A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in meters) that is sold is a function of the selling price p (in dollars per meter), so we can write q = f(p). Then the total revenue earned with selling price p is R(p) = pf(p). (a) What does it mean to say that f(20) = 10,000 O There are 325 total m of fabric and $20 to spend on it. O When the price of fabric is $20/m, 325 m will be sold. O When the price of fabric is $20/m, 10,000 m will be sold. O There are 10,000 total m of fabric and $325 to spend on it. O When the price of fabric is $325/m, 20 m will be sold. What does it mean to say that f'(20) = -325? O As the price of the fabric increases past $325/m, the amount of fabric which will be sold is increasing at a rate of 20 m per (dollar per meter). O As the price of the fabric increases past $20/m, the amount of fabric which will be sold is decreasing at a rate of 325 m per (dollar per meter). O As the price of the fabric decreases past $325/m, the amount of fabric which will be sold is decreasing at a rate of $10,000 per (dollar per meter). O As the price of the fabric decreases past $20/m, the amount of fabric which will be sold is increasing at a rate of $325 per (dollar per meter). O As the price of the fabric decreases past $20/m, the amount of fabric which will be sold is increasing at a rate of 10,000 m per (dollar per meter). (b) Assuming the values in part (a), find R'(20). R'(20) = Interpret your answer. As the price of fabric increases past $ /m, the total revenue is ---Select-- v at $ per (dollar per meter).

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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.