Question

In each exercise, determine the largest positive integer \(r\) such that \(q(h)=O\left(h^{r}\right) .\) [Hint: Determine the first nonvanishing term in the Maclaurin expansion of \(q .\) ] \(q(h)=e^{h}-(1+h)\)

Short answer

Answer: 2

Step-by-step solution

Determine the Maclaurin expansion of \(q(h)\)

In order to find the Maclaurin expansion of \(q(h)\), we will first remind the Maclaurin series for \(e^h\), which is given by: $$e^h = \sum_{k=0}^{\infty}\frac{h^k}{k!} = 1 + h + \frac{h^2}{2!} + \frac{h^3}{3!} + \dots$$ Now, subtracting \((1+h)\) from this series, we obtain the Maclaurin series for \(q(h)\): $$q(h) = e^h - (1+h) = \left(1 + h + \frac{h^2}{2!} + \frac{h^3}{3!} + \dots\right) - (1+h)$$

Determine the first nonvanishing term in the Maclaurin expansion of \(q(h)\)

By subtracting \((1+h)\) from the Maclaurin series of \(e^h\), we can identify the remaining terms as: $$q(h) = \frac{h^2}{2!} + \frac{h^3}{3!} + \dots$$ Because everything summand in the series of q is positive for h>0, \(\frac{h^2}{2!}\) or \(\frac{1}{2} h^2\) is the first nonvanishing term in the Maclaurin expansion of \(q(h)\).

Identify the largest positive integer r

We determined that the first nonvanishing term in the Maclaurin expansion of \(q(h)\) is \(\frac{1}{2}h^2\). This term corresponds to the power \(r=2\). Therefore, the largest positive integer \(r\) such that \(q(h) = O(h^r)\) is \(r=2\).

Key concepts

Differential Equations

Differential equations are mathematical tools used to describe various phenomena in engineering, physics, biology, and many other fields. The general form of a differential equation represents a relationship between a function and its derivatives. They can be classified into different types based on their characteristics including order, linearity, and boundary conditions.

Understanding Differential Equations

Differential equations can range from simple linear equations to complex nonlinear ones. The solutions of these equations can be functions or sets of functions. Some differential equations can be solved exactly in a closed form, while others are only solvable through numerical methods or series approximations like the Maclaurin series. Ideally, students are expected to be familiar with basic methods of solving various types of differential equations, such as separation of variables and integrating factors, and understand concepts like particular and homogeneous solutions.

Boundary Value Problems

Boundary value problems (BVPs) arise when you are dealing with differential equations that specify conditions (boundary values) not just at a single point, but over some range or at multiple points.

Importance of Boundary Values

Such problems require the solution not only to satisfy the differential equation but to also meet the boundary conditions. BVPs are essential in the study of physical systems where initial conditions may not suffice, such as in the study of steady-state heat distribution or vibrations of a drumhead. Solution techniques vary based on whether the BVP is linear or nonlinear, and whether it is defined on a finite or infinite domain. Numerical methods often come into play when analytical solutions are intractable.

Big-O Notation

Big-O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann-Landau or asymptotic notation.

Application in Error Analysis

In the context of Maclaurin series and differential equations, Big-O notation plays a critical role in error analysis, indicating the rate at which the remainder (error term) approaches zero as the series progresses. When we say that a function is within the bound of 'Big-O of something', for example, having error terms of the form \(O(h^r)\), it provides a roof over how the error behaves, allowing one to understand the efficiency and accuracy of approximations in numerical methods. Understanding Big-O notation is crucial for analyzing algorithmic complexity in computer science and error estimation in numerical solutions.