Your college newspaper, The Collegiate Investigator, sells for 90¢ per copy. The cost of producing x copies of an edition is given by C(x) = 60 + 0.10x + 0.001x2 dollars. (a) Calculate the marginal revenue R'(x) and profit P'(x) functions R'(x) = P'(x) = (b) Compute the revenue and profit, and also the marginal revenue and profit, if you have produced and sold 500 copies of the latest edition. revenue $ profit $ marginal revenue $ per additional copy marginal profit $ per additional copy Interpret the results. The approximate profit loss from the sale of the 501st copy is $ . c) For which value of x is the marginal profit zero? x = _____copies Interpret your answer. The graph of the profit function is a parabola with a vertex at x =______ , so the profit is at a maximum when you produce and sell _____ copies.
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.
Your college newspaper, The Collegiate Investigator, sells for 90¢ per copy. The cost of producing x copies of an edition is given by
| revenue | $ | |
| profit | $ | |
| marginal revenue | $ | per additional copy |
| marginal profit | $ | per additional copy |
Interpret the results.
x = _____copies
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