Applications of derivatives

Suppose we throw an object vertically upwards at the initial speed $v_0=80\mathrm{m}/\mathrm{s}$. Determine the time needed for the object to reach the maximum height and determine the corresponding maximum height as well. $$$$ Hint: The vertical upwards motion of a body is the movement composed of uniformly rectilinear motion (vertically upwards) and free fall. The dependence of the instantaneous height of a body on the time is given by the relation $h=v_{0}t-\frac{1}{2}g{t}^2$, where $v_0$ is the magnitude of the initial velocity and $g$ is the gravitational acceleration. In this problem, calculate with the rounded value of $g=10\frac{\mathrm{m}}{{\mathrm{s}}^2}$. We measure time $t$ in second and height $h$ in meters.

An electrical source is characterized by the electromotive force $U_e=40\mathrm{V}$ and the internal resistance $R_i=2\mathrm{\Omega }$. Determine the value of the electric current for which the appliance power will be at its maximum and determine the value of this maximum power as well. $$$$ Hint: The dependence of the power of an appliance ($P$, unit Watt ($\mathrm{W}$)) on the magnitude of the folowing current ($I$, unit Ampere ($\mathrm{A}$)) is given by the relation $P=U_{e}I-R_i{I}^2$. The source properties have a role of parameters: $U_e$ is the electromotive force and $R_i$ is the internal resistance of the source.