Q. Find the linear change in volume dV if the sides of a cube change from 10 to 10.1.

A. We can use the definition of to determine the estimated linear change. In this problem, we start with the cubic volume equation of where is the length of one side of the cube. The derivative is then . Filling in for linear approximation we get . This results in . We can see that the linear approximation is not entirely accurate, hence the name approximation, as .

Related equations and relationships:
If then is increasing. If then is decreasing.

Problem:
A rock thrown upward with velocity 16 ft/sec reaches height at time t.
(a) Find its average speed from to .
(b) Find its average speed from to .
(c) What is at ?

Solution:
To find speed given distance, we need to take the first derivative. Derivatives of polynomials are one of the few things I remember, even though this is covered in the material I’m reading. That equation is: . However, the question isn’t asking for a limit, so the full equation of should be used. Essentially, it is the average of the first derivative at two given points.
This results in:
(a) 8 ft/sec
(b) -8 ft/sec
(c) 0, as it is the instantaneous point of reversal (this one assumes a limit, as t approaches )