A silver dollar is dropped from the top of a building that is 1332 feet tall. Use the position function below for free-falling objects. s(t) = −16t2 + v0t + s0   (a) Determine the position and velocity functions for the coin.   (b) Determine the average velocity on the interval [1, 2].   (c) Find the instantaneous velocities when t = 1 second and t = 2 seconds.   (d) Find the time required for the coin to reach the ground level. (Round your answer to three decimal places.)   (e) Find the velocity of the coin at impact. (Round your answer to three decimal places.)

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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.