Problem 3 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Do the following. (a) Compute the fourth degree Taylor polynomial for \(f(x)\) at \(x=0 .\) (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\). (c) Use \(P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) to approximate \(f(0.1)\) and \(f(0.3) .\) Compare these approximations to those given by a calculator. $$ f(x)=\tan ^{-1} x $$

Step-by-Step Solution

Answer

The fourth degree Taylor polynomial for \(f(x) = tan^{-1}x\) at \(x=0\) is \(P_{4}(x) = x - x^3/3\). It, along with the function itself and the first three Taylor polynomials, can be plotted on a graph. The function \(f(x) = tan^{-1}x\) can be approximated at \(x = 0.1\) and \(x = 0.3\) by substituting these values into the Taylor polynomials.

Step 1: Taylor polynomial

The formula for the Taylor series expansion of a function \(f(x)\) about a point \(a\) is given by: \[\(f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + f''''(a)(x-a)^4/4!\]For \(f(x) = \tan^{-1}(x)\) at \(a = 0\), the derivatives needed for the formula are: \(f'(x) = 1/(1+x^2)\) \(f''(x) = -2x/(1+x^2)^2\) \(f'''(x) = 2(3x^2-1)/(1+x^2)^3\) \(f''''(x) = -24x(1-x^2)/(1+x^2)^4\)At \(x = 0\), these become: \(f'(0) = 1\) \(f''(0) = 0\) \(f'''(0) = -2\) \(f''''(0) = 0\)Substituting these values into the formula, the fourth degree Taylor polynomial \(P_4(x)\) is: \[P_{4}(x) = f(0) + f'(0)x + (f''(0)*x^2)/2 + (f'''(0)*x^3)/6 + (f''''(0)*x^4)/24 = x - x^3/3.\]

Step 2: Graph plotting

Plotting graphs of the function \(f(x) = \tan^{-1}x\) and its Taylor polynomials requires two steps: First, we need to calculate the first three Taylor polynomials: \(P_{1}(x) = x\), \(P_{2}(x) = x\) (as the second derivative at \(x = 0\) is zero), \(P_{3}(x) = x - x^3/3\). Then, by using graphing software, one would plot the functions \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) on the same graph.

Step 3: Function approximation

To approximate \(f(0.1)\) and \(f(0.3)\) we plug these values into the Taylor Polynomials:Results would then be compared with the calculator's values for arctan(0.1) and arctan(0.3). Values might slightly differ due to the approximation nature of the Taylor polynomials.

Key Concepts

Taylor Series Expansion

The Taylor series expansion is a powerful mathematical tool used to approximate complex functions with polynomials, making them easier to handle. It is especially useful when trying to understand or calculate the behavior of functions near a specific point, typically denoted as a. The expansion is based on the idea that any smooth function can be represented as an infinite sum of its derivatives at a single point.

The general form of the Taylor series for a function f(x) about the point a is given by:
\[ f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots \]
Each term of the series involves higher-order derivatives of the function evaluated at the point a, multiplied by the input variable x raised to increasing powers, with each term divided by the factorial of the order of the derivative. The more terms we include, the closer the polynomial approximates the function around the point a.

Function Approximation

Function approximation is a vital concept in mathematics and engineering, where exact solutions are often either impossible or impractical to determine. Approximation allows us to use simpler functions to represent more complicated ones within a particular range, which can be tremendously helpful in calculations, modeling, and problem-solving.

One of the most common approximations is the Taylor polynomial, which gives us a finite expression by truncating the Taylor series after a set number of terms. Although higher-degree Taylor polynomials provide greater accuracy, they also require more computational effort. The trade-off between accuracy and efficiency is a key consideration when choosing the degree of the polynomial for a given function approximation. In computational tools, these approximations enable quick estimations and are widely used in algorithms that solve complex mathematical expressions.

Derivative of Inverse Tangent

Calculating derivatives is a fundamental operation in calculus, and understanding the derivatives of trigonometric functions is particularly important. The inverse tangent function, also known as arctan or tan^-1, is one such function that arises frequently in various mathematical contexts.

The first derivative of the inverse tangent function f(x) = tan^-1(x) is:
\[ f'(x) = \frac{1}{1+x^2} \]
Subsequent derivatives become more complex but are essential for constructing the Taylor series. For example, the second derivative introduces a negative sign and factors of x, indicative of an increasing function complexity. Discovering the derivative pattern is crucial when forming higher-order terms of the series, as demonstrated in the step-by-step solution provided for the fourth-degree Taylor polynomial.

Graphing Functions

Graphing functions is a fundamental technique used to visualize the relationship between variables. It involves plotting a set of points that represent the function's output values for corresponding input values. When comparing a function to its Taylor polynomial approximations, graphing these on the same axes can illustrate how well the polynomials perform relative to the actual function.

The process usually starts by choosing a range of x-values and then calculating the corresponding y-values for each polynomial and the function itself. When graphing the inverse tangent and its Taylor polynomials, we see that the polynomials tend to fit the function closely near the point of expansion but may diverge as we move further away. This visual comparison is crucial for understanding both the power and the limitations of polynomial approximations in representing complex functions.