Calculus Questions

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

Let $f(x)=x+x^{-1}$. Then which of the following is correct?

Solve: ${\int }_0^{\pi /2}\frac{f(x)}{f(x)+f\left(\frac{\pi }{2}-x\right)}dx=?$

A manufacturing company has its variable cost given by $C(x)=x(80-x^2)$ where $x$ is the number of quantities produced. The price per unit is $p(x)=5x$. The ratio of marginal cost and marginal revenue when 5 units were both produced and in demand is:

If Rolle's theorem holds for the function $f(x)=x^3-a{x}^2+bx-4,x\in [1,2]$ with $f^{\prime }(4/3)=0$, then the ordered pair $(a,b)$ is equal to:

If $y=4x-5$ is tangent to the curve $y^2=p{x}^3+q$ at $(2,3)$, then:

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

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 $f:\mathbb{R}\to \mathbb{R}$ 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 $f$ be differentiable for all $x$. If $f(1)=-2$ and $f^{\prime }(x)\ge 2$ for $x\in [1,6]$, then:

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:

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

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:

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

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

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:

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:

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

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:

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

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

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

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

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

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

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:

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:

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

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:

The normal to the curve $x=a(\cos \theta +\theta \sin \theta )$, $y=a(\sin \theta -\theta \cos \theta )$ at any point $\theta$ is such that:

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

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

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

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

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:

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 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:

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

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

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: ${\int }_a^{b}f(x)dx={\int }_a^{b}f(a+b-x)dx$.

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:

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

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:

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

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

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?

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:

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

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

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:

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 $f(2)=f(5)$.

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

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:

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:

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

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

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:

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

$\int {\left\{\frac{(\log x-1)}{1+(\log x{)}^2}\right\}}^{2}dx$ is equal to:

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

1: The set $\{x:f(x)=f^{-1}(x)\}=\{0,-1\}$ where $f(x)=(x+1{)}^2-1,x\ge -1$.
2: $f$ is a bijection.

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

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 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:

${\lim }_{n\to \infty }\left[\frac{1^3+2^3+3^3+\cdots +n^3}{n^4}\right]-{\lim }_{n\to \infty }\left[\frac{1^4+2^4+3^4+\cdots +n^4}{n^5}\right]$ is:

1: $g\circ f$ is differentiable at $x=0$ and its derivative is continuous at that point, where $f(x)=x|x|$ and $g(x)=\sin x$.
2: $g\circ f$ is twice differentiable at $x=0$.

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

If $c$ is a point at which Rolle's theorem holds for the function $f(x)={\log }_e\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $[3,4]$, then the value of $c$ is:

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

If the curve $y=a{x}^2+bx+c$ passes through $(1,2)$ and the tangent to this curve at the origin is $y=x$, then the possible values of $a,b,c$ are:

Find ${\lim }_{n\to \infty }\left(\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\cdots +\frac{1}{3n}\right)$

Let $\frac{d}{dx}F(x)=\frac{e^{\sin x}}{x}$, $x>0$. If ${\int }_1^4\frac{3e^{\sin x^3}}{x}dx=F(k)-F(1)$, then one of the possible values of $k$, is:

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

What is the area bounded by the curves $|y|=1-x^2$?

If $I_1=\int \frac{e^x}{e^x+e^{-x}}dx$ and $I_2=\int \frac{dx}{e^{2x}+1}$, then what is $I_1+I_2$ equal to?