This problem set deals with the problem of non-constant acceleration. Two researchers from Fly By Night Industries conduct an experiment with a sports car on a test track. While one is driving the car, the other will look at the speedometer and record the speed of the car at one- second intervals. Now, these aren't official researchers and this isn't an official test track, so the speeds are in miles per hour using an analog speedometer. The data set they create is: {(1,5), (2, 2), (3, 30), (4, 50), (5, 65), (6,70)} Z = 25 They notice that the acceleration is not a constant value. They decide that a fourth-degree polynomial will be the best to describe the speed of the car as a function of time. The task here is to determine the fourth-degree polynomial that fits this data set the best. 1. Construct the system of normal equations A¹ Ax = A¹b. AT A = A¹b = 2. Solve the system of normal equations. (I don't want you doing this by hand. Use a calculator or app.) x =
This problem set deals with the problem of non-constant acceleration. Two researchers from Fly By Night Industries conduct an experiment with a sports car on a test track. While one is driving the car, the other will look at the speedometer and record the speed of the car at one- second intervals. Now, these aren't official researchers and this isn't an official test track, so the speeds are in miles per hour using an analog speedometer. The data set they create is: {(1,5), (2, 2), (3, 30), (4, 50), (5, 65), (6,70)} Z = 25 They notice that the acceleration is not a constant value. They decide that a fourth-degree polynomial will be the best to describe the speed of the car as a function of time. The task here is to determine the fourth-degree polynomial that fits this data set the best. 1. Construct the system of normal equations A¹ Ax = A¹b. AT A = A¹b = 2. Solve the system of normal equations. (I don't want you doing this by hand. Use a calculator or app.) x =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Set: Non-Constant Acceleration Analysis**
This exercise addresses the challenge of non-constant acceleration. Researchers at Fly By Night Industries conduct an experiment using a sports car on a test track. One researcher drives while the other records speed from an analog speedometer at one-second intervals. Although this is an unofficial study, the collected data in miles per hour is as follows:
Data Set:
\[
\{(1, 5), (2, z), (3, 30), (4, 50), (5, 65), (6, 70)\}
\]
Where \( z = 25 \).
The researchers note that acceleration is not constant and decide that a fourth-degree polynomial would best describe the speed as a function of time. The objective is to find this polynomial that best fits the data.
**Task:**
1. **Construct the System of Normal Equations:**
Use the equation \( A^T A \vec{x} = A^T \vec{b} \).
- Calculate \( A^T A \): (Details should be filled by students based on their calculations)
- Calculate \( A^T \vec{b} \): (Details should be filled by students based on their calculations)
2. **Solve the System of Equations:**
Solve the equations using a calculator or app and find:
- \( \vec{x} = \) (Students are encouraged to calculate this solution with computational tools, not by hand)
**Note:** This practical approach to applying polynomial regression will benefit students in recognizing how mathematical models describe real-world phenomena.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b8a4ad9-b66e-4dbe-8b3f-39bf0adb3cae%2Feff46812-624a-4887-9904-e5c074e3c212%2F238yvbg_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Set: Non-Constant Acceleration Analysis**
This exercise addresses the challenge of non-constant acceleration. Researchers at Fly By Night Industries conduct an experiment using a sports car on a test track. One researcher drives while the other records speed from an analog speedometer at one-second intervals. Although this is an unofficial study, the collected data in miles per hour is as follows:
Data Set:
\[
\{(1, 5), (2, z), (3, 30), (4, 50), (5, 65), (6, 70)\}
\]
Where \( z = 25 \).
The researchers note that acceleration is not constant and decide that a fourth-degree polynomial would best describe the speed as a function of time. The objective is to find this polynomial that best fits the data.
**Task:**
1. **Construct the System of Normal Equations:**
Use the equation \( A^T A \vec{x} = A^T \vec{b} \).
- Calculate \( A^T A \): (Details should be filled by students based on their calculations)
- Calculate \( A^T \vec{b} \): (Details should be filled by students based on their calculations)
2. **Solve the System of Equations:**
Solve the equations using a calculator or app and find:
- \( \vec{x} = \) (Students are encouraged to calculate this solution with computational tools, not by hand)
**Note:** This practical approach to applying polynomial regression will benefit students in recognizing how mathematical models describe real-world phenomena.
Expert Solution
Step 1
For a system: , the formula for least-squares fit is . The matrix: is the transpose of matrix: A.
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