Problem 19 - Calculus: Early Transcendentals

Question

(a) Approximate \( f \) by a Taylor polynomial with degree \( n \) at the number \( a. \) (b) Use Taylor's Inequality to estimate the accuracy of the approximation \( f(x) \approx T_n(x) \) when \( x \) lies in the given interval. (c) Check you result in part (b) by graphing \( \mid R_n(x) \mid . \) \( f (x) = e^{x^{2}}, \) \( a = 0, \) \( n = 3, \) \( 0 \le x \le 0.1 \)

Step-by-Step Solution

Answer

The Taylor polynomial for \( f(x) = e^{x^2} \) at \( a = 0 \), \( n = 3 \) is \( T_3(x) = 1 + x^2 \), and the error remains below 0.00001 for \( 0 \le x \le 0.1 \).

Step 1: Write the Taylor Polynomial Formula

The Taylor polynomial of degree \( n \) for a function \( f(x) \) around a point \( a \) is given by:\[ T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k \] Here, \( n = 3 \), \( a = 0 \), and we need the polynomial for \( f(x) = e^{x^2} \).

Step 2: Compute the Derivatives

Calculate the first four derivatives of \( f(x) = e^{x^2} \) to obtain:- \( f(x) = e^{x^2} \)- \( f'(x) = 2xe^{x^2} \)- \( f''(x) = (4x^2 + 2)e^{x^2} \)- \( f'''(x) = (8x^3 + 12x)e^{x^2} \).

Step 3: Evaluate Derivatives at \( a = 0 \)

Evaluate the derivatives at \( a = 0 \):- \( f(0) = e^{0} = 1 \)- \( f'(0) = 0 \)- \( f''(0) = 2 \)- \( f'''(0) = 0 \).

Step 4: Formulate the Taylor Polynomial

Using the computed derivatives, the Taylor polynomial is:\[ T_3(x) = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 = 1 + \frac{2}{2} x^2 = 1 + x^2 \].

Step 5: Use Taylor's Inequality for Estimation

Taylor's Inequality states that the remainder \( R_n(x) \) satisfies:\[ |R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1} \]where \( M \) is the maximum of \( |f^{(n+1)}(z)| \) over the interval. Here, \( M \approx 12x \) for \( 0 \leq x \leq 0.1 \). Thus:\[ |R_3(x)| \leq \frac{0.12}{4!}(0.1)^4 \approx 0.00001 \].

Step 6: Graph \( |R_3(x)| \)

Graph the function \( |R_3(x) = f(x) - T_3(x)| \) over the interval \( 0 \leq x \leq 0.1 \). Observe that as expected, \( |R_3(x)| \) remains below the estimated accuracy of approximately 0.00001 over the given interval.

Key Concepts

Taylor's Inequality

Taylor's Inequality is a powerful tool used to estimate how accurate a Taylor polynomial approximation is over a specified interval. This is crucial when you want to understand the difference between the true function and its approximation. In simple terms, the inequality provides a bound on the error, known as the remainder, when using the Taylor polynomial.

• The remainder \( R_n(x) \) indicates the difference between the actual function value and the value predicted by the Taylor polynomial.
• The inequality states that \( |R_n(x)| \) is less than or equal to \( \frac{M}{(n+1)!}|x-a|^{n+1} \), where \( M \) is the maximum value of the \( (n+1) \)-th derivative over the interval.

For our exercise, after calculating the derivatives, we use \( M = 12x \) for \( 0 \leq x \leq 0.1 \). Applying the inequality ensures that \( |R_3(x)| \) remains under approximately 0.00001, confirming our polynomial's accuracy within this small range.

Derivative Evaluation

Derivative evaluation is a core part in constructing Taylor polynomials. It involves calculating the derivatives of a function up to a certain degree, which are then evaluated at a specific point to form the polynomial.

• First, compute the necessary derivatives of the function. For instance, the function \( f(x) = e^{x^2} \) requires you to compute up to the third derivative.
• Next, evaluate these derivatives at the expansion point, here it's \( a = 0 \). This step involves plugging \( 0 \) into each derivative function to get values like \( f(0) = 1 \) and \( f''(0) = 2 \).

These values are critical as they directly influence the coefficients of the Taylor polynomial. By understanding and performing derivative evaluation correctly, you create more accurate polynomial approximations.

Polynomial Approximation

Polynomial approximation refers to representing a function with a polynomial that approximates the function's behavior near a certain point. The Taylor polynomial offers a way to achieve this, making complex functions simpler to work with.

• A Taylor polynomial is constructed using the function's derivatives evaluated at a single point, termed the expansion point.
• The polynomial, \( T_n(x) = 1 + x^2 \), for \( n = 3 \), approximates \( f(x) = e^{x^2} \) well near \( x = 0 \).

This reduction to simpler terms like \( 1 + x^2 \) eases computations significantly. Here, only the first three derivatives significantly impact the polynomial's form, each contributing a term to the approximation. Such polynomials serve as a handy tool in both theoretical and practical numerical calculations.

Remainder Estimation

Remainder estimation deals with quantifying the error between the actual function and its Taylor polynomial approximation. It's essential for understanding how closely a polynomial approximates a function over an interval.

• The remainder \( R_n(x) = f(x) - T_n(x) \) provides insight into the approximation's precision.
• Through Taylor's Inequality, we estimate this remainder to ensure it stays within acceptable bounds.

In the exercise, it was shown that \( |R_3(x)| \) remains under 0.00001 when \( x \) is between 0 and 0.1. Graphing \( |R_3(x)| \) further confirms this, illustrating that our polynomial approximation is highly precise in this narrow interval. Understanding remainder estimation allows you to decide when a Taylor polynomial is sufficient for certain purposes or when higher-order terms are necessary.