Problem 9 - Essential Calculus

Question

Use the definition to find the Taylor series (centered at \(c\) ) for the function. $$ f(x)=\sec x, \quad c=0 \text { (first three nonzero terms) } $$

Step-by-Step Solution

Answer

The Taylor series for the function \(f(x) = \sec x\) centered at \(c=0\) for the first three nonzero terms is \(1 + x^2/2\).

Step 1: Calculate the function at the center point

Firstly, calculate the value of \(f(x)\) at the center point \(c=0\). Since \(f(x) = \sec x\), when \(x=0\) it equals 1, i.e. \(f(0) = 1\).

Step 2: Calculate first three derivatives

The next step is to calculate the first three derivatives of the function, and evaluate these at \(c = 0\). The first derivative of \(f(x) = \sec x\) is \(f'(x) = \sec(x)\tan(x)\), substituting \(x = 0\), \(f'(0) = 0\). The second derivative \(f''(x) = \sec(x)\tan^2(x) + \sec^3(x)\), for \(x = 0\), \(f''(0) = 0 + 1 = 1\). The third derivative \(f'''(x) = 2\sec(x)\tan(x)\tan^2(x) + 5\sec^3(x)\tan(x)\), when \(x = 0\), \(f'''(0) = 0\).

Step 3: Build the Taylor series

The Taylor series is built up according to the formula: \(f(c) + f'(c)(x-c)/1! + f''(c)(x-c)^2/2! + f'''(c)(x-c)^3/3!\). Substituting the values from the previous steps and keeping only non-zero terms, the Taylor series becomes: \(1 + 0 + 1*x^2/2! + 0 = 1 + x^2/2\)

Key Concepts

Taylor Series

The Taylor series is a powerful tool in calculus used to approximate functions that might otherwise be very complex or difficult to analyze. By using the Taylor series, a function can be expressed as an infinite sum of terms. Each of these terms consists of the function's derivatives evaluated at a certain point, known as the center.
These series allow you to approximate the behavior of functions near this center point. This can be particularly useful for functions like trigonometric or exponential functions that might not have a simple polynomial form. For a given function \( f(x) \) centered at \( c \), the Taylor series is given by:

• \( f(c) + f'(c)(x-c)/1! + f''(c)(x-c)^2/2! + \ldots \)

Using this series, you can approximate the function to the desired degree of accuracy by deciding how many terms to include. The more terms you use, the closer the approximation will be to the actual function.

Maclaurin Series

The Maclaurin series is a specific type of Taylor series that centers around \( c = 0 \). This simplification allows us to approximate functions near zero. The Maclaurin series is particularly useful when studying functions that have symmetrical properties around zero. The expression for the Maclaurin series is:

• \( f(0) + f'(0)x/1! + f''(0)x^2/2! + \ldots \)

It's a great choice when you need an approximation around zero because it simplifies calculations by eliminating the need to subtract \( c \) from \( x \). In our original exercise, \( f(x) = \sec x \) was expanded using a Maclaurin series.

Function Derivatives

Derivatives are fundamental in calculus as they help find the rate at which a function is changing at any given point. They are essential in Taylor and Maclaurin series because these series use derivatives to build polynomial approximations of the original function.

• The first derivative represents how the function changes, providing the linear approximation at the point.
• The second derivative gives insight into the curvature of the function, helping refine the approximation.
• Higher-order derivatives further enhance the accuracy of the approximation by incorporating more information on the function's behavior.

In the solution of our exercise, we calculated and used the first three derivatives of \( f(x) = \sec(x) \) at \( x = 0 \) to construct the series.

Power Series Expansion

A power series is an infinite series composed of terms that are powers of a variable, often represented as \( \sum_{n=0}^{\infty} a_n (x-c)^n \). Each term in this series represents a part of the function's behavior near the center \( c \).

• The coefficients \( a_n \) usually are derived from the function's derivatives at the center.
• The power series encompasses both Taylor and Maclaurin series as specific cases with specific centers.

The process of expanding a function into a power series can make tough problems much easier to tackle, whether for analyzing limits, solving differential equations, or even simplifying calculations in physics and engineering. In the exercise, the function \( \sec x \) was expanded into a power series around \( c = 0 \), converting it into a usable polynomial form.