Answer the following question.

Question

Month Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec

Temperature 34.0 38.3 49.6 62.0 73.0 82.7 87.1 84.9 77.8 65.3 51.2 37.8

(a)Use a graphing utility to plot the data and find a model for the data of the form

T(t) = a + b sin(ct − d)

where T is the temperature and t is the time in months, with t = 1 corresponding to January. (Round all numerical values to two decimal places.)

(c) Find T' and use a graphing utility to graph the derivative.

Answer 1

I would recommend you use desmos for this. Open up desmos and click the plus symbol in the upper left and select "table". Populate the table with the data but instead of using "Jan" type in "1", for "Feb", use "2", etc to give you a table like this:

x1 y1

1 34.0

2 38.3

3 49.6

4 62.0

5 73.0

6 82.7

7 87.1

8 84.9

9 77.8

10 65.3

11 51.2

12 37.8

Then click on the plus symbol in the upper left hand corner again and this time select "f(x) expression". Enter y₁ ~ a + bsin(cx₁ - d)

To enter y₁, type y, then the underscore then the number 1 then to get out of the subscripting, use the cursor over button. The "~" is the button just left of the number 1 on your keyboard.

When finished you should get this:

You now have the values of a, b, c, and d for the function that models the data. The R² tells how good the fit is (with "1" being perfect). This is actually a pretty good fit at 0.9971.

You can now take the derivative remembering to use the chain rule. To graph the derivative, use "y=" (whatever you got for the derivative)