Problem 27 - Calculus for Biology and Medicine

Question

(a) Write$$h(x, y)=\sin \left(x^{2}+y^{2}\right)$$ as a composition of two functions. (b) For which values of \((x, y)\) is \(h(x, y)\) continuous?

Step-by-Step Solution

Answer

(a) \( f(u) = \sin(u) \), \( g(x, y) = x^2 + y^2 \); (b) \( h(x, y) \) is continuous for all \( (x, y) \in \mathbb{R}^2 \).

Step 1: Understanding Composition of Functions

We need to express the function \( h(x, y) = \sin(x^2 + y^2) \) as a composition of two functions. A composition of functions is when you have \( f(g(x)) \), meaning one function is applied, and then another function is applied to the result.

Step 2: Choosing Inner and Outer Functions

We choose the inner function \( g(x, y) = x^2 + y^2 \). For the outer function, we select \( f(u) = \sin(u) \). This way, \( h(x, y) \) becomes \( f(g(x, y)) = \sin(x^2 + y^2) \). So, the composition is \( (f \circ g)(x, y) \).

Step 3: Expressing h(x, y) as a Composition

To express \( h(x, y) = \sin(x^2 + y^2) \) in terms of composition:\( f(u) = \sin(u) \) and \( g(x, y) = x^2 + y^2 \). Therefore, \( h(x, y) = f(g(x, y)) \) which is \( f(x^2 + y^2) = \sin(x^2 + y^2) \).

Step 4: Understanding Continuity of Functions

A function is continuous everywhere if its formula is composed of continuous functions. The function \( \sin(u) \) is continuous for all real numbers, and \( x^2 + y^2 \) is a polynomial, meaning it's also continuous everywhere.

Step 5: Stating Continuity Conditions

Since \( x^2 + y^2 \) and \( \sin(u) \) are both continuous everywhere, the composition \( \sin(x^2 + y^2) \) is continuous for all \( (x, y) \in \mathbb{R}^2 \). Thus, \( h(x, y) \) is continuous for every point \( (x, y) \) in the plane.

Key Concepts

Composition of Functions

In multivariable calculus, understanding the composition of functions can greatly simplify complex expressions. A composition means that one function is applied to the result of another function. For the function given in the exercise, we have \( h(x, y) = \sin(x^2 + y^2) \).

• The inner function \( g(x, y) = x^2 + y^2 \) takes two variables, squares them, and sums them up.
• The outer function \( f(u) = \sin(u) \) takes the result from \( g(x, y) \) and applies the sine function to it.

So, we can express \( h(x, y) \) as \( f(g(x, y)) = \sin(x^2 + y^2) \), which clearly shows how each part of the function plays its role. This approach of identifying inner and outer functions can be used to tackle similar problems across calculus.

Continuity

Continuity is a fundamental concept in calculus, ensuring that small changes in input lead to small changes in output. A continuous function has no breaks, jumps, or holes in its graph. For the function \( h(x, y) = \sin(x^2 + y^2) \), we must consider the continuity of both the inner and outer functions.

• \( \sin(u) \) is known to be continuous for all real numbers due to its smooth wave-like graph.
• \( x^2 + y^2 \) is a polynomial, and polynomials are continuous everywhere.

Because both these components are continuous everywhere, their composition \( \sin(x^2 + y^2) \) is also continuous for every \( (x, y) \) in the plane \( \mathbb{R}^2 \). This means you can expect \( h(x, y) \) to behave smoothly without any interruptions.

Trigonometric Functions

Trigonometric functions like \( \sin(x) \), \( \cos(x) \), and \( \tan(x) \) are foundational in calculus. They describe the relationships of angles and sides in triangles and produce continuous, periodic graphs.For \( h(x, y) = \sin(x^2 + y^2) \), we specifically use the sine function, which has specific properties:

• It has a range of [-1, 1], meaning no matter what real input it receives, the output will be between -1 and 1.
• Its graph is periodic with a fundamental period of \( 2\pi \), meaning it repeats every \( 2\pi \) units.

Trigonometric functions' continuous and repetitive nature makes them predictable, and they often appear in compositions like \( \sin(x^2 + y^2) \) to model phenomena that are cyclical or wave-like.