Questions without a selected answer in Calculus

Consider the two functions $f_{1}(x)=\dfrac{x^{2}-4}{x-2}$ and $f_{2}(x)=x^{2}-2 x+2$. Which of the following is the value of $\left(f_{1}(x)+f_{2}(x)\right)$ as $x \righ...

Which of the following functions has the highest area under the curve between $x=0$ and $x={10}?$$y=x+8$$y=3 x$$y=2 x+1$$y=x+2$

​​​​The minimum value of the function$$ f(x)=x+\frac{4}{x} $$for $x>0$ is$1$$2$$3$$4$

​​​​If the function$$ f(x)=\left\{\begin{array}{ll}\sin 2 x, & \text { for } x>0 \\a+b x, & \text { for } x \leq 0\end{array}\right.$$where $a$ and $b$ are constants, is ...

Let $a_{0}=0$ and define $a_{n}=\frac{1}{2}\left(1+a_{n-1}\right)$ for all positive integers $n \geq 1$. The least value of $n$ for which $\left|1-a_{n}\right|<\frac{1}{2...

The value of the series $1+\sin x+\cos ^{2} x+\sin ^{3} x+\cdots$ at $x=\frac{\pi}{4}$ is $\_\_\_\_\_\_\_$.$\frac{1}{\sqrt{2}+1}$$\frac{\sqrt{2}}{\sqrt{2}+1}$$\frac{1}{\s...

The value of the limit $\lim _{x \rightarrow \infty} \frac{x}{2} \ln \left(1+\frac{2024}{x}\right)$ is $\_\_\_\_\_\_\_\_$.

The value of $\lim _{x \rightarrow 0}\left[\frac{\cos 2 x-\cos 4 x}{x^{2}}\right]$ is ___________.

If $f(x)=\frac{\sin x+\cos x}{\sin x-\cos x}$, the value of $f^{\prime}(x)$ at $x=0$ is ___________.

If $f(2)=5$ and $(f(x))(f(x+1))=3$ for all real values of $x$, the value of $f(10)$ is ____________.

The maximum value of the function $f(x) = 3x^{2} – 2x^{3}$ for $x>0$ is ________.

For $x_{1}>0$ and $x_{2}>0,$ the value of $\displaystyle{}\lim_{x_{1} \to x_{2}}\frac{x_{1}-x_{2}}{x_{2}\ln\left(\frac{x_{1}}{x_{2}}\right)}$ is __________.

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 ...

$\dfrac{d}{dx} \left [ \ln (2x) \right ]$ is equal to$1/2x \\$$1/x \\$$1 /2 \\$$x$

The sum of the infinite geometric series $1 + 1/3 + 1/3^2+ 1/3^3+ \dots$ (rounded off to one decimal place) is ___________

The value of $\underset{x \to 0} \lim \left [ \dfrac{x- \sin 2x}{x-\sin 5x} \right]$ (rounded off to two decimal places) is __________

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 _________

Calculate the following integral $\int \limits_0^{\pi^2/4} \sin \sqrt{x} dx =$____________.

The solution of $ \underset{x\rightarrow 8 }\lim\left ( \dfrac{x^{2}-64}{x-8} \right )$ is _____________.

A function $f$ is given as :$$f(X)=4X-X^{2}$$The function $f$ is maximized when $X$ is equal to _________________.

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).

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 ...

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,\:\do...

The value of the integral $\displaystyle \int_0^{0.9} \dfrac{dx}{(1-x)(2-x)}$ is ____________

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: $\pi=3.14$

$\underset{x \to 0}{\lim} \dfrac{\sin(x)}{x}$ is ________

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 _________

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 ____________

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$

The limit of the function $e^{-2t}\sin (t)$ as t $\rightarrow\infty$ so, is

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)$

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$ con...

The graph of the function $F(x) =\frac{x}{k_1x^2+k_2x+1}$ for $0<x<\infty$ is

Evaluate $\underset{x\rightarrow\infty }{\lim}x\tan\frac{1}{x}$$\infty$$1$$0$$-1$

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$$180^\circ$$2...