Problem 67 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Find a curve given by a polynominal \(P_{5}(x)\) that provides a smooth transition between two horizontal lines. That is, assume a function of the form \(P_{5}(x)=a_{0}+a_{1} x+a_{2} x^{2}+\) \(a_{3} x^{3}+a_{4} x^{4}+a_{5} x^{5}\), which provides a smooth transition between \(y=0\) for \(x \leq 0\) and \(y=1\) for \(x \geq 1\) in such a way that the function, its derivative, and curvature are all continuous for all values of \(x\). $$ y=\left\\{\begin{array}{ll} 0 & \text { if } \quad x \leq 0 \\ P_{5}(x) & \text { if } \quad 0

Step-by-Step Solution

Answer

Solve the system using boundary and continuity conditions to get coefficients.

Step 1: Establish continuity at endpoints

To ensure continuity at the endpoints, set the polynomial equal to the endpoint values. Thus, solve for the unknowns with these conditions: \(P_{5}(0) = 0\) and \(P_{5}(1) = 1\).

Step 2: Define the function and its derivative

We define the polynomial and its first derivative to create continuity. Start with the polynomial: \[P_{5}(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5\]Differentiate to find:\[P_{5}'(x) = a_1 + 2a_2 x + 3a_3 x^2 + 4a_4 x^3 + 5a_5 x^4\]

Step 3: Establish derivative continuity at endpoints

Ensure the first derivative \(P_{5}'(x)\) is zero at the endpoints (\(x=0\) and \(x=1\)):For \(x = 0\), set \(P_{5}'(0) = 0\).For \(x = 1\), set \(P_{5}'(1) = 0\).

Step 4: Define and establish curvature continuity

The second derivative expresses curvature. Compute it as:\[P_{5}''(x) = 2a_2 + 6a_3 x + 12a_4 x^2 + 20a_5 x^3\]Ensure continuity by setting \(P_{5}''(0) = 0\) and \(P_{5}''(1) = 0\).

Step 5: Solve the system of equations

Formulate and solve the system of equations derived from the conditions in previous steps. This will yield the values of \(a_0, a_1, a_2, a_3, a_4,\) and \(a_5\).

Step 6: Verify solution

Check that the solution satisfies all initial conditions for continuity of the function, first derivative, and curvature across the interval \(x = 0\) and \(x = 1\).

Key Concepts

Smooth transition with polynomials

Creating a smooth transition between two horizontal lines involves crafting a polynomial function that seamlessly connects the lines at specific points. In this case, we're looking for smooth transitions at the endpoints where the polynomials meet these lines. This concept involves ensuring that the polynomial function does not have any sudden jumps or kinks, making the change appear steady and gradual.

• Start with defining the polynomial, ensuring it meets the boundary conditions.
• Check endpoint values, ensuring that the polynomial's value at these points aligns with the horizontal lines.
• Avoid sharp edges or discontinuities in the graph.

Polynomial functions like the one we're considering should satisfy equations at given endpoints. Specifically, at all points from start to finish, the smoothness or continuity means there should be no abrupt changes in the curve as you move along the graph.

Derivative continuity

Derivative continuity ensures that the change rate of our main function is consistent at the endpoints. In mathematical terms, this means that the first derivative of the polynomial must match at these key points. When computing the first derivative, it is vital that at point zero and point one, the derivatives also equate to zero. Simple checks involve:

• Taking the first derivative of the polynomial, denoting how fast the function's value changes.
• Setting the derived function equal to zero at the boundary points.
• Solving these equations to maintain derivative continuity.

These actions help ensure that as the curve sits at each end, there are no "sharp edges," which can occur if the rate of change doesn't match up smoothly from one section of the graph to the next. This process maintains a gradual change, supporting the overall goal of smooth transition.

Curvature continuity

Curvature continuity takes our tasks a step further, demanding that not only the function and its first derivative are smooth, but so is the second derivative. The second derivative gives us insight into the curve's bend or concave quality and tells us about the change within the change.

• The second derivative is computed from the first derivative and describes how the rate of change itself changes.
• You'll need to solve conditions like ensuring this derivative is zero at specific points: typically the same boundary points.
• This step guarantees no unexpected twists or turns in our transition.

Maintaining curvature continuity assures the entire polynomial provides a consistent, sleek path from one horizontal direction to the other without any unexpected deviations in its smoothness. These calculations ultimately contribute to creating a beautiful, seamless bridge connecting our two horizontal lines.