1. A plant manager has worked hard over the past years using various innovative means to increase the number of items produced per employee in his plant and has noticed that the number of items n(t) produced per employee per day over the past 10 years is given by n(t) 9t where t is the number of t+1 years from 10 years ago. Sketch the graph of the function on (0, ∞). That is, find any extrema, all intervals of increase and decrease, intervals of concavity, any points of inflection and determine the behavior at infinity and any singular points. b. Based on your graph, what is happening with the number of items n(t) produced per employee per day? (Note: checking your work with a graphing calculator is fine but you must show all your work to get to each of the answers in the curve sketch) a.
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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A plant manager has worked hard over the past years using various innovative means to increase the number of items produced per employee in his plant and has noticed that the number of items ?(?)
produced per employee per day over the past 10 years is given by ?(?) = 9? where ? is the number of ?+1
years from 10 years ago.
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Sketch the graph of the function on (0, ∞). That is, find any extrema, all intervals of increase and
decrease, intervals of concavity, any points of inflection and determine the behavior at infinity and
any singular points.
-
Based on your graph, what is happening with the number of items ?(?) produced per employee
per day?
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(Note: checking your work with a graphing calculator is fine but you must show all your work to get to each of the answers in the curve sketch)
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