Question

Please research and share two or more different real life applications where differentiation/derivatives apply. You must use reliable sources and cite any source used.

Answer 1

1)To calculate the dynamic amplification factor of a cyclic force, such as a person walking, you need to know the frequency of the force and the vertical displacement of the centre of gravity. If the vertical displacement of the centre of gravity varies according to A cos(wt) where A is half of the vertical displacement of the centre of gravity at each stride and w is the frequency in radians of the stride (there are 2 x pi radian in a cycle or approximately 6.28 radians in a cycle, so to obtain the frequency in radians you multiply the frequency in Hertz by 6.28. The frequency in Hz is the inverse of the duration of a single stride in seconds. You can obtain the duration of a stride by measuring the number of strides you can make in a minute and divide by 60. Anyway, all that sounds complicated, but to obtain the vertical acceleration of your centre of gravity, a = - Aw2 cos(wt) where w2 is the frequency in radians squared. This is because the derivative of A cos(wt) = Aw sin(wt) and the second derivative is -Aw2 cos(wt). Acceleration will be maximum at the same time that the vertical displacement is maximum, so where cos(wt) = 1. Taking absolute values, the dynamic amplification factor is therefore Aw2 (amplitude x frequency in radians squared), which you can multiply by the weight of a person to calculate the dynamic loading of a person walking (or of any other cyclic force). For the average person the dynamic amplification factor is around 4 or 5

2)For example you are taking some path from your home to your school. There is a lot of traffic midway. So your speed due to the trafic is not uniform. Now your school wants to do a survey and wants to find what is the average time taken by a student from their respective homes to the school. Here what they would do is that (I am assuming that the people in the school are quite less) is they would choose some certain time in your travel and find your "instantaneous velocity" at that point. This instantaneous velocity is going to be the slope of you distance time graph. And now since we have equipped ourselves with differentiation we could easily do the survey