1-4 Modeling Periodic Data A set of data is given.
(a) Make a
(b) Find a cosine function of the form
(c) Graph the function you found in part (b) together with the scatter plot. How well does the curve fit the data?
(d) Use a graphing calculator to find the sine function that best fits the data, as in Example 2.
(e) Compare the functions you found in parts (b) and (d). [Use the reduction formula
t | y |
0 | 2.1 |
2 | 1.1 |
4 |
|
6 |
|
8 |
|
10 | 0.6 |
12 | 1.9 |
14 | 1.5 |

(a)
To find:
A scatter plot of the data.
Answer to Problem 1P
Solution:
Explanation of Solution
Calculation:
The scatter plot for the given data is given by
Final statement:

(b)
To find:
A cosine function of the form
Answer to Problem 1P
Solution:
Explanation of Solution
Calculation:
Here, the maximum value is 2.1 and the minimum value is -2.1.
The maximum value occurs at time 0 and the minimum value occurs at time 6.
The vertical shift
The amplitude
The time between consecutive maximum and minimum values is half of one period.
That is,
Since the maximum value of the data occurs at
There is no horizontal shift.
Apply the values in the function
Since cosine is an even function,
Final statement:

(c)
To find:
The graph of the function from part (b).
Answer to Problem 1P
Solution:
Explanation of Solution
Calculation:
The graph of the function
Final statement:

(d)
To find:
The sine function that fits the data.
Answer to Problem 1P
Solution:
Explanation of Solution
Calculation:
Using the given data and the SinReg command on the T1-83 calculator, we get a function of the form
Here,
Apply the values in the function
Final statement:

(e)
To find:
The comparison between the functions from part (b) and (d).
Answer to Problem 1P
Solution:
It is same as (b), corrected to one decimal.
Explanation of Solution
Calculation:
From part (d),
Apply the reduction formula
So,
It is same as (b), corrected to one decimal.
Final statement:
It is same as (b), corrected to one decimal.
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