Recent questions tagged 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 equation $\dfrac{d^{2} y}{d x^{2}}-y=0$ has a solution of the form $y=e^{A x}$. The value(s) of $A$ satisfying this is/are:$0$$1$$-1$$-\infty$

Consider the equation $\dfrac{d y}{d x}=\dfrac{1}{x}$The value of the integral $\int_{1}^{2} d y$ using trapezoidal method and interval $h=0.25$ is $\_\_\_\_$. (rounded o...

From the following plot of $\dfrac{d y}{d x}$ versus $x$ and if $y(2)=5$, the value of $y(3)$ is $\_\_\_\_$. (rounded off to one decimal place) 

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

​​Let $m$ and $n$ be fixed real numbers. If the function $y(t)=C_{1} e^{t}+C_{2} e^{-t}$ is a solution of$$ \frac{d^{2} y}{d t^{2}}+m \frac{d y}{d t}+n y=0$$for any const...

​​​​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 $\_\_\_\_\_\_\_\_$.

Let $y(x)=x^2 \ln x$ for $x>0$, be a solution of $x^2 \frac{d^2 y}{d x^2}+4 y=\alpha x \frac{d y}{d x}$. Then the value of $\alpha$ is $\_\_\_\_\_\_\_\_\_$.

The absolute relative error in evaluating the integral $\int_0^1 x^2 d x$ by the trapezoidal rule with the step size $0.25$ is $\%$ $\_\_\_\_\_\_\_\_$ (rounded off to $2$...

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

The distance between the two points of intersection of $x^{2}+y=7$ and $x+y=7$ (rounded off to two decimal places) 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 ____________.

Consider a nonlinear algebraic equation, $x\ln x + x – 1 = 0.$ Using the Newton-Raphson method, with the initial guess of $x_{0} = 3,$ the value of $x$ after one iteratio...

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

Consider the ordinary differential equation $\dfrac{dy}{dx} = f(x,y) = 2x^{2} – y^{2}.$ If $y(1)=1,$ the value(s) of $y(1.5),$ using the Euler’s implicit method $\left[y_...

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 _____________.

What is the solution of the differential equation $\dfrac{\mathrm{dy} }{\mathrm{d} x}=\dfrac{x}{y}$, with the initial condition, at $x=0, y=1?$$x^{2}=y^{2}+1$$y^{2}=x^{2}...

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 ____________

Consider the equation$V=\dfrac{aS}{b+S+\dfrac{S^2}{c}}$Given $a = 4$, $b = 1$ and $c = 9$, the positive value of $S$ at which $V$ is maximum, will be _________

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 an infinite number of cylinders. The first cylinder has a radius of $1$ meter and height of $1$ meter. The second one has a radius of $0.5$ meter and height of $...