How do you get the equations for the elastic curve using x1 an x2 coordinates?

If you were designing a bumper for a car, would you prefer it to exhibit elastic or plastic deformation? why? consider the functions of a bumper in both a minor "fender-bender" and a major collision.

Thermal stresses are developed in a metal when its a) initial temperature is changed b) final temperature is changed c) density is changed d) thermal deformation is prevented e) expansion is prevented f) contraction is prevented

Which of the following components of a pid controlled accumulates the error over time and responds to system error after the error has been accumulated? a)- proportional b)- derivative c)- integral d)- on/off.