Question

Plot a smoothed "hat" function. The "hat" function \(N(x)\) defined by (3.5) on page 109 has a discontinuity in the derivative at \(x=1\). Suppose we want to "round" this function such that it looks smooth around \(x=1\). To this end, replace the straight lines in the vicinity of \(x=1\) by a (small) cubic curve $$ y=a(x-1)^{3}+b(x-1)^{2}+c(x-1)+d $$ for \(x \in[1-\epsilon, 1+\epsilon]\), where \(a, b, c\), and \(d\) are parameters that must be adjusted in order for the cubic curve to match the value and the derivative of the function \(N(x)\). The new rounded functions has the specification $$ \tilde{N}(x)= \begin{cases}0, & x<0 \\ x, & 0 \leq x<1-\epsilon \\\ a_{1}(x-1)^{3}+b(x-1)+c(x-1)+d_{1}, & 1-\epsilon \leq x<1 \\\ a_{2}(x-1)^{3}+b(x-1)+c(x-1)+d_{2}, & 1 \leq x<1+\epsilon \\ 2-x, & 1+\epsilon \leq x<2 \\ 0, & x \geq 2\end{cases} $$ with \(a_{1}=\frac{1}{3} \epsilon^{-2}, a_{2}=-a_{1}, d_{1}=1-\epsilon+a_{1} \epsilon^{3}, d_{2}=1-\epsilon-a_{2} \epsilon^{3}\), and \(b=c=0 .\) Plot this function. (Hint: Be careful with the choice of \(x\) coordinates!) Name of program file: plot_hat.py.

Short answer

Use a cubic curve to smooth transitions at \( x = 1 \) with a defined piecewise function \( \tilde{N}(x) \). Plot using a dense \( x \) set around \( x = 1 \).

Step-by-step solution

Understand the Function Structure

The goal is to modify the hat function \( N(x) \) to ensure it is smooth around \( x = 1 \) by introducing a cubic curve. The primary function \( N(x) \) is piecewise and is divided into linear segments, which need smoothing between specific intervals \( [1-\epsilon, 1) \) and \( [1, 1+\epsilon) \).

Define the Cubic Polynomial

We use the cubic polynomial \( y = a(x-1)^3 + b(x-1)^2 + c(x-1) + d \) to transition between linear sections. The coefficients are configured as: \( a_1 = \frac{1}{3} \epsilon^{-2} \), \( a_2 = -a_1 \), \( d_1 = 1 - \epsilon + a_1 \epsilon^3 \), \( d_2 = 1 - \epsilon - a_2 \epsilon^3 \), and \( b = c = 0 \).

Calculate Key Points

Choose a small \( \epsilon \) ensuring the function smoothly transitions at \( x=1 \). Compute \( a_1, a_2, d_1, \) and \( d_2 \) using this \( \epsilon \). The value \( \epsilon \) dictates how close the cubic curve approaches the linear sections around \( x = 1 \).

Piecewise Function Specification

The smooth version \( \tilde{N}(x) \) is defined piecewise:- \( \tilde{N}(x) = 0 \), for \( x < 0 \)- \( \tilde{N}(x) = x \), for \( 0 \leq x < 1 - \epsilon \)- \( \tilde{N}(x) = a_1(x-1)^3 + d_1 \), for \( 1-\epsilon \leq x < 1 \)- \( \tilde{N}(x) = a_2(x-1)^3 + d_2 \), for \( 1 \leq x < 1+\epsilon \)- \( \tilde{N}(x) = 2-x \), for \( 1+\epsilon \leq x < 2 \)- \( \tilde{N}(x) = 0 \), for \( x \geq 2 \)

Plot the Function

Use a software tool like Python with the library Matplotlib for plotting. Carefully choose \( x \) values from \( -0.5 \) to \( 2.5 \) to capture the entire function, especially focusing on regions around \( x=1-\epsilon \) and \( x=1+\epsilon \) to visualize the rounding effect. Maintain smoothness in the graph by using a dense set of \( x \) values around \( x=1 \).

Python Implementation

Create a Python file named 'plot_hat.py'. Import necessary libraries, define the function \( \tilde{N}(x) \) using if-elif-else constructs, plot using Matplotlib, and label the graph appropriately. Ensure to test with different \( \epsilon \) values to see the smoothing effect.

Key concepts

Piecewise Functions

Piecewise functions are a fascinating type of function used in mathematics, defined by multiple sub-functions, each applying to a certain interval in the domain. In this context, a piecewise function allows us to craft different behaviors and properties, all integrated into one overarching function. This kind of function is incredibly useful in situations where a single expression is insufficient to describe the entire behavior over its domain.

• For the 'hat' function, we define different mathematical expressions for various ranges of its independent variable, which is typically denoted by \(x\).
• A common example includes simple linear equations in separate intervals to model different stages of a process.

The ability to define functions in pieces is crucial when addressing problems involving abrupt changes, like discontinuities or non-smooth transitions in derivative values. By understanding how each piece contributes to the whole, you can better manage and predict function behavior within its specific intervals.

Cubic Polynomial

Cubic polynomials polynomial are expressions of the form \(y = ax^3 + bx^2 + cx + d\) which incorporate a cubic term, making them highly versatile in modeling smooth transitions within a function.

• They are particularly suited for smoothing because they provide flexible curves, unlike linear or quadratic expressions.
• In smoothing applications, like our 'hat' function, cubic polynomials help round off sharp edges and create continuous transitions between otherwise disjoint function pieces.

In the 'hat' function problem, the cubic polynomial is used to smooth out the sharpness at \(x=1\), ensuring the function and its derivative transition seamlessly across joined intervals. Choosing the right coefficients \(a, b, c,\) and \(d\) is key, as they control the exact shape and position of the cubic curve.
Cubic polynomials are a staple in graphics programming, simulations, and wherever smooth physical or abstract flows are critical.

Function Plotting

Plotting functions visually represent mathematical functions to provide insight into their behavior and characteristics. It is an essential tool in mathematics and science for understanding function properties like continuity, differentiability, and smoothness.

• Using software like Python's Matplotlib, plotting a function involves defining the domain, calculating corresponding range values, and displaying these points on a grid.
• It helps visualize complex functions, confirming analytical solutions, or spotting errors in calculations.

For example, in our 'hat' function exercise, attention should be given to the transitions around \(x = 1\). This ensures the plot accurately presents how cubic smoothing transitions across the domains. A dense selection of \(x\) values ensures the graph reflects even the subtleties in curvature, providing a profound understanding of function smoothness. Plotting isn't just about drawing graphs; it's about clearly conveying function characteristics to support further analysis and insights.

Smoothing Techniques

Smoothing techniques refer to methods used to remove noise or sharp changes in data or functions. This is essential in creating more easily interpreted data, minimizing disruptions while retaining key trends and behaviors.

• In mathematical functions, smoothing often involves using polynomials, such as cubic polynomials, to alleviate abrupt changes or oscillations in the function or its derivatives.
• In our 'hat' function, the specific task was to smooth around \(x=1\) by replacing abrupt linear transitions with a cubic curve, achieving a smooth and continuous form.

These techniques are widely used in signal processing, data visualization, and computational sciences to ensure that the datasets or function descriptions still convey underlying trends despite potential noise or approximations. The art of smoothing is finding the balance between reducing noise and maintaining accuracy, ensuring that smoothing enhances clarity without misleading pesudo-trends.