calculus

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as follows: $$ f(x)=\left\{\begin{array}{cc} c_1 e^x-c_2 \log _e\left(\frac{1}{x}\right), & \text { if } x>0 \\...

For a real number $a$, let $I(a)=\int\limits_{-1}^1\left(3 x^2-a x+1\right) d x$. Which of the following statements is/are true?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as follows: $$ f(x)=\left(\frac{|x|}{2}-x\right)\left(x-\frac{|x|}{2}\right) $$ Which of the following statements is/are true?

The value of x such that x > 1, satisfying the equation $\int_{1}^{x} t \ln t \, dt = \frac{1}{4}$ is

Consider the given function $f(x)$. $f(x) = \begin{cases} ax + b & \text{for } x < 1 \\ x^3 + x^2 + 1 & \text{for } x \ge 1 \end{cases}$ If the function is differentiable everywher...

Consider the given function $f(x)$. $$f(x)=\left\{\begin{array}{cc} a x+b & \text { for } x If the function is differentiable everywhere, the value of $b$ must be _________ (Rounde...

The value of $x$ such that $x>1$, satisfying the equation $\int_1^x t \ln t d t=\frac{1}{4}$ is

Let f: R → R be a function such that f(x) = max{x,x³}, x ∈ R, where R is the set of all real numbers. The set of all points where f(x) is NOT differentiable is

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x) = \max \{x, x^3\}, x \in \mathbb{R}$, where $\mathbb{R}$ is the set of all real numbers. The set of all po...

Let $f(x)$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x) = 1 - f(2 - x)$ Which one of the following options is the CORRECT value of $\int_0^2 f(x) dx$?

Let $$f(x) = {x^3} + 15{x^2} - 33x - 36$$ be a real-valued function. Which of the following statements is/are TRUE?

The value of the definite integral $$\int\limits_{ - 3}^3 {\int\limits_{ - 2}^2 {\int\limits_{ - 1}^1 {(4{x^2}y - {z^3})dz\,dy\,dx} } } $$ is ___________. (Rounded off to the neare...

A function y(x) is defined in the interval [0, 1] on the x-axis as $$y(x) = \left\{ \matrix{ 2\,if\,0 \le x Which one of the following is the area under the curve for the interval...

The value of the following limit is _____________. $$\mathop {\lim }\limits_{x \to {0^ + }} {{\sqrt x } \over {1 - {e^{2\sqrt x }}}}$$

Suppose that f : R → R is a continuous function on the interval [-3, 3] and a differentiable function in the interval (-3, 3) such that for every x in the interval, f'(x) ≤ 2. If f...

Consider the following expression $$\mathop {\lim }\limits_{x \to -3} \frac{{\sqrt {2x + 22} - 4}}{{x + 3}}$$ The value of the above expression (rounded to 2 decimal places) is ___...

Consider the functions I. $${e^{ - x}}$$ II. $${x^2} - \sin x$$ III. $$\sqrt {{x^3} + 1} $$ Which of the above functions is/are increasing everywhere in [0,1]?

Compute $$\mathop {\lim }\limits_{x \to 3} {{{x^4} - 81} \over {2{x^2} - 5x - 3}}$$

The value of $$\int_0^{\pi /4} {x\cos \left( {{x^2}} \right)dx} $$ correct to three decimal places (assuming that $$\pi = 3.14$$ ) is ________.

If $f(x) = R \sin (\frac{\pi x}{2}) + S$, $f' (\frac{1}{2}) = \sqrt{2}$ and $\int_0^1 f(x)dx = \frac{2R}{\pi}$, then the constants R and S are, respectively

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

The value of $$\mathop {\lim }\limits_{x \to 1} {{{x^7} - 2{x^5} + 1} \over {{x^3} - 3{x^2} + 2}}.$$

If $$f\left( x \right)\,\,\, = \,\,\,R\,\sin \left( {{{\pi x} \over 2}} \right) + S.f'\left( {{1 \over 2}} \right) = \sqrt 2 $$ and $$\int_0^1 {f\left( x \right)dx = {{2R} \over \p...

$$\mathop {\lim }\limits_{x \to 4} {{\sin \left( {x - 4} \right)} \over {x - 4}} = \_\_\_\_\_\_\_.$$

Let $$\,\,f\left( x \right) = {x^{ - \left( {1/3} \right)}}\,\,$$ and $${\rm A}$$ denote the area of the region bounded by $$f(x)$$ and the $$X-$$axis, when $$x$$ varies from $$-1$...

$$\,\int\limits_{1/\pi }^{2/\pi } {{{\cos \left( {1/x} \right)} \over {{x^2}}}dx = } $$ __________.

The value of $$\mathop {\lim }\limits_{x \to \alpha } {\left( {1 + {x^2}} \right)^{{e^{ - x}}}}\,\,$$ is

If for non-zero $$x,$$ $$af\left( x \right) + bf\left( {{1 \over x}} \right) = {1 \over x} - 25$$ where $$a \ne b$$ then $$\int\limits_1^2 {f\left( x \right)dx} \,$$ is

$$\,\,\mathop {\lim }\limits_{x \to \infty } \,{x^{1/x}}\,\,$$ is

The function $$f(x) =$$ $$x$$ $$sinx$$ satisfies the following equation: $$f$$"$$\left( x \right) + f\left( x \right) + t\,\cos \,x\,\, = \,\,0$$. The value of $$t$$ is ______ .

The value of the integral given below is $$$\int_0^\pi {{x^2}\,\cos \,x\,dx} $$$

Let the function $$f\left( \theta \right) = \left| {\matrix{ {\sin \,\theta } & {\cos \,\theta } & {\tan \,\theta } \cr {\sin \left( {{\pi \over 6}} \right)} & {\cos \left( {{\pi \...

A function $$f(x)$$ is continuous in the interval $$\left[ {0,2} \right]$$. It is known that $$f(0)$$ $$=$$ $$f(2)$$ $$= -1$$ and $$f(1)$$ $$ = 1$$. Which one of the following stat...

Which one of the following functions is continuous at $$x = 3$$?

Which one of the following functions is continuous at $$x=3?$$

Consider the function $$f\left( x \right) = \sin \left( x \right)$$ in the interval $$x \in \left[ {\pi /4,\,\,7\pi /4} \right].$$ The number and location(s) of the local minima of...

What is the value of $$\mathop {\lim }\limits_{n \to \infty } {\left[ {1 - {1 \over n}} \right]^{2n}}?$$

$$\int\limits_0^{\pi /4} {\left( {1 - \tan x} \right)/\left( {1 + \tan x} \right)dx} $$ $$\,\,\,\,\,\,$$ evaluates to

$$\mathop {\lim }\limits_{x \to \infty } {{x - \sin x} \over {x + \cos \,x}}\,\,Equals$$

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve $$3{x^4} - 16{x^3} + 24{x^2} + 37$$ is

If $$\,\,\,\,f\,\,\,\,\left( x \right)$$ is defined as follows, what is the minimum value of $$f\,\left( x \right)$$ for $$x \in \left( {0,2} \right)$$ ? $$$f\left( x \right) = \le...

The value of $$\int\limits_0^3 {\int\limits_0^x {\left( {6 - x - y} \right)dxdy\,\,\,} } $$ is _____.

What is the value of $$\int\limits_0^{2\pi } {{{\left( {x - \pi } \right)}^3}\left( {\sin x} \right)dx} $$

$$\mathop {Lim}\limits_{x \to 0} \,{{Si{n^2}x} \over x} = \_\_\_\_.$$

The value of the integral is $${\rm I} = \int\limits_0^{{\raise0.5ex\hbox{$\scriptstyle \pi $} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}} {{{\cos }^2}x\,dx} $$

(a) Find the points of local maxima and minima, if any, of the following function defined $$0 \le x \le 6.\,\,\,{x^3} - 6{x^2} + 9x + 15$$ (b) Integrate $$\,\,\,\int\limits_{ - \pi...

Find the points of local maxima and minima if any of the following function defined in $$0 \le x \le 6,$$ $$\,\,\,\,f\left( x \right) = {x^3} - 6{x^2} + 9x + 15.$$

Consider the function $$y = \left| x \right|$$ in the interval $$\left[ { - 1,1} \right]$$. In this interval, the function is

What is the maximum value of the function $$f\left( x \right) = 2{x^2} - 2x + 6$$ in the interval $$\left[ {0,2} \right]$$?

If at every point of a certain curve, the slope of the tangent equals $${{ - 2x} \over y}$$ the curve is

$$\mathop {Lim}\limits_{x \to \infty } {{{x^3} - \cos x} \over {{x^2} + {{\left( {\sin x} \right)}^2}}} = \_\_\_\_\_\_.$$

The value of the double integral $$\int\limits_0^1 {\int\limits_x^{{1 \over x}} {{x \over {1 + {y^2}}}\,\,dx\,\,dy = \_\_\_\_\_.} } $$