Problem 21 - 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 \(4 .\) $$ \sin x^{2} $$

Step-by-Step Solution

Answer

Plot sin(x^2) and polynomials P1(x)=0, P2(x)=x^2, P3(x)=x^2, P4(x)=x^2-x^4/2.

Step 1: Understand the Function and Maclaurin Series

The function we need to analyze is \( \sin(x^2) \). The Maclaurin series is a way to approximate functions using polynomials centered at \( x = 0 \). The general form of a Maclaurin series for a function \( f(x) \) is given by \( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \). In this exercise, we want to approximate \( \sin(x^2) \) using Maclaurin polynomials of different orders (1, 2, 3, and 4).

Step 2: Find the Derivatives at x = 0

Calculate the derivatives necessary for constructing the Maclaurin polynomials:- **First derivative**: \( f'(x) = 2x \cos(x^2), \; f'(0) = 0 \)- **Second derivative**: \( f''(x) = 2\cos(x^2) - 4x^2 \sin(x^2), \; f''(0) = 2 \)- **Third derivative**: \( f'''(x) = -12x \sin(x^2) - 8x^3 \cos(x^2), \; f'''(0) = 0 \)- **Fourth derivative**: \( f^{(4)}(x) = -12\sin(x^2) - 64x^2 \cos(x^2), \; f^{(4)}(0) = -12 \)

Step 3: Construct the Maclaurin Polynomials

Using the derivatives, construct the Maclaurin polynomials:- **Order 1 Polynomial**: \( P_1(x) = 0 \)- **Order 2 Polynomial**: \( P_2(x) = 0 + \frac{2}{2!} x^2 = x^2 \)- **Order 3 Polynomial**: \( P_3(x) = x^2 \)- **Order 4 Polynomial**: \( P_4(x) = x^2 + \frac{-12}{4!} x^4 = x^2 - \frac{x^4}{2} \)

Step 4: Plot the Function and Polynomials

Plot the original function \( \sin(x^2) \) and the Maclaurin polynomials on the same axes. Use different colors or styles for each to distinguish them. - Function: \( \sin(x^2) \)- \( P_1(x) \): Constant at zero line - \( P_2(x) \): Parabola \( x^2 \)- \( P_3(x) \): Matches \( P_2(x) \) as no new terms are added- \( P_4(x) \): Parabola with a slight correction term, \( x^2 - \frac{x^4}{2} \)

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables raised to non-negative integer powers and coefficients. They can represent a broad range of mathematical behaviors, making them highly versatile in calculations and analysis.
Polynomials appear in many forms and degrees:

• Linear polynomial: A polynomial of degree 1, e.g., \(a_0 + a_1x\).
• Quadratic polynomial: A polynomial of degree 2, e.g., \(a_0 + a_1x + a_2x^2\).
• Cubic polynomial: A polynomial of degree 3, e.g., \(a_0 + a_1x + a_2x^2 + a_3x^3\).

With polynomials, we can easily perform arithmetic operations, differentiate, and integrate, making them ideal for approximating more complicated functions like \( \sin(x^2) \). All Maclaurin polynomials derived from a function are polynomials tailored to match a specific degree of accuracy at \(x=0\). Each polynomial provides a step closer to the actual behavior of the function with higher-degree polynomials offering more precise approximations.

Derivatives

Derivatives form the backbone of calculus, measuring how a function changes as its input changes. The concept of a derivative is central when constructing polynomial approximations like the Maclaurin series.
When approximating a function using the Maclaurin series, we take successive derivatives of the function:

• The first derivative tells us the slope or rate of change at any point on the function.
• The second derivative gives us the rate of change of the rate of change, indicating concavity.
• Higher derivatives continue this trend, providing additional detail on how the function behaves.

The calculation of derivatives at \(x=0\) allows us to determine the coefficients of each term in the Maclaurin polynomial. For a function like \( \sin(x^2) \), the derivatives give insight into the behavior around \(x=0\) and enable us to calculate the Maclaurin polynomials of various orders.

Function Approximation

Function approximation is the process of representing complex functions with simpler ones. This approach makes mathematical analysis more manageable and forms a crucial aspect of numerical methods and engineering practices.
The Maclaurin series is a common technique for function approximation around \(x=0\).

• A series of polynomials sums up to approximate a function.
• Low-order polynomials provide rough estimates, while higher-order polynomials yield closer approximations.

The essence of Maclaurin series is to match the value, slope, curvature, and higher level natures of the function near \(x = 0\). For \( \sin(x^2) \), the Maclaurin polynomials aim to mirror the function’s behavior by incorporating derivatives at the origin, allowing us to substitute a complex trigonometric function with a series of manageable polynomial expressions. This approach simplifies understanding, computation, and graphical representation, offering a clear idea of how \( \sin(x^2) \) behaves near \(x = 0\).