Problem 1 - Advanced Calculus

Question

For each of the following pairs of functions, determine its highest order of contact at the indicated point: a. \(f(x)=x^{2}\) and \(g(x)=\sin x\) for all \(x ; x_{0}=0\). b. \(f(x)=e^{x^{2}}\) and \(g(x)=1+2 x^{2}\) for all \(x ; x_{0}=0\). c. \(f(x)=\ln x\) and \(g(x)=(x-1)^{3}+\ln x\) for all \(x>0 ; x_{0}=1\). d. \(f(x)=\ln x\) and \(g(x)=(x-1)^{200}+\ln x\) for all \(x>0 ; x_{0}=1\).

Step-by-Step Solution

Answer

The highest order of contact at the indicated point for each pair is: a) 1, b) 2, c) 3, d) 200.

Step 1 Title - Determine the Order of Contact for Pair a

First, calculate the Taylor series of both functions around the point of interest, which is at \(x_0 = 0\). The Taylor series expansion of \(f(x) = x^2\) about \(x = 0\) is simply \(f(x) = x^2\), as it is already a polynomial. For \(g(x) = \sin x\), the Taylor series expansion about \(x = 0\) is \(g(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\). Determine the highest order of the polynomial term that both functions have in common, which is the first term, \(x\). Therefore, the highest order of contact is 1 (linear contact).

Step 2 Title - Determine the Order of Contact for Pair b

For the pair \(f(x) = e^{x^2}\) and \(g(x) = 1 + 2x^2\), find the Taylor series around \(x_0 = 0\). The Taylor series of \(e^{x^2}\) is \(1 + x^2 + \frac{(x^2)^2}{2!} + \dots\), and the Taylor series of \(g(x)\) around \(x_0 = 0\) is already a polynomial: \(g(x) = 1 + 2x^2\). The highest order of the polynomial term they have in common is \(x^2\), which means the highest order of contact is 2 (quadratic contact).

Step 3 Title - Determine the Order of Contact for Pair c

For the functions \(f(x) = \ln x\) and \(g(x) = (x-1)^3 + \ln x\), we expand around \(x_0 = 1\). The Taylor series for \(\ln x\) about \(x = 1\) is simply \(f(x) = 0 + (x - 1) - (x - 1)^2/2 + (x - 1)^3/3 - \dots\). The function \(g(x)\) is already a sum of a polynomial and \(\ln x\). The highest order they share is the cubic term, so the highest order of contact is 3.

Step 4 Title - Determine the Order of Contact for Pair d

The functions are \(f(x) = \ln x\) and \(g(x) = (x-1)^{200} + \ln x\). Using the same expansion for \(\ln x\) as in the previous step, the highest order of the polynomial term that \(f(x)\) and \(g(x)\) have in common is the \(200\)th term because the terms from \((x - 1)\) up to \((x - 1)^{199}\) in the Taylor expansion of \(\ln x\) are non-zero, and \(g(x)\) has all of those terms plus the \(200\)th term. Therefore, the highest order of contact is 200.

Key Concepts

Taylor Series Expansion

When we talk about Taylor Series Expansion, we are delving into a fundamental concept in calculus that allows us to approximate complex functions with polynomials. A Taylor series represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Imagine you're trying to sketch a curve but can only use straight lines or simple parabolas; Taylor series give you more makeshift 'straight lines' that can bend just enough to trace the curve more accurately.

For example, the sine function can seem complicated, but near the point where the angle is zero, you can approximate it with a simple polynomial using its Taylor series. In the given exercise, the Taylor expansion for \(g(x) = \sin x\) around the point \(x_0 = 0\) is \(g(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\), which starts to build up the sine curve from its derivatives at zero.

This concept is crucial for determining the 'order of contact' between functions, which essentially means how closely two functions 'touch' or 'agree' with each other at a given point. By comparing their Taylor series expansions, we identify the highest degree to which the polynomials match, which is the highest order of contact. Therefore, a solid grasp of Taylor series expansion is vital for analyzing the behavior of functions around specific points.

Calculus of Polynomial Functions

When discussing calculus of polynomial functions, we're essentially looking at the study of algebraic expressions that involve sums of powers of a variable. These mathematical expressions are prevalent in calculus because they're easy to work with and differentiate. What's fascinating is that polynomials are smooth and continuous everywhere, and their behavior is well-understood, which makes them excellent for approximations.

In our textbook problem, we encounter polynomials within the realm of Taylor series. The function \(f(x) = x^2\) is itself a polynomial and thus its own Taylor expansion. Similarly, \(g(x) = 1 + 2x^2\) is already a polynomial that represents the Taylor expansion at the point \(x_0 = 0\). Here's the thing: calculus allows us to take these well-behaved polynomial functions and find their derivatives and integrals with ease, which helps us understand how functions act and change over input values. The calculus of polynomials provides the groundwork for orders of contact, enabling us to ascertain the precise degree of tangency between curves at specific points—a key concept for advancing in understanding more intricate math and physics problems.

Analyzing Function Behavior

Understanding how functions behave is like being a detective who examines clues to predict what happens next in a sequence of events. In mathematics, analyzing function behavior often involves scrutinizing the graphical representation of a function to comprehend its continuity, limits, differentiability, and points of inflection. The main goal is to predict how the function will act under various conditions.

This exploration of function behavior is exemplified in our exercise when considering the 'orders of contact' between pairs of functions. Here, the ability to anticipate a function's behavior is aligned with knowing its derivatives at a certain point, since the derivatives provide information about how functions change. For instance, the natural logarithm function, \(f(x) = \ln x\), behaves quite differently from the polynomial \(g(x) = (x-1)^3 + \ln x\), but by looking at how their derivatives align at the point \(x_0 = 1\), we can determine their order of contact.

In sum, rigorous analysis of the behavior of functions enables students to solve complex problems by breaking them down into more manageable pieces and understanding the subtle nuances of function interaction. This type of investigation empowers students to tackle a wide variety of challenges in calculus and beyond.