All Tasks - Asymptote

A billboard parallel to a highway is $4$m high and the background is at the eye level of a passing motorist. Let $α$ be the angle it subtends at the motorist's eyes, and let $h$ be the distance at which it is placed. Use differentials to determine an approximation for the angle, $α$, knowing that the distance, $h$, from the road is $4.16$m.

A TV crew monitors the lift-off of a rocket, at 1.6 km from the launch pad and at a height h. Consider the function $\theta\left(h\right)=arctcot\left(\frac{1.6}{h}\right)$ which gives the angle $\theta$ of elevation of the chamber with respect to the height of the rocket, h. Use differentials to calculate an approximation to the angle when the rocket is at a height of 5.1 km.

Applying the inverse function derivative theorem, to calculate the derivative of the derivative $\displaystyle\frac{dy}{dx}$ of the function $y=\mbox{arccot}(x)$ we get:

Find the abscissa of the intersection point ​​between the curves $y=-\dfrac{3\pi}{2}\text{arccot}\left(\dfrac{2x+1}{3}\right)$ and $y=\ln e^{-\frac{3\pi^2}{8}}$.

The domain ($D_f$) and the range ($D'_f$) of function $f(x)= 2 \arcsin\left(\dfrac{x}{3}\right)$ are, respectively,

The domain ($D_f$) and the range ($D'_f$) of function $f(x)= \dfrac{\pi}{2} -2 \arcsin(1-2x)$ are, respectively,