Problem 60 - 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 $$F(x)=\left\\{\begin{array}{ll}{0} & {\text { if } x \leqslant 0} \\ {P(x)} & {\text { if } 0< x <1} \\ {1} & {\text { if } x \geqslant 1}\end{array}\right.$$ is continuous and has continuous slope and continuous curvature. (b) Use a graphing calculator or computer to draw the graph of \(F .\)

Step-by-Step Solution

Answer

The polynomial of degree 5 is \( P(x) = -6x^5 + 15x^4 - 10x^3 \).

Step 1: Understanding Requirements for Continuity

To ensure continuity of the function, we need \(F(x)\) to be continuous at points where sections meet: at \(x=0\), \(x=1\), and across the polynomial \(P(x)\). This means: \(\lim_{{x \to 0^+}} P(x) = 0\), and \(\lim_{{x \to 1^-}} P(x) = 1\), plus actual values at endpoints.

Step 2: Conditions for Continuous Slope

The slope of \(F(x)\), which is its first derivative, must also be continuous. This implies that \(P'(0) = 0\) and \(P'(1) = 0\) for smooth joining at \(x=0\) and \(x=1\).

Step 3: Conditions for Continuous Curvature

Continuous curvature means the second derivative \(P''(x)\) should be continuous as well. Thus, \(P''(0) = 0\) and \(P''(1) = 0\).

Step 4: Construct Polynomial

Since \(P(x)\) is degree 5, we write it as \(P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f\). Using conditions from previous steps, we derive equations for these coefficients.

Step 5: Applying Boundary Conditions

Using conditions at \(x=0\), we have: \(f=0\), \(e=0\), and \(d=0\). From \(x=1\): \(a+b+c+d+e+f=1\). Differentiating \(P(x)\) and using slope conditions at \(x=0\) and \(x=1\), solve for the coefficients.

Step 6: Solve the System of Equations

We construct a system of linear equations from boundary conditions and derivatives: 1. \(a+b+c+d+e=1\)2. \(5a+4b+3c+2d=-2\)3. \(20a+12b+6c=0\). These provide enough information to solve for \(a,b,c,d,e\).

Step 7: Final Polynomial and Check

Solve equations to find each coefficient. Verify results meet all conditions. Suppose: \(a=-6, b=15, c=-10, d=0, e=0, f=0\). Thus, \(P(x) = -6x^5 + 15x^4 - 10x^3 \). Check each point and derivative condition for correctness.

Step 8: Graphing Function

Utilize a graphing calculator or software like Desmos or GeoGebra to plot \(F(x)\). Inspect the plot to verify the function transitions smoothly between sections as designed.

Key Concepts

Polynomial Functions

Polynomial functions are algebraic expressions made up of terms consisting of a variable raised to an integer power and multiplied by a coefficient. In the given exercise, we are dealing with a polynomial of degree 5, which means the highest power of the variable is 5. These functions are continuous everywhere on their domain because they don't have gaps or asymptotes.

A degree 5 polynomial can be expressed generally as \( P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f \). Each of the coefficients \(a, b, c, d, e, \) and \(f\) is crucial in determining the function's behavior.

• The coefficients dictate the shape and direction of the graph.
• The polynomial's leading term (with the highest power) impacts the end behavior.

In this task, the polynomial needs specific coefficients for smooth transitions, requiring careful calculation using the conditions of continuity, slope, and curvature.

Derivatives

Derivatives represent the rate of change of a function and are critical in understanding how a function behaves. The first derivative, \(P'(x)\), gives the slope of the tangent to the function at any point, telling us how fast the function's value is changing. For our polynomial, ensuring that the derivative is continuous is key to smoothing transitions between track segments.

When we take the second derivative, \(P''(x)\), it tells us about the curvature of the function or how the slope is changing. Smooth curvature indicates the absence of sharp bends or jumps in the track.

• A continuous first derivative ensures that there's no abrupt change in direction.
• A continuous second derivative guarantees that curvature changes are gradual.

Meeting these conditions required that we derive and check certain values, such as \(P'(0) = 0\) and \(P'(1) = 0\), ensuring a seamless blend at both ends of the track.

Graphing Functions

Graphing functions is a fundamental technique to visualize mathematical relationships. It's especially useful in confirming the correctness of a constructed polynomial like \(F(x)\). Once we have derived the polynomial, we use graphing to verify its smoothness at the specified transition points.

Using graphing tools, such as a graphing calculator or software like GeoGebra, the function's behavior over specified intervals can be inspected.

• Graphing reveals visual evidence of continuity at \(x=0\) and \(x=1\).
• It allows you to see if the function's shape aligns with expectations based on calculated slopes and curvatures.

Effective graphing will show no visible disruptions in the line at the transitions, validating your work precisely aligns with design constraints.

Curvature

Curvature in calculus refers to how quickly a curve changes direction. In this exercise, it's important to ensure the track's curvature changes smoothly to avoid mechanical stresses on the railroad.

The polynomial's second derivative, \(P''(x)\), captures this curvature. When graphing \(P(x)\), continuous curvature is indicated by a smooth, bend-free curve meaning no sharp turns. We emphasize the need for \(P''(0) = 0\) and \(P''(1) = 0\) to ensure the curvature doesn't jump between sections of the track.

• A small curvature indicates a gentle slope change, while larger values indicate sharp turns.
• Ensuring smooth curvature means keeping the transition seamless, aligning with physical requirements for track safety.

By designing a track with calculated continuous curvature, we make sure trains stay safely aligned with the track line at all speeds.