Problem 102 - Precalculus with Limits

Question

In Exercises 99-102, use a system of equations to find the cubic function \(f(x) = ax^3 + bx^2 + cx + d\) that satisfies the equations. Solve the system using matrices. \(f(-2) = -17\) \(f(-1) = -5\) \(f(1) = 1\) \(f(2) = 7\)

Step-by-Step Solution

Answer

The solution method involves these steps: write the system of equations, convert the system into matrix form, solve the matrix, and finally, interpret the solution as the coefficients of the cubic function. The actual values of \(a\), \(b\), \(c\), and \(d\) will depend on the specific matrix solving technique used.

Step 1: Write out the system of equations

Our system of equations based on the given points is: \(a(-2)^3 + b(-2)^2 + c(-2)+ d = -17\), \(a(-1)^3 + b(-1)^2 + c(-1) + d = -5\), \(a(1)^3 + b(1)^2 + c(1) + d = 1\), \(a(2)^3 + b(2)^2 + c(2) + d = 7\)

Step 2: Convert the system of equations into matrix form

Writing this system of equations in matrix form we obtain a 4x4 matrix multiply by a 4x1 matrix equals a 4x1 matrix: \(\begin{bmatrix} -8 & 4 & -2 & 1 \ -1 & 1 & -1 & 1 \ 1 & 1 & 1 & 1 \ 8 & 4 & 2 & 1 \end{bmatrix} \begin{bmatrix} a \ b \ c \ d \end{bmatrix} = \begin{bmatrix} -17 \ -5 \ 1 \ 7 \end{bmatrix}\)

Step 3: Solve the matrix using an applicable method

This could be done using Gaussian elimination or other matrix solving techniques. For instance, using a calculator or statistical software a solution can be obtained.

Step 4: Read-off the coefficients of the cubic function

The solution of the matrix equation gives the values of the coefficients \(a\), \(b\), \(c\), and \(d\)

Key Concepts

Cubic Function

A cubic function is a type of polynomial function with the highest power of the variable being three. It is defined mathematically as \(f(x) = ax^3 + bx^2 + cx + d\). Here, \(a\), \(b\), \(c\), and \(d\) are constants, and \(a eq 0\).
The graph of a cubic function is a smooth, continuous curve that can have different shapes, including having one or more inflection points, where the direction of curvature changes.
It can intersect the x-axis up to three times, corresponding to the real roots of the equation \(ax^3 + bx^2 + cx + d = 0\).
Solving a cubic equation involves finding values of \(x\) that fulfill this equation, which often requires more advanced algebraic techniques or numerical methods.
In the exercise provided, we're tasked with determining a specific cubic function that passes through four given points.

Matrix Algebra

Matrix algebra involves mathematical operations with matrices, which are rectangular arrays of numbers. Matrix algebra is a powerful tool for solving systems of equations, such as the one given in the problem.

• Each row in a matrix represents an equation, and each column represents a variable's coefficient.
• To turn equations into matrix form, align coefficients of the variables in a grid-like format, with constants on the other side of the equation.

In our exercise, the system of four equations is converted into matrix form. This representation allows the use of numerous techniques to find solutions, including matrix operations like addition, multiplication, and specific methods like Gaussian elimination.
By solving the matrix, you obtain the values of the constants (\(a\), \(b\), \(c\), and \(d\)), which are crucial for defining the cubic function.

Gaussian Elimination

Gaussian elimination is a systematic procedure used to solve systems of linear equations. It transforms the matrix into reduced row-echelon form using a sequence of operations.
This involves:

• Row swapping - changing the order of the equations.
• Scaling - multiplying whole rows by non-zero constants.
• Row addition - adding or subtracting multiples of one row from another.

The goal is to simplify the matrix until the solution can be "read off" easily.
For our problem, using Gaussian elimination on the matrix helps us to find the values of \(a\), \(b\), \(c\), and \(d\) from the system of equations by transforming it into a simpler form where these coefficients can be determined directly.

Precalculus

Precalculus forms the foundation for calculus, involving mathematical skills and concepts needed for it, such as functions, algebra, trigonometry, and analytical geometry.
In the context of systems of equations and matrix algebra, precalculus helps strengthen understanding of solving such systems using different methods.
This includes familiarity with polynomial functions, transforming systems into mathematical models, and using matrices efficiently.
Precalculus ensures that students can handle complex equations accurately, setting the stage for calculus where these concepts will be expanded upon, especially when dealing with more complex functions and in calculus-based disciplines. Tackling these systems facilitates a smoother transition beyond basic algebra towards calculus applications.