calculus

Let $f(x) = \begin{vmatrix} x^3 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^2 & p^3 \end{vmatrix}$ where $p$ is a constant. The value of $\frac{d^3}{dx^3} f(x)$ at $x = 0$ is

Let $f(x)$ be a continuous function defined in $[0,2] \rightarrow \mathbb{R}$ and satisfying the equation $\int_{0}^{2} f(x)[x - f(x)]dx = \frac{2}{3}$. The value of $f(1)$ is

Consider differential equation $\frac{dy}{dx} + xy = x$ with the condition as y = 0 at x = 0. The value of y at x = 1.0 is ________ (rounded off to two decimal places).

The value of $\lim_{x\to\infty} (x - \sqrt{x^2 + x})$ is equal to

Consider the function given below and pick one or more CORRECT statement(s) from the following choices. f(x) = x³ − (15/2)x² + 18x + 20

The maximum value of the function h(x) = −x³ + 2x² in the interval [-1,1.5] is equal to ________ (rounded off to 1 decimal place).

Consider the function given below and pick one or more CORRECT statement(s) from the following choices. $$ f(x)=x^3-\frac{15}{2} x^2+18 x+20 $$

Integration of $\ln (x)$ with $x$ i.e. $$ \int \ln (x) d x= $$__________

$$ \text { The value of }\mathop {\lim }\limits_{x \to \infty } \left(x-\sqrt{x^2+x}\right) \text { is equal to }$$

The maximum value of the function $h(x)=-x^3+2 x^2$ in the interval $[-1,1.5]$ is equal to _________ . (rounded off to 1 decimal place)

Which one of the following options is the correct Fourier series of the periodic function $f(x)$ described below: $$ f(x)=\left\{\begin{array}{cl} 0 & \text { if }-2

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

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

The expression for computing the effective interest rate $(i_{eff})$ using continuous compounding for a nominal interest rate of 5% is $i_{eff} = \lim\limits_{m \to \infty} \left(1...

For the integral $I = \int_{-1}^{1} \frac{1}{x^2} dx$, which of the following statements is TRUE?

Consider that a force P is acting on the surface of a half-space (Boussinesq's problem). The expression for the vertical stress ($\sigma_z$) at any point (r, z), within the half-sp...

The following function is defined over the interval [-L, L]: f(x) = px 4 + qx 5 . If it is expressed as a Fourier series, $\rm f(x)=a_0 +\displaystyle\sum^\infty_{n=1} \left\{a_n \...

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

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

Let max {a, b} denote the maximum of two real numbers a and b. Which of the following statement(s) is/are TRUE about the function f(x) = max{3 - x, x - 1} ?

The Fourier cosine series of a function is given by : $$f(x) = \sum\limits_{n = 0}^\infty {{f_n}\cos nx} $$ For f(x) = cos 4 x, the numerical value of (f 4 + f 5 ) is _________. (r...

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

The value of $\lim_{x\to\infty} \frac{x \ln(x)}{1+x^2}$ is

The unit normal vector to the surface X²+Y²+Z²-48=0 at the point (4, 4, 4) is

The value (round off to one decimal place) of $\int_{-1}^{1} x e^{|x|} dx$ is

Consider the limit: $\lim_{x\to 1} \left(\frac{1}{\ln x} - \frac{1}{x-1}\right)$ The limit (correct up to one decimal place) is _________

The value of $\int_0^1 e^x dx$ using the trapezoidal rule with four equal subintervals is

The values of abscissa (x) and ordinate (y) of a curve are as follows: X y 2.0 5.00 2.5 7.25 3.0 10.00 3.5 13.25 4.0 17.00 By Simpson's 1/3rd rule, the area under the curve (round...

Numerically integrate, $f(x) = 10x - 20x^2$ from lower limit $a = 0$ to upper limit $b = 0.5$. Use Trapezoidal rule with five equal subdivisions. The value (in units, round off to...

The value of $\lim_{x\to\infty} \frac{x^2-5x+4}{4x^2+2x}$ is

The area of an ellipse represented by an equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is

Which one of the following is correct?

The following inequality is true for all x close to 0. 2 - x²/3 < (x sin x) / (1 - cos x) < 2 What is the value of lim (x sin x) / (1 - cos x) ?

For a small value of h, the Taylor series expansion for f(x+h) is

Which one of the following is NOT a correct statement?

Consider the hemi-spherical tank of radius 13 m as shown in the figure (not drawn to scale). What is the volume of water (in m³) when the depth of water at the centre of the tank i...

At the point x = 0, the function f(x) = x³ has

The divergence of the vector field V = x² i + 2y³ j + z⁴ k at x = 1, y = 2, z = 3 is

lim (tanx / (x^2 - x)) is equal to

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

For the function f(x)=a+bx, 0 ≤ x ≤ 1, to be a valid probability density function, which one of the following statements is correct?

Consider the following definite integral: $I = \int_0^1 \frac{(\sin^{-1} x)^2}{\sqrt{1-x^2}} dx$ The value of the integral is

The infiltration rate f in a basin under ponding condition is given by f = 30+10e⁻²ᵗ, where, f is in mm/h and t is time in hour. Total depth of infiltration (in mm, up to one decim...

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

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

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 tangent point would be

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

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 equal to

The quadratic approximation of $$f\left( x \right) = {x^3} - 3{x^2} - 5\,\,$$ at the point $$x=0$$ is

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

Consider the following complex function $$f\left( z \right) = {9 \over {\left( {z - 1} \right)\left( {z + 2} \right)}}.$$ Which of the following is ONE of the residues of the above...

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

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

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

There is no value of $$x$$ that can simultaneously satisfy both the given equations. Therefore, find the 'least squares error' solution to the two equations, i.e., find the value o...

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

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\{ {\matrix{ {{{\lambda \,\cos x} \...

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

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 $$h.$$ The equation of the parabola...

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

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

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

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

The Laplace transform of the following function is $$$f\left( t \right) = \left\{ {\matrix{ {\sin t} & {for\,\,0 \le t \le \pi } \cr 0 & {for\,\,t > \pi } \cr } } \right.$$$

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{$\scriptstyle 2$}}}^{{\raise0....

Limit of the following series as $$x$$ approaches $${\pi \over 2}$$ is $$f\left( x \right) = x - {{{x^3}} \over {3!}} + {{{x^5}} \over {5!}} - {{{x^7}} \over {7!}} + - - - - - $$

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

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

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

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

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

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

For the differential equation $$f\left( {x,y} \right){{dy} \over {dx}} + g\left( {x,y} \right) = 0\,\,$$ to be exact is

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 $$f\left( b \right) - f\left( a \right) = \left( {b - a} \right)\,\,f'\left( \varepsilon \right)$$ for $$f\left( x \right...