The human body has circadian rhythms where the body temperature oscillates about each day. In addition, a women's body temperature varies about the same over her menstrual cycle during the month. Before the advent of birth control, some women used their body temperature as an indicator of when peak fertility (ovulation) occurred. It has been shown that most women have a rapid increase in body temperature at the time of ovulation, so changes in body temperature give some indication of fertility.

a. Below is a Table of body temperatures from one female subject taken at the same time each day over one month.

t
T(t)
t
T(t)
t
T(t)
0
36.69
10
36.31
20
37.12
1
36.62
11
36.53
21
37.22
2
36.44
12
36.64
22
37.14
3
36.43
13
36.41
23
37.09
4
36.41
14
36.49
24
37.02
5
36.55
15
36.83
25
36.97
6
36.59
16
36.82
26
36.79
7
36.41
17
36.93
27
36.88
8
36.16
18
36.97
28
36.73
9
36.39
19
36.92


Use the table of data to find the best fitting cubic polynomial for this particular subject. Assume that the cubic polynomial has the form:

then Trendline gives the coefficients: a0, a1, a2, and a3.

Find the derivative of the function, T '(t). Also, find the critical points of the function T(t). List the critical points, and determine the temperature at each of the critical points.

Find the second derivative of the function, T ''(t). Use the second derivative to determine if each of the critical points are minimums or maximums, i.e., find the value of T ''(t) at the critical points and use the sign of these values to determine if the critical point is a minimum or maximum of the function.

Now find the point of inflection, ti, which is the point where the function T(t) is changing the fastest, i.e., when T '(t) is at an extrema. Since we want an extrema of the derivative we look at when T "(t) = 0. Give the point of inflection, ti, and find the rate of change at the point of inflection, T '(ti).

Based on the discussion above, give the time that the model predicts is the peak time of fertility.


b. In your Lab Report, create a graph of the data and the function T(t) on the domain . Clearly, mark the minimum and maximum points (t, T(t)) and label the point of inflection. On a separate graph, plot its derivative, T '(t), and second derivative, T ''(t). Write a paragraph describing what the model is predicting near ovulation (based on your knowledge of when ovulation usually occurs in the menstrual cycle).