Recent questions and answers in Calculus

Let $f(x)=\left|\begin{array}{ccc}x^{3} & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^{2} & p^{3}\end{array}\right|$ where $p$ is a constant.The value of $\dfrac{d^{3}}{d x^{3...

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

​​​​The sum of the following infinite series is:$$ \frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\cdots $$$\pi$$1+e$$e-1$$e$

​​​​​​The function $f(x)=x^{3}-27 x+4,1 \leq x \leq 6$ hasMaxima pointMinima pointSaddle pointInflection point

​​​​​A vector field $\vec{p}$ and a scalar field $r$ are given by\[\begin{aligned}\vec{p} & =\left(2 x^{2}-3 x y+z^{2}\right) \hat{\imath}+\left(2 y^{2}-3 y z+x^{2}\right...

For the integral$$I=\int_{-1}^{1} \frac{1}{x^{2}} d x$$which of the following statements is $\text{TRUE?}$ $I=0$ $I=2$$I=-2$ The integral does not converge

For the function $f(x)=e^{x}|\sin x|; x \in \mathbb{R},$ which of the following statements is/are TRUE? The function is continuous at all $x$The function is differentiabl...

$\displaystyle{}\int \left( x-\frac{x^{2}}{2} + \frac{x^{3}}{3} – \frac{x^{4}}{4} + \dots \right) dx$ is equal to$\frac{1}{1+x} + $ Constant$\frac{1}{1+x^{2}} + $ Consta...

Consider the polynomial $f(x) = x^{3} – 6x^{2} + 11x – 6$ on the domain $S$ given by $1 \leq x \leq 3.$ The first and second derivatives are $f’(x)$ and $f’’(x).$Consider...

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

The value of $\lim \limits_{x\rightarrow \infty } \dfrac{x \:\text{ln}\left ( x \right )}{1+x^{2}}$ is$0$$1.0$$0.5$$\infty$

The unit normal vector to the surface $X^{2} + Y^{2} + Z^{2} – 48 = 0$ at the point $(4, 4, 4)$ is $\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}$$\frac{1}{\sq...

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

A function is defined in Cartesian coordinate system as $f(x,y)=xe^{y}$. The value of the directional derivative of the function ($\textit{in integer}$) at the point $(2,...

The volume determined from $\int \int \int _{V}\:8\:xyz\:dV$ for $V=\left [ 2,3 \right ]\times \left [ 1,2 \right ]\times \left [ 0,1 \right ]$ will be ($\textit{in integ...

The area of an ellipse represented by an equation $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ is$\dfrac{\pi ab}{4} \\$$\dfrac{\pi ab}{2} \\$$\pi ab \\$$\dfrac{4\pi ab}{3}$

A continuous function $f(x)$ is defined. If the third derivative at $x_i$ is to be computed by using the fourth order central finite-divided-difference scheme (with step ...

If $C$ represents a line segment between $(0,0,0)$ and $(1,1,1)$ in Cartesian coordinate system, the value (expressed as integer) of the line integral$$\int_C [(y+z)dx+(x...

The value of $$\lim_{x\to\infty}\dfrac{\sqrt{9x^2+2020}}{x+7}\:\text{is}$$$\dfrac{7}{9}$$1$$3$indeterminable

Velocity distribution in a boundary layer is given by $\dfrac{u}{U_\infty} = \sin\large \left( \dfrac{\pi}{2}\dfrac{y}{\delta} \right)$, where $u$ is the velocity at vert...

Let $w=f(x,y)$, where $x$ and $y$ are functions of $t$. Then, according to the chain rule, $\dfrac{dw}{dt}$ is equal to$\dfrac{dw}{dx} \dfrac{dx}{dt} + \dfrac{dw}{dy} \df...

The divergence of the vector field $V=x^2 i + 2y^3 j + z^4 k$ at $x=1, \: y=2, \: z=3$ is ________

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

Consider the following definite integral: $I= \displaystyle{} \int_0^1 \dfrac{(\sin ^{-1}x)^2}{\sqrt{1-x^2}} dx$. The value of the integral is$\dfrac{\pi ^3}{24} \\$$\dfr...

For a small value of $h$, the Taylor series expansion for $f(x+h)$ is$f(x)+h{f}’ (x) + \dfrac{h^2}{2!}{f}’’(x) + \dfrac{h^3}{3!}{f}’'’(x)+\dots \infty \\$$f(x)-h{f}’ (x) ...

Which one of the following is NOT a correct statement?The function $\sqrt[x]{x}, \: (x>0)$, has the global maxima at $x=e$The function $\sqrt[x]{x}, \: (x>0)$, has the gl...

Consider two functions: $x=\psi \text{ ln } \phi$ and $y= \phi \text{ ln } \psi$. Which one of the following is the correct expression for $\frac{\partial \psi}{\partial ...

The following inequality is true for all $x$ close to $0$.$$2-\dfrac{x^2}{3} < \dfrac{x \sin x}{1- \cos x} <2$$What is the value of $\underset{x \to 0}{\lim} \dfrac{x \si...

What is curl of the vector field $2x^2y \textbf{i} + 5z^2 \textbf{j} – 4yz \textbf{k}$?$6z \textbf{i} + 4x \textbf{j} – 2x^2 \textbf{k}$$6z \textbf{i} - 8xy \textbf{j} + ...

The optimum value of the function $f(x)=x^2-4x+2$ is$2$ (maximum)$2$ (minimum)$ – 2$ (maximum)$ – 2$ (minimum)

What is the value of $\underset{x \rightarrow 0 \\ y \rightarrow 0}{\lim} \dfrac{xy}{x^2+y^2}$?$1$$-1$$0$Limit does not exist

A circular curve of radius R connects two straights with a deflection angle of $60^{\circ}$. The tangent length is$0.577 \: R$$1.155 \: R$$1.732 \:R$$3.464 \: R$

The angle of intersection of the curves $x^{2}=4y$ and $y^{2} = 4x$ at point $(0,0)$ is$0^{\circ}$$30^{\circ}$$45^{\circ}$$90^{\circ}$