Recent questions tagged calculus

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$(...

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}$

The area between the parabola $x^2=8y$ and the straight line $y=8$ is _______.

The value of $\int_0^{\infty} \dfrac{1}{1+x^2} dx + \int _0^{\infty} \dfrac{\sin x}{x} dx$ is$\dfrac{\pi}{2} \\$$\pi \\$$\dfrac{3 \pi}{2} \\$$1$

The area of the region bounded by the parabola $y=x^2+1$ and the straight line $x+y=3$ is$\dfrac{59}{6} \\$$\dfrac{9}{2} \\$$\dfrac{10}{3} \\$$\dfrac{7}{6}$

The magnitudes of vectors $\textbf{P, Q}$ and $\textbf{R}$ are $100$ kN, $250$ kN and $150$ kN, respectively as shown in the figure.The respective values of the magnitude...

Let $x$ be a continuous variable defined over the interval $(- \infty, \infty)$, and $f(x) = e^{-x-e^{-x}}$. The integral $g(x)= \int f(x) \: dx$ is equal to$e^{e^{-x}}$$...

$\underset{x \to 0}{\lim} \bigg( \dfrac{\tan x}{x^2-x} \bigg)$ is equal to _________

While minimizing the function $f(x)$, necessary and sufficient conditions for a point, $x_0$ to be a minima are:$f’(x_0) 0$ and $f’’(x_0)=0$$f’(x_0) < 0$ and $f’’(x_0)=0...

Given, $i=\sqrt{-1}$, the value of the definite integral, $I= \int_0^{\pi/2} \dfrac{\cos x+ i \sin x}{\cos x – i \sin x} dx$ is:$1$$-1$$i$$-i$

$\underset{x \to \infty}{\lim} \bigg( 1+ \dfrac{1}{x} \bigg)^{2x} $ is equal to$e^{-2}$$e$$1$$e^2$

Consider the following complex function:$$f(z) = \frac{9}{(z-1)(z+2)^2}$$Which of the following is one of the residues of the above function?$-1$$9/16$$2$$9$

The directional derivative of the field $u(x,y,z) = x^2-3yz$ in the direction of the vector $(\hat{i}+\hat{j}-2 \hat{k})$ at point $(2, -1, 4)$ is _________.

The expression $\displaystyle{} \lim_{a \to 0} \dfrac{x^a-1}{a}$ is equal to$\log x$$0$$x \log x$$\infty$

$z=\dfrac{2-3i}{-5+i}$ can be expressed as$-0.5-0.5i$$-0.5+0.5i$$0.5-0.5i$$0.5+0.5i$

A particle moves along a curve whose parametric equations are: $x=t^3+2t$, $y=-3e^{-2t}$ and $z=2 \sin (5t)$, where $x$, $y$ and $z$ show variations of the distance cover...

$\underset{x \to \infty}{\lim} \bigg( \dfrac{x+\sin x}{x} \bigg)$ equals to$- \infty$$0$$1$$\infty$

For a portion of national highway where a descending gradient of $1$ in $25$ meets with an ascending gradient of $1$ in $20$, a valley curve needs to be designed for a ve...

For a portion of national highway where a descending gradient of $1$ in $25$ meets with an ascending gradient of $1$ in $20$, a valley curve needs to be designed for a ve...

The solution for $\int_0^{\pi/6} \cos^4 3 \theta \sin^3 6 \theta \: d \theta$ isA. $0$B. $\dfrac{1}{15}$C. $1$D. $\dfrac{8}{3}$

For the parallelogram OPQR shown in the sketch, $\overrightarrow{OP}=a \hat{i}+b \hat{j}$ and $\overrightarrow{OR}=c \hat{i} +d \hat{j}$. The area of the parallelogram is...

The infinite series $1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!} + \dots $ corresponds to$\sec x$$e^x$$\cos x$$1+\sin^2x$

The value (up to two decimal places) of a line integral $\int_C \overrightarrow{F}(\overrightarrow{r}) . d\overrightarrow{r}$, for $ \overrightarrow{F}(\overrightarrow{r...

The value of the integral $\int_0^{\pi} x \cos^2 x \: dx$ is$\pi^2/8$$\pi^2/4$$\pi^2/2$$\pi^2$

At the point $x= 0$, the function $f(x) = x^3$ haslocal maximumlocal minimumboth local maximum and minimumneither local maximum nor local minimum