5.3a Homework: Constructing Linear Functions Directions: Determine whether the situations represented below are linear or nonlinear. For the situations that are linear, construct a function that models the relationship between the two quantities. Be sure to define your variables. 1. The graph below shows the amount of revenue a company will make selling t-shirts dependent on the price of each t-shirt. Consider the relationship between price of each shirt and revenue made. Is the data linear? Why or why not? Revenue ($) 2000 1800 1600 If yes, construct a function to model the relationship between the two variables. Be sure to define your variables. 1400 1200 1000 800 600 400 200 1 2 34 5 67 8 9 10 Price ($) When Camilo opened his email this morning he had 140 unread emails. The table below shows the numb of remaining unread emails Camilo has in his inbox. Assume that Camilo does not receive any new emai while he is reading his email. Consider the relationship between time and the number of unread emails. Is the data linear? Why or why not? # of Unread Emails Time (hours) 180 0.5 160 1 140 If yes, construct a function to model the relationship between the two quantities. Be sure to define your variables. 100 te 2.5 80 4. 20 4.5 2.
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.
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