Question

39\. Calculate \(d y\) for \(y=a^{x}\) by noting that \(d y=a^{x+d x}-a^{x}=\) \(a^{x}\left(a^{d x}-1\right)\) and then expanding \(a^{d x}-1\) into the power series \(\ln a d x+\frac{(\ln a)^{2} d x^{2}}{2}+\cdots\) (Euler).

Short answer

Given the step by step solution provided, write a short answer question based on the solution: Question: Calculate the differential, \(dy\), for the function \(y = a^x\) using the given expression \(dy = a^x (a^{dx} - 1)\) and expand it into a power series using Euler's method.

Step-by-step solution

Recall the concept of differentials and power series.

In calculus, differentials are a way of approximating the change in a function as the input undergoes an infinitesimal change. Given a function \(y = f(x)\), the differential \(dy\) is the change in \(y\) when \(x\) changes by a small amount \(dx\). A power series is a series of the form \(\sum_{n=0}^{\infty} c_n (x - a)^n\), where \(c_n\) are the coefficients, and is used to approximate a function to any desired precision.

Finding the power series of \(a^{dx} - 1\).

We can write the power series of \(a^{dx} - 1\) as a power series of \(\ln a \cdot dx\), since that's what we have in Euler's method. Recall that the power series of natural exponential function is \(e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\). We can write \(a^{dx} = e^{\ln a \cdot dx}\), where we have used the property \(a^x = e^{(\ln a) \cdot x}\). Thus, the power series of \(a^{dx} - 1\) is: \(a^{dx} - 1 = e^{\ln a \cdot dx} - 1 = \sum_{n=0}^{\infty} \frac{(\ln a \cdot dx) ^n}{n!} - 1\)

Substitute the power series in the expression for \(dy\)

Now that we have the power series for \(a^{dx} - 1\), we can substitute the result back into the expression for \(dy\): \(dy = a^x (a^{dx} - 1) = a^x (\sum_{n=0}^{\infty} \frac{(\ln a \cdot dx)^n}{n!} - 1)\) To simplify further, we can distribute \(a^x\) within the series and combine it with the \(- 1\) term: \(dy = a^x (\sum_{n=1}^{\infty} \frac{(\ln a \cdot dx)^n}{n!}) = a^x (\ln a \cdot dx + \frac{(\ln a)^2 \cdot dx^2}{2} + \cdots)\) Thus, we have found the differential \(dy\) for \(y = a^x\) as a power series using Euler's method.

Key concepts

Differentials

In the realm of differential calculus, the concept of differentials plays a key role. It allows us to estimate the change in a function's output, known as 'y', when there is a tiny change in the input, denoted by 'dx'. This approximated change is expressed as 'dy'. Mathematically, if we have a function represented by 'y = f(x)', the differential 'dy' is used to approximate the change in 'y' when 'x' is incremented by an infinitesimal quantity 'dx'.

Imagine increasing the value of 'x' by just a hair's breadth; the corresponding movement in 'y' along the curve is what we call 'dy'. This concept is a cornerstone in calculus, with deep ties to derivatives, and provides the foundation for many applications and methods, such as Euler's method.

Power Series Expansion

A powerful tool in calculus is the power series expansion, which allows us to express functions as an infinite sum of terms based on powers of a variable. The general form for a power series centered at 'a' is written as \(\sum_{n=0}^\infty c_n (x - a)^n\), where 'c_n' represents the coefficients, 'x' is the variable, and 'n' is a non-negative integer that increases with each term.

This tool can approximate complex functions with a series that is often much simpler to work with. Power series can also converge to the exact value of a function within a certain range around 'a'. By using power series, we can handle otherwise intractable calculations, such as finding the differential of a function using Euler's method, which involves expanding exponential changes into an infinite series.

Natural Exponential Function

The natural exponential function, represented as 'e^x', is one of the most important functions in mathematics, especially in the context of calculus. It is defined by the infinite power series \(e^x = \sum_{n=0}^\infty \frac{x^n}{n!}\), where 'n!' denotes the factorial of 'n'.

The beauty of this function lies in its property of being its own derivative, meaning that the rate at which 'e^x' grows is proportional to its current value. The power series expansion of the natural exponential function can be used to approximate 'e^x' for any real number 'x', thereby providing us with a precise understanding of growth patterns. It enables us to deal with continuous growth scenarios and is a pivotal part of financial mathematics, population dynamics, and other fields requiring the modeling of exponential growth or decay.

Euler's Method

Euler's method is a fundamental numerical procedure used to approximate solutions to differential equations. Named after the Swiss mathematician Leonhard Euler, this method bridges the conceptual gap between differentials and the practical need for calculation.

The essential idea of Euler's method is to start from a known point on a function's graph and take small steps along the tangent line, which represents the instantaneous rate of change. By repeatedly applying this step-by-step process, we can traverse the curve and estimate points that we cannot readily calculate.

This iterative process is an invaluable technique for advancing the understanding of systems described by differential equations, especially when an explicit solution is difficult or impossible to obtain directly. Such applications are found in various fields, from physics to economics, where dynamic changes are continuously modeled and studied.