Problem 29 - Calculus for Biology and Medicine

Question

(a) Write $$h(x, y)=e^{x y}$$ as a composition of two functions. (b) For which values of \((x, y)\) is \(h(x, y)\) continuous?

Step-by-Step Solution

Answer

(a) \(h(x, y) = e^{xy}\) is \(f(g(x, y))\) with \(g(x, y) = xy\) and \(f(u) = e^u\). (b) \(h(x, y)\) is continuous for all \((x, y) \in \mathbb{R}^2\).

Step 1: Identify the Inner and Outer Functions

To express \(h(x, y) = e^{xy}\) as a composition of two functions, we first identify the inner function and the outer function. Here, the inner function can be \(g(x, y) = xy\), which takes \((x, y)\) and outputs a single real number. The outer function is \(f(u) = e^u\), which takes the output of \(g\) as its input. Therefore, we can write \(h(x, y)\) as a composition of \(f\) and \(g\): \(h(x, y) = f(g(x, y)) = e^{xy}\).

Step 2: Evaluation of Continuity

A function of two variables, like \(h(x, y)\), is continuous at \((x_0, y_0)\) if for every sequence \((x_n, y_n)\) converging to \((x_0, y_0)\), the sequence \(h(x_n, y_n)\) converges to \(h(x_0, y_0)\). Exponential functions, as well as the product of real numbers, are continuous everywhere in their respective domains. Hence, the function \(h(x,y) = e^{xy}\) is continuous everywhere in \(\mathbb{R}^2\), meaning for all real \(x\) and \(y\).

Key Concepts

Exponential Function

An exponential function is a mathematical function of the form \( f(x) = a^{x} \), where \( a \) is a constant and \( x \) is the variable. In our example, \( h(x, y) = e^{xy} \), the base \( e \) is a well-known mathematical constant approximately equal to 2.718. It is referred to as Euler's number. Exponential functions have graceful properties that make them valuable in mathematics and real-world applications.

• They grow rapidly for positive exponents, but decay fast with negative exponents.
• Exponentials can model compound interest, population growth, and radioactive decay.

In calculations, exponential functions often reveal patterns or serve as solutions to complex differential equations. Understanding their simplicity and versatility can greatly enhance problem-solving skills in calculus.

Continuity

Continuity is an essential concept in calculus that describes how functions behave as you zoom in at a point. A function is continuous at a point if there is no "jump" or "hole" at that point. More formally, a function is continuous at a point if the limit of the function as it approaches the point equals the function's value at that point.

• This means no abrupt changes in the function's output.
• If a function is continuous throughout its domain, it provides a smooth curve or surface.

In our function \( h(x, y) = e^{xy} \), both the exponential function and the multiplication of \( x \) and \( y \) are continuous across the entire plane \( \mathbb{R}^2 \). Therefore, \( h(x, y) \) remains continuous everywhere applicable, making it predictable and easier to analyze along any selected path within its domain.

Function Composition

Function composition involves creating a new function by applying one function to the results of another. Given two functions \( f(u) \) and \( g(x, y) \), their composition \( h(x, y) = f(g(x, y)) \) involves first calculating \( g(x, y) \) and then applying \( f \) to this result. The composition allows for building more complex functions based on simpler, well-understood components.

• In this exercise, the inner function is \( g(x, y) = xy \).
• The outer function is \( f(u) = e^u \).

Understanding function composition is vital because it creates a cascade of transformations and calculations using basic building blocks. This modular approach simplifies handling multi-step processes, common in calculus problems, as demonstrated by decomposing \( h(x, y) = e^{xy} \) into its parts.