Problem 43 - Thomas Calculus

Question

Let \(f(x)\) have derivatives through order \(n\) at \(x=a .\) Show that the Taylor polynomial of order \(n\) and its first \(n\) derivatives have the same values that \(f\) and its first \(n\) derivatives have at \(x=a\) .

Step-by-Step Solution

Answer

The Taylor polynomial and its derivatives match \( f(x) \) and its derivatives at \( x = a \).

Step 1: Understanding the Taylor Polynomial

The Taylor polynomial of order \( n \) for a function \( f(x) \) at \( x = a \) is given by \( P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n \). This polynomial approximates \( f(x) \) near \( x = a \).

Step 2: Evaluating the Taylor Polynomial at x = a

Evaluate \( P_n(a) \), which becomes \( f(a) + f'(a)(a-a) + \frac{f''(a)}{2!}(a-a)^2 + \ldots + \frac{f^{(n)}(a)}{n!}(a-a)^n \). All terms involving \((x-a)\) vanish, leaving \( P_n(a) = f(a) \).

Step 3: Deriving the First Derivative of Taylor Polynomial

Find the first derivative of \( P_n(x) \). For example, if \( P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots + \frac{f^{(n)}(a)}{n!}(x-a)^n \), then \( P_n'(x) = f'(a) + f''(a)(x-a) + \frac{f^{(3)}(a)}{2!}(x-a)^2 + \ldots + \frac{f^{(n)}(a)}{(n-1)!}(x-a)^{n-1} \).

Step 4: Evaluating the First Derivative at x = a

Set \( x = a \) for the first derivative. This gives \( P_n'(a) = f'(a) \) as all terms involving \((x-a)\) again vanish. Thus, the first derivative of the Taylor polynomial at \( x = a \) matches the first derivative of \( f \) at \( x = a \).

Step 5: Generalizing to Higher Derivatives

Similarly, by repeatedly differentiating and setting \( x = a \), we find that each successive derivative \( P_n^{(k)}(a) = f^{(k)}(a) \) for \( k = 2, 3, \ldots, n \), as all terms involving \((x-a)\) vanish after differentiation.

Key Concepts

Derivatives

Derivatives are fundamental in calculus, assisting us in understanding the rate at which a function changes. When we discuss a derivative, commonly denoted as \( f'(x) \), it represents the slope of the tangent line to the curve of \( f(x) \) at a given point. More generally, derivatives can be seen as a tool for assessing how a small change in \( x \) impacts the value of \( f(x) \).
Here are some key points about derivatives:

• They provide a measure of sensitivity in a function's output relative to changing inputs.
• The first derivative gives the slope of the function, while higher-order derivatives can reveal more about the function’s behavior.
• At a point of tangency, the derivative equals the slope of the tangent line, illustrating an instantaneous rate of change.

Given their importance, derivatives are central to techniques like Taylor polynomials, ensuring precise function approximations.

Function Approximation

Function approximation is the process of finding a function that closely represents the behavior of a more complicated or unknown function. Taylor polynomials are a powerful method for function approximation, particularly for smooth functions at a certain point.
Why use Taylor polynomials for approximation?

• They provide a polynomial of finite degree that approximates a function, making calculations easier than using an infinite series.
• These polynomials can be evaluated by knowing the derivatives of the function at a specific point, usually making it computationally simpler.
• Taylor polynomials are highly accurate near the point of expansion, \( x = a \).

By using such approximations, you can analyze a function's behavior locally while avoiding the complexity of handling the full function.

Higher Order Derivatives

Higher order derivatives extend beyond the first derivative, providing deeper insights into a function's behavior. The \( n \)-th derivative of a function is denoted by \( f^{(n)}(x) \).
Here are some points to remember about higher order derivatives:

• They help in understanding a function's curvature and concavity, by analyzing how the slope of the slope changes.
• The second derivative \( f''(x) \) informs us about the acceleration of the function, while the third and subsequent derivatives provide additional layers of analysis.
• In Taylor polynomials, higher order derivatives are crucial for determining the coefficients of the polynomial terms, ensuring the polynomial matches the rate of change to a required order.

This concept makes it possible to approximate functions accurately by aligning all derivatives up to the \( n \)-th order, resulting in a closer mimicking of the function's behavior near \( x = a \).