A polynomial function that fits the data in each table, using finite differences and a system of equations; and
It has been determined that
It has further been evaluated that
Given:
The following tables:
Concept used:
The degree of a polynomial that fits a given data set can be obtained by repeatedly analyzing the differences between the data values corresponding to equally spaced values of the independent variable.
Calculation:
Repeatedly calculating the difference until the differences are constant, for the first table, it follows that:
1 | 4 | |||
2 | 9 | |||
3 | 26 | |||
4 | 57 | |||
5 | 104 | |||
6 | 169 |
It can be seen from the above table that constant finite difference is obtained after three rounds of finding differences.
This implies that
Then, it can be written that,
From the given table,
Put
Put the previously obtained values in these expressions to get,
Subtracting equation (1) from equation (2),
Subtracting equation (2) from equation (3),
Subtracting equation (3) from equation (4),
Subtracting equation (5) from equation (6),
Subtracting equation (6) from equation (7),
Subtracting equation (8) from equation (9),
Simplifying,
Put
Simplifying,
On further simplification,
Put
Simplifying,
On further simplification,
Put
Simplifying,
On further simplification,
Put
This is the required polynomial function.
Similarly, repeatedly calculating the difference until the differences are constant, for the second table, it follows that:
1 | -2 | |||
2 | -2 | |||
3 | 12 | |||
4 | 52 | |||
5 | 130 | |||
6 | 258 |
It can be seen from the above table that constant finite difference is obtained after three rounds of finding differences.
This implies that
Then, it can be written that,
From the given table,
Put
Put the previously obtained values in these expressions to get,
Subtracting equation (10) from equation (11),
Subtracting equation (11) from equation (12),
Subtracting equation (12) from equation (13),
Subtracting equation (14) from equation (15),
Subtracting equation (15) from equation (16),
Subtracting equation (17) from equation (18),
Simplifying,
Put
Simplifying,
On further simplification,
Put
Simplifying,
On further simplification,
Put
Simplifying,
On further simplification,
Put
This is the required polynomial function.
It has been determined that
Then,
So,
This is the required expression.
Conclusion:
It has been determined that
It has further been evaluated that
Chapter 2 Solutions
Holt Mcdougal Larson Algebra 2: Student Edition 2012
- Algebra and Trigonometry (6th Edition)AlgebraISBN:9780134463216Author:Robert F. BlitzerPublisher:PEARSONContemporary Abstract AlgebraAlgebraISBN:9781305657960Author:Joseph GallianPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
- Algebra And Trigonometry (11th Edition)AlgebraISBN:9780135163078Author:Michael SullivanPublisher:PEARSONIntroduction to Linear Algebra, Fifth EditionAlgebraISBN:9780980232776Author:Gilbert StrangPublisher:Wellesley-Cambridge PressCollege Algebra (Collegiate Math)AlgebraISBN:9780077836344Author:Julie Miller, Donna GerkenPublisher:McGraw-Hill Education