Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot on the same axes the given function along with the Maclaurin polynomials of orders \(1,2,3\), and \(4 .\) $$ e^{-x^{2}} $$

Step-by-Step Solution

Answer

The function and its Maclaurin polynomials of orders 1, 2, 3, and 4 are plotted together.

Step 1: Understand the Exercise

We are asked to plot the function \(e^{-x^2}\) along with its Maclaurin polynomials of orders 1, 2, 3, and 4. A Maclaurin series is a Taylor series expansion of a function about 0.

Step 2: Find the Maclaurin Series

The Maclaurin series for a function \(f(x)\) is given by \(\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n\). For \(f(x) = e^{-x^2}\), this will result in a series expansion.

Step 3: Compute the First Few Derivatives

Calculate the necessary derivatives of \(f(x)\). \(f(x) = e^{-x^2}\), \(f'(x) = -2xe^{-x^2}\), \(f''(x) = (4x^2 - 2)e^{-x^2}\). Evaluate these at \(x = 0\) to get the coefficients for the series.

Step 4: Find the Maclaurin Polynomials

Evaluate derivatives at \(x = 0\): - \(f(0) = 1\), - \(f'(0) = 0\), - \(f''(0) = -2\). Calculate the series: - Order 0: \(P_1(x) = 1\) - Order 1: \(P_2(x) = 1\) - Order 2: \(P_2(x) = 1 - x^2\) - Order 3: \(P_3(x) = 1 - x^2\) - Order 4: \(P_4(x) = 1 - x^2 + \frac{x^4}{2}\).

Step 5: Plot the Functions

Using a graphing tool or software, plot \(e^{-x^2}\) and the polynomials \(P_1(x) = 1\), \(P_2(x) = 1\), \(P_3(x) = 1 - x^2\), and \(P_4(x) = 1 - x^2 + \frac{x^4}{2}\) on the same axes for some range, such as \([-2, 2]\). Observe how the polynomials approximate the function.

Key Concepts

Taylor Series Expansion

The Taylor Series Expansion is a key mathematical tool that helps approximate functions using polynomials. By expanding a function into a series of terms based on its derivatives at a single point, we can create a polynomial that closely resembles the original function.
For a function centered at zero, the expansion is known as the Maclaurin series. Given a function \( f(x) \), its Taylor series can be expressed as:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots \)
• When \( a = 0 \), the series simplifies to the Maclaurin series: \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \ldots \)

The beauty of the Taylor Series lies in its ability to approximate any smooth function to any desired degree of accuracy by considering more terms of the expansion. This becomes particularly useful in computational applications, where simple polynomials are preferred for their ease of calculation.

Derivative Evaluation

Derivative Evaluation is critical in computing the Taylor or Maclaurin series. Derivatives provide the coefficients for each term in the polynomial expansion. Calculating these derivatives can sometimes be straightforward, but they can also become quite complex.
The derivatives of a function \( f(x) \) give rate information about how \( f \) changes at a point. In the context of a Maclaurin series, these derivatives are evaluated at \( x = 0 \). For example, if \( f(x) = e^{-x^2} \):

• \( f(0) = 1 \) (function value at zero)
• \( f'(0) = 0 \) (first derivative at zero)
• \( f''(0) = -2 \) (second derivative at zero)

These evaluations form the series terms. With each specific order of derivative providing increasingly nuanced information about the function's behavior, each derivative acts as a building block for constructing a polynomial that approximates the original function's graphical shape.

Polynomial Approximation

Polynomial Approximation is a powerful technique for representing complex functions as sums of simpler polynomial expressions. This allows for smoother calculations and simulations when working with these functions.
Consider a function like \( e^{-x^2} \), which can be difficult to graph and compute exactly. By approximating it with a Maclaurin polynomial, you use a simpler polynomial function that behaves similarly near \( x = 0 \):

• Order 0: \( P_1(x) = 1 \)
• Order 1: \( P_2(x) = 1 \)
• Order 2: \( P_3(x) = 1 - x^2 \)
• Order 4: \( P_4(x) = 1 - x^2 + \frac{x^4}{2} \)

The degree of the polynomial corresponds to how many terms from the series are used, and the accuracy of the approximation improves with higher orders. Each additional term in these polynomials corrects and fine-tunes the approximation to match the behavior of the original function, especially near the expansion point.

Graphing Polynomials

Graphing Polynomials, particularly in the context of Taylor or Maclaurin series, offers a visual understanding of how well a series can approximate a function. By plotting the original function alongside its polynomial approximations, you can compare how closely the polynomials match the function's behavior.
For instance, to approximate \( e^{-x^2} \) using Maclaurin polynomials of varying degrees (like orders 1, 2, 3, and 4), one would plot each polynomial approximation of \( e^{-x^2} \) and set them against the actual curve of \( e^{-x^2} \). This comparison helps observe key characteristics:

• The fit is better near \( x = 0 \), the center of the series expansion.
• Higher order polynomials provide more accurate approximations over a larger area.
• The approximation tends to diverge from the original function as you move further from zero.

Through visual analysis, it becomes apparent how the polynomial's accuracy increases with its degree, making graphing an invaluable educational tool in understanding polynomial approximations and series expansion concepts.