Problem 23 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Introduction to Error Analysis: Let \(f(x)=e^{x}\) and let \(P_{k}(x)\) be its \(k\) th degree Taylor polynomial about \(x=0\). Graph \(R_{k}(x)=f(x)-P_{k}(x)\) for \(k=1,2, \ldots, 5 .\)

Step-by-Step Solution

Answer

The graphs of the residual functions \(R_k(x)\) show that as \(k\) increases from 1 to 5, the difference between \(f(x) = e^x\) and its Taylor polynomial approximation decreases. This demonstrates that a higher degree Taylor polynomial provides a more accurate approximation of the function \(f(x) = e^x\).

Step 1: Compute Taylor Polynomials

Start by calculating the Taylor polynomial of degree \(k\) for the function \(f(x)=e^x\) around \(x=0\) using the formula: \[ P_{k}(x) = \sum_{n=0}^{k} \frac{f^{(n)}(0) \cdot x^n}{n!} \] where \( f^{(n)}(0) \) is the \(n^{th}\) derivative of \(f\) evaluated at 0. Since the derivative of \( f(x) = e^x \) is \( f'(x) = e^x \), and \( f^{(n)}(0) = e^0=1 \) for all \(n\) => \[ P_k(x) = \sum_{n=0}^{k} \frac{x^n}{n!} \]

Step 2: Compute Residual Functions

Calculate the residual (error) function \( R_k(x) = f(x) - P_k(x) \) for each \( k \) from 1 to 5. This represents the difference between the original function \( f(x) \) and its Taylor polynomial approximation \( P_k(x) \).

Step 3: Graph the Residual Functions

Now, we graph each \( R_k(x) \) on the same set of axes. This will illustrate how the difference between the actual function and its Taylor polynomial approximation decreases as \(k\) increases from 1 to 5.

Key Concepts

Taylor Polynomial

When you're tackling higher-level calculus, one of the quintessential tools at your disposal is the Taylor polynomial. Think of a Taylor polynomial as an approximation of a more complicated function, like a stunt double in a movie that resembles the actor closely enough for specific scenes. For the function f(x) = e^x, which is our protagonist, its Taylor polynomial P_k(x) stands in for the function, but only near a specific point—in our case, x=0.

Creating this mathematical stunt double involves using derivatives of f(x) at that point to build the polynomial's terms. Every additional term in the Taylor polynomial brings it closer to the original function, much like adding more makeup to make a stunt double look more like the leading actor. The formula contains these terms summed up, where the powers of x grow and each is divided by the factorial of its power to keep the approximation in balance.

Residual Function

Now, how do we measure the difference between the real deal and our stunt double? Enter the residual function, R_k(x), which is essentially the 'error' of the approximation. By subtracting the Taylor polynomial P_k(x) from our actual function f(x), we're left with R_k(x). This tells us how much the original and the approximation differ.

To understand it better, imagine you're measuring how much a rubber band stretches compared to its original length. The residual function serves the same purpose—it stretches and shrinks as you go further away or closer to the point of approximation, respectively. It's crucial to graph R_k(x) for different values of k because it visually demonstrates how the Taylor polynomial becomes a better stand-in for the function with each additional term.

Derivative of Exponential Function

The exponential function, f(x) = e^x, has a unique and fascinating property—its derivative with respect to x is itself! That means if you're asked to differentiate this function once, twice, or even a dozen times, you'll always get e^x again.

When crafting a Taylor polynomial, these derivatives evaluated at x=0 give us the coefficients of the polynomial. Since the value of e^0 is always 1, this makes computations neater. The consistency and pattern of the derivatives make the exponential function a textbook example for demonstrating Taylor polynomials—it's like a clear, predictable rhythm in music, making it easier for students to follow along and grasp the underlying principles of Taylor series expansion.

Factorial Notation

Factorials are the mathematical equivalent of multi-level tower stacking games—the fun is in seeing how high you can go. In mathematical terms, a factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. For example, 3! = 3 × 2 × 1 = 6.

In the Taylor polynomial formula, factorials play an essential role in balancing the magnitude of each term. As the power of x increases, dividing by a factorial ensures that the terms don't explode in value and keeps the approximation grounded. Understanding factorial notation is key to recognizing why each term in the Taylor polynomial has the influence it does, much like understanding each ingredient's role in a recipe influences the final dish.