Problem 54 - Essential Calculus Early Transcendentals

Question

Let's consider the problem of designing a railroad track to make a smooth transition between sections of straight track. Existing track along the negative \(x\) -axis is to be joined smoothly to a track along the line \(y=1\) for \(x \geqslant 1\) (a) Find a polynomial \(P=P(x)\) of degree 5 such that the function \(F\) defined by $$F(x)=\left\\{\begin{array}{ll}{0} & {\text { if } x \leqslant 0} \\ {P(x)} & {\text { if } 0

Step-by-Step Solution

Answer

The polynomial is \( P(x) = 6x^5 - 15x^4 + 10x^3 \), ensuring smooth transitions.

Step 1: Define the Polynomial Requirements

The polynomial \( P(x) \) must satisfy the following conditions for continuity and smoothness:1. \( P(0) = 0 \) for continuity at \( x = 0 \).2. \( P(1) = 1 \) for continuity at \( x = 1 \).3. \( P'(0) = 0 \) to ensure a smooth slope at \( x = 0 \).4. \( P'(1) = 0 \) to ensure a smooth slope at \( x = 1 \).5. \( P''(0) = 0 \) for continuous curvature at \( x = 0 \).6. \( P''(1) = 0 \) for continuous curvature at \( x = 1 \).

Step 2: General Form of the Polynomial

Since the polynomial \( P(x) \) is of degree 5, it can be expressed as:\[ P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \]We have 6 conditions to satisfy, which means we need 6 equations. Each of the above conditions will provide one equation.

Step 3: Apply Conditions to Form Equations

Using the conditions:1. Set \( P(0) = 0 \), we get: \[ f = 0 \]2. Set \( P(1) = 1 \), we get: \[ a + b + c + d + e + f = 1 \]3. Set \( P'(0) = 0 \), giving: \[ e = 0 \]4. Set \( P'(1) = 0 \), giving: \[ 5a + 4b + 3c + 2d + e = 0 \]5. Set \( P''(0) = 0 \), resulting in: \[ d = 0 \]6. Set \( P''(1) = 0 \), we get: \[ 20a + 12b + 6c + 2d = 0 \]

Step 4: Solve the Equations

With the simplified system of equations, solve for the coefficients \( a, b, c, d, e, f \):1. \( f = 0 \)2. \( e = 0 \)3. \( d = 0 \)Substitute these into our equations:\[ a + b + c = 1 \]\[ 5a + 4b + 3c = 0 \]\[ 20a + 12b + 6c = 0 \]Solve these:- From \( 20a + 12b + 6c = 0 \), simplifying gives \( 10a + 6b + 3c = 0 \).- Subtract this equation from \( 5a + 4b + 3c = 0 \), get \( 5a + 2b = 0 \), so \( b = -\frac{5}{2}a \).- Substitute in \( a + b + c = 1 \), gives \( a - \frac{5}{2}a + c = 1 \) which simplifies to \( -\frac{3}{2}a + c = 1 \).- After solving, we find: \[ a = 6, \ b = -15, \ c = 10 \]

Step 5: Verify the Polynomial

Substitute the values back to verify:\[ P(x) = 6x^5 - 15x^4 + 10x^3 \]Check each condition:- \( P(0) = 0 \)- \( P(1) = 1 \)- \( P'(x) = 30x^4 - 60x^3 + 30x^2; P'(0) = 0, P'(1) = 0 \)- \( P''(x) = 120x^3 - 180x^2 + 60x; P''(0) = 0, P''(1) = 0 \)

Step 6: Graphical Representation of F(x)

Use a graphing calculator or software to plot \( F(x) \). - For \( x \leqslant 0 \): plot the line \( y=0 \).- For \( 0 < x < 1 \): plot \( y = P(x) = 6x^5 - 15x^4 + 10x^3 \).- For \( x \geqslant 1 \): plot the line \( y=1 \).Ensure all segments align smoothly as calculated.

Key Concepts

Continuity in Calculus

Continuity in calculus is an essential concept that ensures no abrupt jumps or breaks occur in a function.
When looking at polynomial functions, such as in the exercise where a railroad track must join smoothly, continuity is vital.
A function is continuous over an interval if there are no breaks, jumps, or holes, making the curve flow without interruptions.

To achieve continuity in this scenario:

• The polynomial function must have no breaks within its defined pieces, matching perfectly at the boundaries.
• For the function \( F(x) \) in the problem, it must be continuous at \( x = 0 \) and \( x = 1 \). This means that as \( x \) approaches these points from either side, the function's value does not change suddenly.
• The conditions such as \( P(0) = 0 \) and \( P(1) = 1 \) ensure that the endpoints connect seamlessly.

Ensuring continuity is the first step in creating functions that represent real-world actions, like the smooth journey of a train along its tracks.

Smooth Transition in Curves

To avoid sudden changes in direction when polynomial functions describe real-world paths, smooth transitions in curves are crucial.
In this context, a smooth transition indicates that not only does the curve meet at designated points, but it also does so with consistent slope and curvature.

Considerations for smooth transitions include:

• Continuity of the slope means that the rate at which the function's value changes is steady and doesn't cause abrupt directional changes.
• In the solution, conditions like \( P'(0) = 0 \) and \( P'(1) = 0 \) ensure the slope is level when transitioning at the boundaries, thus there are no sharp angles.
• Additionally, ensuring continuous curvature with \( P''(0) = 0 \) and \( P''(1) = 0 \) provides that the bend or curve transitions smoothly, avoiding sharp turns.

This smoothness is akin to a roller coaster that transitions gently between heights, providing comfort and safety.

Degree of a Polynomial

The degree of a polynomial is a key characteristic that determines the polynomial's shape and complexity.
It indicates the highest power of the variable in the polynomial equation.
In our example, the polynomial is of degree 5, which gives it five roots and sufficient flexibility to meet all continuity and smooth transition requirements.

Importance of Polynomial Degree:

• A degree 5 polynomial can bend and flex sufficiently to create the needed shape as seen in \( P(x) = 6x^5 - 15x^4 + 10x^3 \).
• The degree dictates how a polynomial behaves at the ends of its graph. Higher degree polynomials can better match complex real-world applications, like smooth railroad transitions.
• Such polynomials are powerful tools because they allow for smooth joining of differing endpoints due to their inherent flexibility.

The higher degree gives the polynomial a greater ability to meet many constraints while still maintaining a smooth and continuous transition.