Calculus Questions

For a certain curve $y=f(x)$ satisfying $\frac{d^{2}y}{d{x}^2}=6x-4$, $f(x)$ has a local minimum value $5$ when $x=1$. The global maximum value of $f(x)$ if $0\le x\le 2$, is:

What is the interval over which the function $f(x)=6x-x^2,x>0$ is increasing?

The value of $c$ in Lagrange's theorem for the function $|x|$ in the interval $[-1,1]$ is:

A piecewise-defined function is defined as:
$$f(x)=\{\begin{array}{ll}-1& \text{if }x\le 0\\ ax+b& \text{if }0<x<1\\ 1& \text{if }x\ge 1\end{array}$$
Where $a$ and $b$ are constants. If the function is continuous everywhere, what is the value of $a$?

What is $\int e^x\left(\sqrt{x}+\frac{1}{2\sqrt{x}}\right)dx$ equal to?

The area enclosed between the curves $y^2=x$ and $y=|x|$ is:

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function defined by $f(x)=\frac{1}{e^x+2e^{-x}}$.Statement-1: $f(c)=\frac{1}{3}$, for some $c\in \mathbb{R}$.Statement-2: $0<f(x)\le \frac{1}{2\sqrt{2}}$, for all $x\in \mathbb{R}$.

The area of the region bounded by the curves $y=|x-1|$ and $y=3-|x|$ is:

Let $f:\mathbb{R}\to \mathbb{R}$ be a positive increasing function with ${\lim }_{x\to \infty }\frac{f(3x)}{f(x)}=1$. Then ${\lim }_{x\to \infty }\frac{f(2x)}{f(x)}=$

Let $f:(-1,1)\to \mathbb{R}$ be a differentiable function with $f(0)=-1$ and $f^{\prime }(0)=1$. Let $g(x)=[f(2f(x)+2){]}^2$. Then $g^{\prime }(0)=$

Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$, x-axis and the ordinates $x=\pi /4$ and $x=\beta >\pi /4$ is $\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta$. Then $f(\pi /2)$ is:

Let $f(x)$ be a function satisfying $f^{\prime }(x)=f(x)$ with $f(0)=1$ and $g(x)$ be a function that satisfies $f(x)+g(x)=x^2$. Then the value of the integral ${\int }_0^{1}f(x)g(x)dx$, is:

An object, moving with a speed of $6.25\text{m/s}$, is decelerated at a rate given by:
$$\frac{dv}{dt}=-2.5\sqrt{v}$$
where $v$ is the instantaneous speed. The time taken by the object, to come to rest, would be:

The value of ${\lim }_{x\to 2}\frac{\sqrt{1-\cos \{2(x-2)\}}}{x-2}$:

If $f$ is a real-valued differentiable function satisfying $|f(x)-f(y)|\le (x-y{)}^2$ for $x,y\in \mathbb{R}$ and $f(0)=0$, then $f(1)$ equals:

$\frac{d^{2}x}{d{y}^2}$ equals:

If $\frac{dy}{dx}=y+3>0$ and $y(0)=2$, then $y(\ln 2)$ is equal to:

The limit ${\lim }_{n\to \infty }\frac{1}{n}\left[{\sec }^2\frac{1}{n^2}+{\sec }^2\frac{4}{n^2}+\cdots +{\sec }^{2}1\right]$ equals:

The value of ${\int }_{-\pi }^{\pi }\frac{2x(1+\sin x)}{1+{\cos }^{2}x}dx$ is:

The value of ${\int }_0^{1}8\frac{\ln (1+x)}{1+x^2}dx$ is:

The domain of the function $f(x)=\frac{1}{\sqrt{|x|-x}}$ is:

If the equation $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x=0,a_1\ne 0,n\ge 2$, has a positive root $x=\alpha$, then the equation $n{a}_n{x}^{n-1}+(n-1)a_{n-1}{x}^{n-2}+\cdots +a_1=0$ has a positive root, which is:

Let $I$ be the purchase value of an equipment and $V(t)$ be the value after it has been used for $t$ years. The value $V(t)$ depreciates at a rate given by differential equation $\frac{dV(t)}{dt}=-k(T-t)$, where $k>0$ is a constant and $T$ is the total life in years of the equipment. Then the scrap value $V(T)$ of the equipment is:

Let $P(h,k)$ be a point on the curve $y=x^2+7x+2$, nearest to the line $y=3x-3$. Then the equation of the normal to the curve at $P$ is:

The area bounded by the curves $y=|x|-1$ and $y=-|x|+1$ is:

The area of the region enclosed by the curves $y=x$, $x=e$, $y=1/x$ and the positive $x$ -axis is:

The minimum value of $e^{(2x^2-2x+1){\sin }^{2}x}$ is:

In the interval $(-4,4)$, how many extrema does the following function have?
$$f(x)={\int }_{-10}^x(t^4-4)e^{-4t}dt$$

If the integral $\int \frac{5\tan x}{\tan x-2}dx=x+a\ln |\sin x-2\cos x|+k$, then $a$ is equal to:

Evaluate the following integral:
$${\int }_1^e{x}^2\ln (x)dx$$

The function $y=\frac{1}{1+x^2}$ is decreasing in the interval:

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

The real number $x$ when added to its inverse gives the minimum value of the sum at $x$ equal to:

The marginal cost is much less than the marginal revenue for a product. The company selling the product should:

Let $a,b\in \mathbb{R}$ be such that the function $f$ given by $f(x)=\ln |x|+b{x}^2+ax,x\ne 0$ has extreme values at $x=-1$ and $x=2$.Statement 1: $f$ has local maximum at $x=-1$ and at $x=2$.Statement 2: $a=\frac{1}{2}$ and $b=-\frac{1}{4}$.

If $y=1+\frac{1}{x}+\frac{1}{x^2}+\frac{1}{x^3}+\dots \infty$ with $|x|>1$, then $\frac{dy}{dx}$ is:

The angle between the curves $y=\sin x$ and $y=\cos x$ at their point of intersection in the first quadrant is:

If $g(x)={\int }_0^x{\cos }^{4}tdt$, then $g(x+\pi )$ equals:

If $f(x)=x^n$, then the value of $f(1)-\frac{f^{\prime }(1)}{1!}+\frac{f^{\prime \prime }(1)}{2!}-\frac{f^{\prime \prime \prime }(1)}{3!}+\cdots +(-1{)}^n\frac{f^n(1)}{n!}$ is:

Consider the function $f(x)=|x-2|+|x-5|,x\in \mathbb{R}$.Statement 1: $f^{\prime }(4)=0$ Statement 2: $f$ is continuous in $[2,5]$, differentiable in $(2,5)$ and $f(2)=f(5)$.

A particle of mass $m$ is at rest at the origin at time $t=0$. It is subjected to a force $F(t)=F_0{e}^{-bt}$ in the $x$ direction. Its speed $v(t)$ is depicted by which of the following curves?

The values of $p$ and $q$ for which the function
$$f(x)=\{\begin{array}{ll}\frac{\sin (p+1)x+\sin x}{x},& x<0\\ q,& x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}},& x>0\end{array}$$
is continuous for all $x$ in $\mathbb{R}$, are:

If $I_1={\int }_0^1{2}^{x^2}dx,I_2={\int }_0^1{2}^{x^3}dx,I_3={\int }_1^2{2}^{x^2}dx$ and $I_4={\int }_1^2{2}^{x^3}dx$, then:

If $\int f(x)dx=\Psi (x)$, then $\int x^{5}f(x^3)dx$ is equal to:

The intercepts on $x$ -axis made by tangents to the curve $y={\int }_0^x|t|dt$, $x\in \mathbb{R}$, which are parallel to the line $y=2x$, are equal to:

Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$, $f^{\prime }(2)=5$, $f^{\prime }(x)\ge 1$ and $f^{\prime \prime }(x)\ge 4$ for all $x\in (1,6)$, then:

For $x\in (0,5\pi /2)$, define $f(x)={\int }_0^{x}t\sin tdt$. Then $f$ has:

If $f(1)=1$, $f^{\prime }(1)=3$, then the derivative of $f(f(f(x)))+(f(x){)}^2$ at $x=1$ is:

Let $f(x)=\sqrt{x-1}+\sqrt{x+24-10\sqrt{x-1}}$ be a real valued function for $1<x<26$. Then $f^{\prime }(x)$ is:

The population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{dp(t)}{dt}=0.5p(t)-450$. If $p(0)=850$, then the time at which the population becomes zero is:

The area enclosed between the curve $y={\log }_e(x+e)$ and the coordinate axes is:

${\lim }_{x\to \pi }\frac{\tan [1-\sin (x/2)]}{[\pi -2x{]}^2}$ is:

The area (in square units) bounded by the curves $y=\sqrt{x}$, $2y-x+3=0$, $x$ -axis, and lying in the first quadrant is:

The real number $k$ for which the equation $2x^3+3x+k=0$ has two distinct real roots in $[0,1]$:

If $f:\mathbb{R}\to \mathbb{R}$ is a function defined by $f(x)=[x]\cos \left(\frac{2x-1}{2}\pi \right)$, where $[x]$ denotes the greatest integer function, then $f$ is:

${\lim }_{x\to 0}\frac{(1-\cos 2x)(3+\cos x)}{x\tan 4x}$ is equal to:

At present, a firm is manufacturing $2000$ items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx}=100-12\sqrt{x}$. If the firm employs $25$ more workers, then the new level of production of items is:

If ${\lim }_{x\to 0}\frac{\log (3+x)-\log (3-x)}{x}=k$, the value of $k$ is:

Statement - I: The value of the integral ${\int }_{\pi /6}^{\pi /3}\frac{dx}{1+\sqrt{\tan x}}$ is equal to $\pi /6$.Statement - II: ${\int }_a^{b}f(x)dx={\int }_a^{b}f(a+b-x)dx$.

If $y=\sec ({\tan }^{-1}x)$, then $\frac{dy}{dx}$ at $x=1$ is equal to:

Let $\alpha$ and $\beta$ be the distinct roots of $a{x}^2+bx+c=0$, then ${\lim }_{x\to \alpha }\frac{1-\cos (a{x}^2+bx+c)}{(x-\alpha {)}^2}$ is equal to:

The parabolas $y^2=4x$ and $x^2=4y$ divide the square region bounded by the lines $x=4,y=4$ and the coordinate axes into three parts. If $S_1,S_2,S_3$ are respectively the areas of these parts numbered from top to bottom, then $S_1:S_2:S_3$ is:

Let $f(a)=g(a)=k$ and their $n$ th derivatives $f^n(a),g^n(a)$ exist and are not equal for some $n$. Further if ${\lim }_{x\to a}\frac{f(a)g(x)-f(a)g(a)-g(a)f(x)+g(a)f(a)}{f(x)-g(x)}=4$, then the value of $k$ is:

If $f(x)=\{\begin{array}{ll}x{e}^{-\left(\frac{1}{|x|}+\frac{1}{x}\right)},& x\ne 0\\ 0,& x=0\end{array}$ then $f(x)$ is:

If $x\frac{dy}{dx}=y(\log y-\log x+1)$, then the solution of the equation is:

If the function $f(x)=2x^3-9a{x}^2+12a^{2}x+1$, where $a>0$, attains its maximum and minimum at $p$ and $q$ respectively such that $p^2=q$, then $a$ equals:

If $f(y)=e^y$, $g(y)=y$ for $y>0$ and $F(t)={\int }_0^{t}f(t-y)g(y)dy$, then:

${\int }_0^{\pi }[\cot x]dx$, where $[\cdot ]$ denotes the greatest integer function, is equal to:

Find the absolute minimum value of the function $y=3x^2-4$ in the interval $[-1,5]$.

For real $x$, let $f(x)=x^3+5x+1$, then:

The solution of $\int \frac{\log x}{x^2}dx$ is:

The number that exceeds its square by the greatest amount is:

Suppose $f(x)$ is differentiable at $x=1$ and ${\lim }_{h\to 0}\frac{f(1+h)}{h}=5$, then $f^{\prime }(1)$ equals:

A spherical iron ball $10\text{ cm}$ in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50{\text{ cm}}^3\text{/min}$. When the thickness of ice is $5\text{ cm}$, then the rate at which the thickness of ice decreases, is:

If $f(a+b-x)=f(x)$, then ${\int }_a^{b}xf(x)dx$ is equal to:

The differential equation which represents the family of curves $y=c_1{e}^{c_{2}x}$, where $c_1$ and $c_2$ are arbitrary constants is:

What is the area under the curve $f(x)=x{e}^x$ above the x-axis and between $x=0$ and $x=1$?

If $f(x)=2\ln (e^{x/2})$, what is the area bounded by $f(x)$ for the interval $[0,2]$ on the x-axis?

Let $y$ be an implicit function of $x$ defined by $x^{2x}-2x^x\cot y-1=0$. Then $y^{\prime }(1)$ equals:

Let $f(x)=\{\begin{array}{ll}3x-4,& 0\le x\le 2\\ 2x+l,& 2<x\le 3\end{array}$. If $f$ is continuous at $x=2$, then what is the value of $l$?

The value of ${\int }_{-1}^3[{\tan }^{-1}(\frac{x}{x^2+1})+{\tan }^{-1}(\frac{x^2+1}{x})]dx$ is:

The value of ${\lim }_{x\to 0}\frac{{\int }_0^{x^2}{\sec }^{2}tdt}{x\sin x}$ is:

The area of the region bounded by the parabola $(y-2{)}^2=x-1$, the tangent to the parabola at the point $(2,3)$ and the x-axis is:

The function $y=f(x)$ has relative minima where:

Find the area of the parabola $y^2=4ax$ bounded by its latus rectum.

Given $P(x)=x^4+a{x}^3+b{x}^2+cx+d$ such that $x=0$ is the only real root of $P^{\prime }(x)=0$. If $P(-1)<P(1)$, then in the interval $[-1,1]$: