On August 4, 2007, NASA's Phoenix Mars Lander was launched into space to search for life in the icy northern region of the planet Mars; it touched dowr on Mars on May 25, 2008. As the ship raced into space, its jet fuel tanks dropped off when they were used up. The distance one of the tanks falls in t seconds is modeled by the function s(t) = 16t2 where s is measured in feet per second and t = 0 is the instant the tank left the ship. Use the function s to find the average speed of a tank during the following time intervals. (a) The first 10 seconds after separation from the ship 1600 X ft/s (b) The first 50 seconds after separation from the ship 40000 X ft/s

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Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.