Question

13 The graph below approximates the rate of change of the price of tomatoes over a 60-month period, where p(t) is the price of a pound of tomatoes and is time (in months). 14 15 16 17 18 19 20 21 22 23 0.07 0.06 p'(t) 0.05 0 15 24 0.04 30 0.06 0 -0.02 0 0.06 25 26 27 0.03 45 p'(t) (dollars per month) 0.02 60 0.01 28 0 0 10 20 30 40 -0.01 60 50 70 -0.02 -0.03 t(months) 29 30 31 32 33 34 35 36 37 38 39 40 Write a brief description of the graph of y- p(), including a discussion of local extrema and inflection points. Answer in this textbox. Answer: 41 42 43

anyone who understands advanced math, please help!

Answer 1

The given graph approximates the rate of change of the price of tomatoes over a 60-month period, where p(t) is the price of a pound of tomatoes and t is the time, in months.

Thus, we are given with a graph of the function y=p'(t), along with a table of the values of p'(t) at different values of t as well.

From the graph of p'(t), we can see that p'(t) starts at 0.06 and then decreases gradually to -0.02 and then increases again back to 0.06. Now, p'(t) gives us the approximate rate of change of the price of tomatoes, y=p(t), at any point of time, t. So, a large value of p'(t) tells us that the rate of change of the price at that point is large, and the sign, + or -, of p'(t) tells us whether this change is positive or negative, that is, when p'(t) is positive, the price p(t) is increasing, and when p'(t) is negative, the price price p(t) is decreasing. When p'(t) is 0, this means that the price is neither increasing nor decreasing, thus the prince p(t) has a local extrema at that point, t.

So, armed with this knowledge, we can give a very concrete description of y=p(t).

The function p(t) increases rapidly initially, and as t becomes larger, the slope(how fast the curve is changing) becomes flatter(indicating that the function is changing very slowly), and eventually, the function reaches an extreme point at t=15 when p'(t)=0, and after t=15, p'(t) becomes negative, thus p(t) becomes decreasing. So, before t=15 the function was increasing, and after t=15, the function is decreasing, hence, t=15 is a local maxima of the function. Thus, the price of tomatoes was highest at 15 months.Now, after t=15, p'(t) keeps decreasing in value till t=30, where p'(t) reaches a minimum, and then p'(t) starts increasing again. Thus, the slope of p(t) stops decreasing at t=30 and starts increasing again. Hence t=30 is an inflection point of the function p(t), that is, it is the point where the slope of the function starts increasing again, and it increases till t=45, where it once again becomes 0. Now, this time, to the left of t=45, p'(t) is negative, and to the right, p'(t) is positive, hence the function p(t) is decreasing before t=45 and increasing after t=45, so the function has a local minimum at this point. Thus, the price of tomatoes was minimum at 45 months. After t=45, the function p'(t) goes on increasing till it reaches 0.06, which is the same as where it started at t=0, thus, the rate of change of the price of tomatoes becomes the same as the initial rate of change after 60 months.

Thus, all this information gives us a rough idea about a sketch of the function, which gives:

64 54 31 2 14 0 20 40 60