Calculus Questions

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

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

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

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.

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 $c\in (0,1)$:

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?

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

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

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

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?

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:

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

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