# Recent questions tagged calculus

1
The Cartesian coordinates $(x,y)$ of a point $A$ with polar coordinates $\left ( 4, \pi/4 \right)$ is $\left( \sqrt{3}, 2 \sqrt{2} \right ) \\$ $\left( 2, 2 \sqrt{3} \right ) \\$ $\left( 2\sqrt{2}, \sqrt{3} \right ) \\$ $\left( 2 \sqrt{2}, 2 \sqrt{2} \right )$
2
$\dfrac{d}{dx} \left [ \ln (2x) \right ]$ is equal to $1/2x \\$ $1/x \\$ $1 /2 \\$ $x$
3
The sum of the infinite geometric series $1 + 1/3 + 1/3^2+ 1/3^3+ \dots$ (rounded off to one decimal place) is ___________
4
The value of $\underset{x \to 0} \lim \left [ \dfrac{x- \sin 2x}{x-\sin 5x} \right]$ (rounded off to two decimal places) is __________
5
If the area of a triangle with the vertices $(k,0), (2,0)$ and $(0, -2)$ is $2$ square units, the value of $k$ is _________
6
Calculate the following integral $\int \limits_0^{\pi^2/4} \sin \sqrt{x} dx =$____________.
7
The solution of $\underset{x\rightarrow 8 }\lim\left ( \dfrac{x^{2}-64}{x-8} \right )$ is _____________.
8
A function $f$ is given as : $f(X)=4X-X^{2}$ The function $f$ is maximized when $X$ is equal to _________________.
9
An infinite series $S$ is given as: $S=1+2/3+3/9+4/27+5/81+\:.\dots$ (to infinity) The value of $S$ is ______________________ (round off to $2$ decimal places).
10
A function $f$ is as follows: $f(x) = \begin{cases} 15 & \text{if }x<1 \\ cx& \text{if } x\geq 1 \end{cases}$ The function $f$ is a continuous function when $c$ is equal to ____________________ (answer in an integer).
11
Which of the following are geometric series? $1,\:6,\:11,\:16,\:21,\:26,\:\dots$ $9,\:6,\:3,\:0,\:-3,\:-6,\:\dots$ $1,\:3,\:9,\:27,\:81,\:\dots$ $4,\:-8,\:16,\:-32,\:64,\:\dots$ $P$ and $Q$ only $R$ and $S$ only $Q$ and $S$ only $P, Q$ and $R$ only
12
The value of the integral $\displaystyle \int_0^{0.9} \dfrac{dx}{(1-x)(2-x)}$ is ____________
13
The surface area (in $m^2$) of the largest sphere that can fit into a hollow cube with edges of length $1$ meter is ______ Given data: $\pi=3.14$
14
The angle (in degrees) between the vectors $\overrightarrow{x}= \hat{i}-\hat{j}+2 \hat{k}$ and $\overrightarrow{y} = 2 \hat{i} – \hat{j}-1.5 \hat{k}$ is _________
15
Consider the following infinite series: $1+ r+r^2 +r^3+ \dots \dots \infty$ If $r = 0.3$, then the sum of this infinite series is ____________
16
The limit of the function $\bigg (1 + \dfrac{x}{n} \bigg )^n$ as $n \to \infty$ is $\ln x$ $\ln \dfrac{1}{x}$ $e^{-x}$ $e^x$
17
The limit of the function $e^{-2t}\sin (t)$ as t $\rightarrow\infty$ so, is
18
If $y =x^x$, then $\frac{dy}{dx}$ is $x^x(x-1)$ $x^{x-1}$ $x^x(1 + \log x)$ $e^x(1 + \log x)$
19
Which of the following statements is true for the series given below? $S_n=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+\dots+\frac{1}{\sqrt{n}}$ $S_n$ converges to $\log(\sqrt{n})$ $S_n$ converges to $\sqrt{n}$ $S_n$ converges to $\exp(\sqrt{n})$ $S_n$ diverges
20
The graph of the function $F(x) =\frac{x}{k_1x^2+k_2x+1}$ for $0<x<\infty$ is
21
Evaluate $\underset{x\rightarrow\infty }{\lim}x\tan\frac{1}{x}$ $\infty$ $1$ $0$ $-1$
22
If $u=\log (e^x+e^y),$ then $\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=$ $e^x+e^y$ $e^x-e^y$ $\frac{1}{e^x+e^y}$ $1$
If $1+r + r^2+ r^3 +\dots \infty = 1.5$, then, $1 + 2r + 3r^2 + 4r^3 + \dots \infty =$ (up to two dcimal places) ________
Calculate the following integral (up to two decimal places) $\displaystyle \int_0^1 (x + 3)(x + 1)dx = \text{ ___________}$
Which one of the following is the solution for $\cos^2 x + 2 \cos x + 1 = 0$, for values of $x$ in the range of $0^\circ < x < 360^\circ$ $45^\circ$ $90^\circ$ $180^\circ$ $270^\circ$