Cx)=x+2 こメ 1=x Example 4 Suppose that a construction zone can allow 50 cars per hour to pass through and that cars arrive randomly at a rate of x cars per hour. Then the average number of cars waiting in line to get through the construction zone can be estimated by xam3 x² 2500 - 50x N(x) = (a) Evaluate N(20), N(40), and N(49). (b) Explain what happens to the length of the line as x approaches 50. (c) If the average number of cars waiting in line is 1, find the average of cars per hour. (d) Find and interpret the vertical asymptotes.
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.
Suppose that a construction zone can allow 50 cars per hour to pass through and that cars arrive randomly at a rate of X cars per hour. Then the average number of cars waiting in line to get through the construction zone can be estimated by:
N(x)=x2/2500-50x
Evaluate N(20), N(40), and N(49).Explain what happens to the length of the line as X approaches 50.If the average number of cars waiting in line is 1, find the average of cars per hour.Find and interpret the vertical asymptotes.
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