Calculus Questions

The value of the integral ${\int }_{-\pi /2}^{\pi /2}(x\cos x)dx$ is:

What is the solution of the differential equation $\ln \left(\frac{dy}{dx}\right)-a=0$?

The function $f(x)=\cos x-2px$ is monotonically decreasing for:

The sum of two numbers is fixed. Their product is maximum when:

The value of $I={\int }_{-1}^1{e}^{|x|}dx$ is equal to:

The solution of the differential equation $du=(16+u^2)dx$ is:

If $f(x)=\{\begin{array}{ll}K{x}^2& \text{if }x\le 2\\ 3& \text{if }x>2\end{array}$ is continuous at $x=2$, then the value of $K$ is:

At which point does the line $\frac{x}{a}+\frac{y}{b}=1$ touch the curve $y=b{e}^{-x/a}$?

The value of $c$ in $(0,2)$ satisfying the Mean Value Theorem for the function $f(x)=x(x-1{)}^2,x\in [0,2]$ is equal to:

If $f(x)=\frac{1}{1-x}$, then the derivative of the composite function $f[f\{f(x)\}]$ is equal to:

Complete solution set of $|\frac{x^2}{x-1}|\le 1$ is given by:

The critical points of the function $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$ are:

The integral ${\int }_0^{\pi /2}|\sin x-\cos x|dx$ is equal to:

If $f(x)={\log }_x(\log x)$, then $f^{\prime }(x)$ at $x=e$ is:

$I=\int \frac{\sin x}{\sin x-\cos x}dx$ is equal to:

If $y=(1+x)(1+x^2)(1+x^4)\cdots (1+x^{2n})$, then the value of $\frac{dy}{dx}$ at $x=0$ is:

The value of $a$ so that $f(x)=\sin x-\cos x-ax+b$ decreases for all real values of $x$ is given by:

The cost of a cricket ball is Rs 500. If the price remains constant, its marginal revenue in Rs will be:

If $y=|\cos x-\sin x|$, then $\frac{dy}{dx}$ at $x=\pi /4$ is:

For the function $f(x)=x+\frac{1}{x},x\in [1,3]$, the value of $c$ for Lagrange's Mean Value Theorem is:

If $\int \frac{dx}{x-x^3}=A\ln \left|\frac{x^2}{1-x^2}\right|+c$, then the value of $A$ is:

The maximum value of the function $f(x)=(x-1)(x-2)(x-3)$ is:

For the function $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$, $f(x)$ has:

The local maximum value of $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$ is:

The normal to the curve $5x^5-10x^3+x+2y+6=0$ at $A(0,-3)$ meets the curve again at two points $B$ and $C$. The length of the line segment $BC$ is:

A function $f:\mathbb{R}\to \mathbb{R}$ satisfies the equation $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb{R}$; $f(x)\ne 0$. Suppose that the function $f(x)$ is differentiable at $x=0$ and $f^{\prime }(0)=2$. If $f^{\prime }(x)=\lambda f(x)$, then the value of $\lambda$ is:

Let $f(x)=x-[x]$ for every real number $x$, where $[x]$ is the greatest integer less than or equal to $x$. Then the value of ${\int }_{-1}^{1}f(x)dx$ is:

The curves $f(x)=\sin x$ and $g(x)=\cos x$ intersect infinitely many times, forming bounded regions of equal areas. The area of one such region is:

For the function $f(x)=\frac{x^2}{x^2-1}$, the area formed by the tangent to the curve at $P(2,\frac{4}{3})$ with the coordinate axes is:

The solution of $\frac{dy}{dx}+2y\tan x=\sin x$ is:

The value of '$c$ ' in Rolle's Theorem for the function $f(x)=\cos \frac{x}{2}$ on the interval $[\pi ,3\pi ]$ is:

The solution of the differential equation $\frac{dy}{dx}=\frac{y}{x}+\frac{f(y/x)}{f^{\prime }(y/x)}$ is:

Consider the function $f(x)=[x+1]-[x]+\frac{1}{x^2-1}$, where $[\cdot ]$ represents the greatest integer function. The function $f(x)$ is decreasing in the interval:

The value of ${\int }_0^1|5x-3|dx$ is:

The function $f(x)=x^3-3\log x$ has:

The normal to the curve $y(x-2)(x-3)=x+6$ at the point where the curve intersects the $y$ -axis passes through the point:

Derivative of $x^2$ w.r.t. $x^3$ is:

Let the function $f:[-7,0]\to \mathbb{R}$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime }(x)\le 2$ for all $x\in (-7,0)$, then for all such functions $f$, $f(-1)+f(0)$ lies in the interval:

The function $f(x)=x^2-4x$, $x\in [0,4]$ attains minimum value at:

Find the area between the curve $y=\sin x$ and the $x$ -axis from $x=-\pi /3$ to $x=\pi /3$.

If $f(x)=a{\tan }^{-1}x+2b\log (1+x)+x+1$ has critical points at $x=0$ and $x=2$, then the values of the constants $a$ and $b$ are respectively:

The area under the curve $y=x^4$ and the lines $x=1,x=5$ and the $x$ -axis is:

What is the area bound by the curve $xy=m^2$, $x$ -axis and the ordinates $x=p$ and $x=q$ where $p>q>0$?

If $f(x)$ is an increasing function and $g(x)$ is a decreasing function, then which of the following is correct for the real values $p$ and $q$? Given that for these values of $x$, both functions are defined.

If the tangent to the curve $y=x+\sin y$ at a point $(a,b)$ is parallel to the line joining $(0,3/2)$ and $(1/2,2)$, then:

Find the area bounded by the curves $y\ge x^2$ and $y=|x|$.

If at a point on the curve $y=f(x)$ the slope of the line tangent to the curve is equal to $2xy$, then the function $f$ is:

${\int }_0^{\pi /2}\log (\tan x)dx$ is equal to:

If $f(x)=\{\begin{array}{ll}0,& x=0\\ x-3,& x>0\end{array}$, then the function $f(x)$ is:

If $y=\cos ({\sin }^{2}x)$, then find the value of $\frac{dy}{dx}$ at $x=\pi /2$.

A cricket bat manufacturing firm assesses its variable cost to be $2x(x-40)$, where $x$ is the number of bats produced, and the fixed cost incurred on storage is $Rs.1200$. How many bats should be manufactured for the minimum total cost?

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

What is the area of one of the loops between the curve $y=c\sin x$ and the x-axis?

Let $f$ be a differentiable function such that $f(1)=2$ and $f^{\prime }(x)=f(x)$ for all $x\in \mathbb{R}$. If $h(x)=f(f(x))$, then $h^{\prime }(1)$ is equal to:

Correct evaluation of $\int \frac{x}{(x-2)(x-1)}dx$ (where $p$ is an arbitrary constant) is:

Calculate the value of:
$${\int }_{1/2}^2\frac{1}{x}\sin \left(x-\frac{1}{x}\right)dx$$

If ${\lim }_{n\to \infty }{\sum }_{k=1}^n\left[{\left(\frac{3k}{n}\right)}^2+2\right]\frac{3}{n}={\int }_0^{6}f(x)dx$, then:

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

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:

Evaluate:
$$\int \frac{1}{e^x+e^{-x}}dx$$

If $x=3\tan t$ and $y=3\sec t$, then the value of $\frac{d^{2}y}{d{x}^2}$ at $t=\frac{\pi }{4}$ is:

The minimum area bounded by circle $x^2+y^2=9$ and straight line $x+y=3$ is:

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

The maximum slope of the curve $y=\frac{1}{2}{x}^4-5x^3+18x^2-19x$ occurs at the point:

If the line $ax+by+c=0$ is a tangent to the curve $xy=4$, then:

If ${\int }_0^{2\pi }|x\sin x|dx=k\pi$, then the value of $k$ is equal to:

If the slope of the tangent to curve $y=x^3$ at a point is equal to the ordinate of the point, then the point is:

Let $g(x)$ be the inverse of an invertible function $f(x)$ which is differentiable at $x=c$, then $g^{\prime }(f(c))$ equals:

Find the area under the curve $y=\cos x$ in the interval $0\le x\le \pi /2$.

Let $f:(0,+\infty )\to \mathbb{R}$ and $F(x)={\int }_0^{x}f(t)dt$. If $F(x^2)=x^2(1+x)$, then $f(4)$ equals:

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

Extract all possible values of $b$ for which the function $f(x)=\{\begin{array}{ll}7x-x^3+\log (b^2-4b+4),& 0\le x<3\\ x-9,& x\ge 3\end{array}$ has local maxima at $x=3$:

Area bounded by the region $R\equiv \{(x,y):y^2\le x\le |y|\}$ is:

Consider the following statements:I. The derivative, where the function attains maxima or minima, is zero.II. If a function is differentiable at a point, then it must be continuous at that point.Which of the above statement(s) is/are correct?

A manufacturing firm has its variable cost given by $C(x)=x(12-x^3)$. The price per unit is $p(x)=6x-2$. Find the marginal revenue when $3$ units were in demand.

Find the area of the region (in sq. units) bounded by the curve $y=x,y=x^2+2$ for $x\in (-1,1)$.

The area bounded by the curve $x^2=ky,k>0$, and the line $y=3$ is $12{\text{ units}}^2$. Then $k$ is:

Let $f(x)=e^x$, $g(x)={\sin }^{-1}x$ and $h(x)=f(g(x))$, then $\frac{h^{\prime }(x)}{h(x)}$ is:

Find the area under the curve $y=x+2$, bounded by the x-axis and the y-axis.

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:

If $x^e=e^{x^2+y^2}$, then find $\frac{dy}{dx}$.

If $f$ and $g$ are differentiable functions in $(0,1)$ satisfying $f(0)=2=g(1)$, $g(0)=0$ and $f(1)=6$, then for some $c\in (0,1)$:

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

Find the value of $f(x)$ when:
$$f(x)={\int }_0^x\frac{dt}{t^2+9}$$

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

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:

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: