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Chapter 8: Problem 88

Use a system of equations to find the cubic function \(f(x)=a x^{3}+b x^{2}+c x+d\) that satisfies the equations. Solve the system using matrices. $$\begin{aligned} &f(-2)=-17\\\ &f(-1)=-5\\\ &f(1)=1\\\ &f(2)=7 \end{aligned}$$

Short Answer

Expert verified
The solution to the given system of equations is \(a=1\), \(b=1\), \(c=-1\), and \(d=0\). Therefore, the cubic function that satisfies the given conditions is \(f(x)=x^3 + x^2 - x\).

Step by step solution

01

Create a system of equations

The conditions in the problem imply that the function \(f(x)\) evaluates to a given value at the points \(x=-2, -1, 1, 2\). Substituting these values into the equation \(f(x)=ax^3 + bx^2 + cx + d\) gives the following system of equations: \[\begin{align*}-8a + 4b -2c + d = -17 \-a + b - c + d = -5 \a + b + c + d = 1 \8a + 4b + 2c + d = 7\end{align*}\]
02

Represent the system as a matrix

This system of equations can be written as a 4x4 matrix where the coefficients of \(a, b, c, d\) form the elements of the matrix, and the values on the right-hand side of the equations form the constant vector, \([-17, -5, 1, 7]\). The matrix equation becomes \[A * [a, b, c, d] = [-17, -5, 1, 7]\]Where \[A = \begin{bmatrix}-8 & 4 & -2 & 1\-1 & 1 & -1 & 1\1 & 1 & 1 & 1 \8 & 4 & 2 & 1 \end{bmatrix}\]
03

Solve the matrix equation

In order to find the values of \(a, b, c, d\), one can use methods such as Gaussian Elimination or Cramer's rule. The solution to this matrix equation gives the coefficients for the cubic function. For this example, Gaussian elimination is used, and the results are \(a=1, b=1, c=-1, d=0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

System of Equations
A system of equations is a set of equations with multiple variables that are solved together. Each equation in the system provides a condition that the solution set must satisfy. In our scenario, we have a cubic function, \(f(x) = ax^3 + bx^2 + cx + d\), that needs to meet specific criteria at certain points. Each criterion forms an equation.
  • \(f(-2) = -17\)
  • \(f(-1) = -5\)
  • \(f(1) = 1\)
  • \(f(2) = 7\)
We substitute these \(x\) values into the cubic equation to form a system of four equations. This system, which will be used later, helps us find coefficients \(a, b, c,\) and \(d\) that define our cubic function.
Matrix Representation
Matrix representation is a way of structuring a system of equations into a compact form using matrices. A matrix is a rectangular array of numbers. Here, our goal is to represent the system of equations derived from the cubic function into a matrix form. Each row in the matrix corresponds to one equation in the system.
This involves creating two matrices:
  • Matrix \(A\) which contains the coefficients of \(a, b, c,\) and \(d\)
  • A column matrix representing the results of equations \([-17, -5, 1, 7]\)
Matrix \(A\) looks like this: \[A = \begin{bmatrix}-8 & 4 & -2 & 1 \ -1 & 1 & -1 & 1 \ 1 & 1 & 1 & 1 \ 8 & 4 & 2 & 1 \end{bmatrix}\]Together with the results, the matrix equation \(A\mathbf{x} = \mathbf{b}\), where \(\mathbf{x} = [a, b, c, d]'\) and \(\mathbf{b} = [-17, -5, 1, 7]'\), can be solved for \(\mathbf{x}\).
Gaussian Elimination
Gaussian Elimination is a method for solving systems of linear equations. The process transforms a matrix into an upper triangular form, making it easier to find the solutions through back substitution.
We start with the matrix equation \(A\mathbf{x} = \mathbf{b}\) and apply the following steps:
  • Eliminate lower elements below the main diagonal by subtracting a multiple of rows from each other.
  • Once in upper triangular form, solve for the last variable (\(d\)) first.
  • Substitute back to find other unknowns (\(c\), \(b\), and \(a\)).
This method is effective for this exercise as it sequentially reduces the complexity, allowing us to solve for the coefficients \(a, b, c,\) and \(d\). In our solution, this method yields \(a=1\), \(b=1\), \(c=-1\), and \(d=0\).
Cramer's Rule
Cramer's Rule is a mathematical theorem used for solving systems of linear equations with as many equations as unknowns, employing determinants. It's an alternative to Gaussian elimination.
Though not used in this particular solution, Cramer's Rule could have been applied given it works when the matrix is square (same number of equations and unknowns) and non-singular (determinant is not zero).
The key steps involve:
  • Calculating the determinant of the coefficient matrix \(\text{det}(A)\).
  • Replacing each column of the coefficient matrix with the result vector and finding the determinant of each new matrix.
  • Solving for each variable \(x_i\) using the formula \(x_i = \frac{\text{det}(A_i)}{\text{det}(A)}\), where \(A_i\) is the modified matrix with the \(i\)-th column replaced.
Cramer's Rule provides a straightforward, though computationally intensive, solution to find \(a, b, c,\) and \(d\) for smaller systems.

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Most popular questions from this chapter

Four test plots were used to explore the relationship between wheat yield \(y\) (in bushels per acre) and amount of fertilizer applied \(x\) (in hundreds of pounds per acre). The results are given by the ordered pairs \((1.0,32),(1.5,41),(2.0,48),\) and (2.5,53) (a) Find the least squares regression line \(y=a x+b\) for the data by solving the system for \(a\) and \(b\) \(\left\\{\begin{array}{l}4 b+7.0 a=174 \\ 7 b+13.5 a=322\end{array}\right.\) (b) Use the regression feature of a graphing utility to confirm the result in part (a). (c) Use the graphing utility to plot the data and graph the linear model from part (a) in the same viewing window. (d) Use the linear model from part (a) to predict the yield for a fertilizer application of 160 pounds per acre.

Determine whether the statement is true or false. Justify your answer. Solving a system of equations graphically using a graphing utility always yields an exact solution.

Determine whether the statement is true or false. Justify your answer. Exploration Find a pair of \(3 \times 3\) matrices \(A\) and \(B\) to demonstrate that \(|A+B| \neq|A|+|B|\)

Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{aligned} x+2 y &=7 \\ 2 x+y &=8 \end{aligned}\right.$$

Write the system of linear equations represented by the augmented matrix. Then use back-substitution to find the solution. (Use the variables \(x, y,\) and \(z,\) if applicable.) $$\left[\begin{array}{rrrrr} 1 & -1 & 4 & \vdots & 0 \\ 0 & 1 & -1 & \vdots & 2 \\ 0 & 0 & 1 & \vdots & -2 \end{array}\right]$$
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