All Tasks - Asymptote

The depicted sequence of matchstick figures is continued. Cross all terms that can be used to determine the number of matches needed at the step n can be determined.

The solution of the equation $2\text{arccot}(x+1)=\dfrac{\pi}{2}$ is:

Let $f$, $g$, and $h$ be 3 real functions of real variable. If $f$ is defined by $f(x)=g(h(x))$, $g$ is defined by $g(x)=arccot(x)$, and $h$ is a function where the values that $h$ and $h^{\prime}$ take for $x=1$ and $x=2$ are defined in the table, then f'(1) is equal to

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: