Consider an object whose position is given by the function $p\left(t\right)=5t^2-10t$, where $p$ is in meters and $t$ is in seconds. Find the velocity and acceleration of the object at $t=4$.

A basketball is shot upwards from the ground with an initial velocity of $40\text{ ft/s}$. The height (in feet) of the basketball $t$ seconds after it is shot is given by the equation $h\left(t\right)=-16t^2+40t$. What is the maximum height reached by the basketball?

The total surface area of a cylinder having radius $r$ and height $h$ is given as $A=2\pi r\left(r+h\right)$. Calculate the value of $\frac{dr}{dh}$ if $A=500\pi$, $r=5$ and $h=12$.

Determine the equation of the normal line to the following curve at the given point:

$x^4-x{y}^2+y^4=24$; $(2,-2)$

A drone is moving horizontally at a constant speed of $200$ meters per minute at an altitude of $300$ meters. It flies directly above an observer standing on the ground. Determine the rate at which the angle of elevation (between the ground and the line from the observer to the drone) is changing $4$ seconds later.

A cylindrical swimming pool with a radius of $2$ meters and a depth of $5$ meters is being emptied. If the water level is dropping at a rate of $3$ cm/min, at what rate is the water leaving the pool in cubic meters per minute?

Determine the linear approximation of $h(y) = \sqrt{4y - 5}$ at the point $a = 2$.

Use linear approximations to estimate the value of $e^{-0.05}$. Choose an appropriate value of $a$ that minimizes the error.

A cylindrical water tank with a fixed radius of $10\text{ cm}$ is being drained. Calculate the change in the water volume when the water level decreases from $30\text{ cm}$ to $29.5\text{ cm}$. Use the formula for the volume of a cylinder, $V\left(h\right)=\pi r^{2}h$.

Provide the differential expression $dy=f^{\prime }\left(x\right)dx$ for the following function:

$f\left(x\right)=4\ln (2-3x)$