The following table shows the percentage P = P(d) of the American food dollar that was spent on eating away from home (at restaurants, for example) as a function of the date d. d = Year P = Percent spent away from home 1969 25% 1989 30% 2009 34% (a) Find P(2009). % Explain what P(2009) means. Before 2009 Americans spent P(2009) percent of their food dollars eating out.Americans spent (2009 times P) percent of their food dollars eating out. After 2009 Americans spent P(2009) percent of their food dollars eating out.In 2009 Americans spent an average of P(2009) dollars eating out each week.In 2009 Americans spent P(2009) percent of their food dollars eating out. (b) What does P(1999) mean? The expression P(1999) is the average amount, in dollars, that Americans spent each week eating away from home in 1999.The expression P(1999) is the percent of the American food dollar spent eating away from home in 1999. The expression P(1999) is the percent of the American food dollar spent eating away from home times 1999.The expression P(1999) is the percent of the American food dollar spent eating away from home before 1999.The expression P(1999) is the percent of the American food dollar spent eating away from home after 1999. Estimate P(1999). % (c) What is the average rate of change per year in percentage of the food dollar spent away from home for the period from 1989 to 2009? % points per year
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.
The following table shows the percentage P = P(d) of the American food dollar that was spent on eating away from home (at restaurants, for example) as a function of the date d.
| d = Year | P = Percent spent away from home |
|---|---|
| 1969 | 25% |
| 1989 | 30% |
| 2009 | 34% |
%
Explain what P(2009) means.
(b) What does P(1999) mean?
Estimate P(1999).
%
(c) What is the average rate of change per year in percentage of the food dollar spent away from home for the period from 1989 to 2009?
% points per year
(d) What does P(2007) mean?
Estimate P(2007). (Hint: Your calculation in part (c) should be useful.)
%
(e) Predict the value of P(2012).
%
Explain how you made your estimate.
| P(2012) | = |
34 +
|
||
| = |
Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If you want remaining sub-parts to be solved, then please resubmit the whole question and specify those sub-parts you want us to solve.
Consider the given:
(a) Explain what P(2009) means.
P(2009) means percentage of the American food dollars that was spent on eating away from the home in 2009.
From above table:
Americans spent P(2009) percent of their food dollars eating out is-
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