in the interval $$\left[ { - 3,3} \right]$$ is

To evaluate the double integral $$\int\limits_0^8 {\left( {\int\limits_{y/2}^{\left( {y/2} \right) + 1} {\left( {{{2x - y} \over 2}} \right)dx} } \right)dy,\,\,} $$ we make the sub...

Let $$f\left( x \right) = x{e^{ - x}}.$$ The maximum value of the function in the interval $$\left( {0,\infty } \right)$$ is

A function $$y = 5{x^2} + 10x\,\,$$ is defined over an open interval $$x=(1,2).$$ At least at one point in this interval, $${{dy} \over {dx}}$$ is exactly

The maximum value of $$f\left( x \right) = {x^3} - 9{x^2} + 24x + 5$$ in the interval $$\left[ {1,6} \right]$$ is

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

At $$t=0,$$ the function $$f\left( t \right) = {{\sin t} \over t}\,\,$$ has

The value of the quantity, where $$P = \int\limits_0^1 {x{e^x}\,dx\,\,\,} $$ is

If $$(x, y)$$ is continuous function defined over $$\left( {x,y} \right) \in \left[ {0,1} \right] \times \left[ {0,1} \right].\,\,\,$$ Given two constants, $$\,x > {y^2}$$ and $$\,...

The integral $$\,\,{1 \over {2\pi }}\int\limits_0^{2\Pi } {Sin\left( {t - \tau } \right)\cos \tau \,d\tau \,\,\,} $$ equals

Consider the function $$f\left( x \right) = {\left( {{x^2} - 4} \right)^2}$$ where $$x$$ is a real number. Then the function has

The expression $$V = \int\limits_0^H {\pi {R^2}{{\left( {1 - {h \over H}} \right)}^2}dh\,\,\,} $$ for the volume of a cone is equal to _________.

For the function $$f\left( x \right) = {x^2}{e^{ - x}},$$ the maximum occurs when $$x$$ is equal to

If $$S = \int\limits_1^\infty {{x^{ - 3}}dx} $$ then $$S$$ has the value

The area enclosed between the parabola $$y = {x^2}$$ and the straight line $$y=x$$ is _______.

The value of $$\,\,\,\int\limits_1^2 {{1 \over x}\,\,\,dx\,\,\,\,} $$ computed using simpson's rule with a step size of $$h=0.25$$ is

$$\mathop {Lim}\limits_{\theta \to 0} \,{{\sin \,m\,\theta } \over \theta },$$ where $$m$$ is an integer, is one of the following :

If $$f(0)=2$$ and $$f'\left( x \right) = {1 \over {5 - {x^2}}},$$ then the lower and upper bounds of $$f(1)$$ estimated by the mean value theorem are ______.

$$\mathop {Lim}\limits_{x \to \infty } \,x\sin {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle x$}} = \_\_\_\_\_.$$

The integration of $$\int {{\mathop{\rm logx}\nolimits} \,dx} $$ has the value

The volume generated by revolving the area bounded by the parabola $${y^2} = 8x$$ and the line $$x=2$$ about $$y$$-axis is