Question

Find a curve given by a polynomial \(P_{5}(x)\) that provides a smooth transition between \(y=0\) for \(x \leq 0\) and \(y=x\) for \(x \geq 1\)

Short answer

The polynomial is \( P_5(x) = x^5 - 3x^4 + 3x^3 \).

Step-by-step solution

Determine the Polynomial Form

Since we need a smooth transition, the polynomial must be of degree 5, written as \( P_5(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \). This degree allows enough flexibility to adjust values and derivatives at the endpoints.

Apply Boundary Conditions

For a smooth transition, use the conditions: \( P_5(0) = 0 \), \( P_5(1) = 1 \), where \( P_5 \) coincides with 0 for \( x \leq 0 \) and with \( x \) for \( x \geq 1 \). Hence, \( f = 0 \) and \( a + b + c + d + e + f = 1 \).

Ensure Continuity of First Derivatives

For continuity of first derivatives, \( P_5'(0) = 0 \) and \( P_5'(1) = 1 \). The first derivative is \( P_5'(x) = 5ax^4 + 4bx^3 + 3cx^2 + 2dx + e \). At \( x = 0 \), \( e = 0 \) and at \( x = 1 \), \( 5a + 4b + 3c + 2d + e = 1 \).

Ensure Continuity of Second Derivatives

For continuity of second derivatives, \( P_5''(0) = 0 \) and \( P_5''(1) = 0 \). The second derivative is \( P_5''(x) = 20ax^3 + 12bx^2 + 6cx + 2d \). At \( x = 0 \), \( d = 0 \) and at \( x = 1 \), \( 20a + 12b + 6c + 2d = 0 \).

Solve the System of Equations

Using the equations derived from above steps: \[\begin{aligned}f &= 0,\ a + b + c + d + e &= 1,\ e &= 0,\ d &= 0,\ 5a + 4b + 3c &= 1,\ 20a + 12b + 6c &= 0.\end{aligned}\] Solve this system of linear equations to find \( a, b, c \).

Substitute Values and Write the Polynomial

From solving the equations, substitute the values of \( a, b, c, d, e, f \) into the polynomial form to get \( P_5(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \).

Key concepts

Degree of Polynomial

In the context of polynomial curve fitting, the degree of a polynomial is crucial. It refers to the highest power of the variable in the polynomial, determining the polynomial's shape and flexibility. For complex shapes, a higher degree may provide the necessary curvature to fit the desired form.

In this exercise, we use a 5th-degree polynomial, represented as \( P_5(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \). This degree is chosen to provide enough flexibility for the desired smooth transition between lines. A polynomial of degree 5 allows adjusting both values and derivatives at the endpoints, essential in achieving the conditions required.

Why degree 5? It ensures continuity of derivatives, which smooths the transition between distinct linear behaviors. A lower degree might not accommodate the continuous changes in slope and value across the boundary.

Boundary Conditions

Boundary conditions are essential for defining the specific behavior of a polynomial at certain points. They specify the values the function must take at particular boundaries, ensuring the curve fits seamlessly with any pre-defined lines or curves.

In this scenario, our boundary conditions are \( P_5(0) = 0 \) and \( P_5(1) = 1 \). These conditions are dictated by the requirements that the polynomial matches \( y = 0 \) for \( x \leq 0 \) and \( y = x \) for \( x \geq 1 \). This implies the polynomial at these boundaries should start at point 0 and rise to meet the line at point 1.

By establishing these boundary conditions, we ensure the polynomial begins and ends accurately. They act as fixed anchors that the curve must pass through, defining its start and endpoint on the graph.

Continuity of Derivatives

The continuity of derivatives is a key concept when ensuring a smooth transition between curves or lines using a polynomial. This continuity involves the first and second derivatives of the polynomial.

For this polynomial, first derivative continuity is ensured by having \( P_5'(0) = 0 \) and \( P_5'(1) = 1 \). The first derivative represents the slope of the curve, and matching these at the boundaries maintains a consistent slope.

Next, the second derivative continuity, with \( P_5''(0) = 0 \) and \( P_5''(1) = 0 \), ensures that changes in the slope (curvature) are smooth. Second derivatives relate to the concavity of the function, indicating how the slope itself changes. Smoothing them out prevents abrupt changes.

This meticulous matching of derivatives across boundaries prohibits any sudden transitions, crafting a polynomial curve that merges consistently across segments.

System of Linear Equations

A system of linear equations arises naturally when dealing with polynomial curve fitting, especially when applying conditions like continuity and boundary limits. Solving these systems helps pinpoint the exact coefficients of the polynomial.

In this exercise, we derived several linear equations from our conditions:
\[\begin{aligned}f &= 0,\ a + b + c + d + e &= 1,\ e &= 0,\ d &= 0,\ 5a + 4b + 3c &= 1,\ 20a + 12b + 6c &= 0.\end{aligned}\]

Here, each equation encapsulates one of the conditions for our polynomial. By solving this system, we find values for \( a, b, \) and \( c \), the unknowns. Linear equations are ideal in these scenarios as they are straightforward to solve, often using substitution or elimination methods.

Successfully solving this system provides the polynomial coefficients needed to ensure the desired curve meets all specified conditions without deviation.