Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 18
Let \(x_{0}
Short Answer
Expert verified
The polynomial is: \[ q(x) = ax^2 + bx + c \]. Ensure x-values create a non-singular system for uniqueness.
Step by step solution
01
Polynomial Assumption
Assume the polynomial to be of the form: \[ q(x) = ax^2 + bx + c \] where a, b, and c are constants to be determined.
02
Apply Initial Conditions
Use the given conditions to create a system of equations: 1. \[ q(x_0) = y_0 \rightarrow a(x_0)^2 + b(x_0) + c = y_0 \]2. \[ q^{\text{'}}(x_1) = y_1^{\text{'}} \rightarrow 2ax_1 + b = y_1^{\text{'}} \]3. \[ q(x_2) = y_2 \rightarrow a(x_2)^2 + b(x_2) + c = y_2 \]
03
Form System of Equations
Equations derived from Step 2:1. \[ ax_0^2 + bx_0 + c = y_0 \]2. \[ 2ax_1 + b = y_1^{\text{'}} \]3. \[ ax_2^2 + bx_2 + c = y_2 \]
04
Solve for Constants
Solve the system of equations to find values of a, b, and c. Use substitution or matrix methods to solve the linear systems.
05
Check Uniqueness
Check if the polynomial is uniquely determined by these equations. The system is uniquely determined if the determinant of the coefficient matrix does not equal zero. Here, ensure the x-values and conditions do not lead to a singular matrix.
06
Summary
Compile the found coefficients into the polynomial expression: \[ q(x) = ax^2 + bx + c \]
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
polynomial interpolation
Polynomial interpolation involves finding a polynomial that passes through a given set of points.
In this exercise, we need a polynomial of degree at most 2, meaning a quadratic polynomial. We generally represent a quadratic polynomial by: ax^2 + bx + c
Given points or conditions tell us where the polynomial should pass through or what slope it should have at specific points.
For this exercise, the conditions are given by: q(x0) = y0, q'(x1) = y1', and q(x2) = y2.
We use these conditions to form a system of equations, which we will then solve to find the coefficients 'a', 'b', and 'c' of our polynomial. This process ensures that our polynomial perfectly fits the given conditions.
In this exercise, we need a polynomial of degree at most 2, meaning a quadratic polynomial. We generally represent a quadratic polynomial by: ax^2 + bx + c
Given points or conditions tell us where the polynomial should pass through or what slope it should have at specific points.
For this exercise, the conditions are given by: q(x0) = y0, q'(x1) = y1', and q(x2) = y2.
We use these conditions to form a system of equations, which we will then solve to find the coefficients 'a', 'b', and 'c' of our polynomial. This process ensures that our polynomial perfectly fits the given conditions.
derivative constraint
A derivative constraint is a condition involving the derivative of the polynomial.
In our example, we are given q'(x1) = y1', which means the first derivative of q(x) at x = x1 equals y1'. The derivative constraint affects the slope of our polynomial at a specific point. For a quadratic polynomial q(x) = ax^2 + bx + c, its derivative is q'(x) = 2ax + b.
So the condition q'(x1) = y1' translates to: 2ax1 + b = y1'. This equation helps us set up our system of equations and is critical in solving for a particularly fitting polynomial.
In our example, we are given q'(x1) = y1', which means the first derivative of q(x) at x = x1 equals y1'. The derivative constraint affects the slope of our polynomial at a specific point. For a quadratic polynomial q(x) = ax^2 + bx + c, its derivative is q'(x) = 2ax + b.
So the condition q'(x1) = y1' translates to: 2ax1 + b = y1'. This equation helps us set up our system of equations and is critical in solving for a particularly fitting polynomial.
system of equations
A system of equations consists of multiple equations that we need to solve simultaneously.
Our quadratic polynomial interpolation involves setting up 3 equations based on given conditions: q(x0) = y0, q'(x1) = y1', q(x2) = y2.
We write these conditions in terms of 'a', 'b', and 'c': 1. ax0^2 + bx0 + c = y0 2. 2ax1 + b = y1' 3. ax2^2 + bx2 + c = y2
These three equations make up our system of equations. Solving this system will give us specific values for 'a', 'b', and 'c', which define our quadratic polynomial.
Our quadratic polynomial interpolation involves setting up 3 equations based on given conditions: q(x0) = y0, q'(x1) = y1', q(x2) = y2.
We write these conditions in terms of 'a', 'b', and 'c': 1. ax0^2 + bx0 + c = y0 2. 2ax1 + b = y1' 3. ax2^2 + bx2 + c = y2
These three equations make up our system of equations. Solving this system will give us specific values for 'a', 'b', and 'c', which define our quadratic polynomial.
uniqueness of solutions
Uniqueness of solutions indicates whether our system of equations has exactly one solution.
This is important because multiple solutions or no solutions mean we can't find a definite polynomial fitting the given conditions.
To ensure uniqueness, we need to check if the determinant of the system's coefficient matrix is non-zero. In other words, the matrix based on our (x0, x1, x2) values should not be singular.
If the determinant is zero, then the system is either dependent (infinite solutions) or inconsistent (no solution). But if the determinant is non-zero, it means there is a unique solution, and thus a unique polynomial that meets our constraints.
This is important because multiple solutions or no solutions mean we can't find a definite polynomial fitting the given conditions.
To ensure uniqueness, we need to check if the determinant of the system's coefficient matrix is non-zero. In other words, the matrix based on our (x0, x1, x2) values should not be singular.
If the determinant is zero, then the system is either dependent (infinite solutions) or inconsistent (no solution). But if the determinant is non-zero, it means there is a unique solution, and thus a unique polynomial that meets our constraints.
