Problem 30 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

(a) \( h(x, y) = f(g(x, y)) \) with \( g(x, y) = y-x \) and \( f(u) = \cos(u) \). (b) Continuous for all \((x, y) \in \mathbb{R}^2\).

Step 1 - Identify inner and outer functions

We need to express the given function \( h(x, y) = \cos(y-x) \) as a composition of two functions. Let's identify what the inner function \( g \) and the outer function \( f \) could be. Notice that \( y-x \) seems to serve as the input for the \( \cos \) function, so we can define an inner function \( g(x, y) = y - x \).

Step 2 - Define outer function

Now that we have defined the inner function \( g(x, y) = y - x \), the outer function \( f(u) = \cos(u) \) uses the output from \( g \). Therefore, we can write \( h(x, y) = f(g(x, y)) = \cos(y-x) \).

Step 3 - State composition of functions

Hence, we can express \( h(x, y) = \cos(y-x) \) as a composition of two functions: \( h(x, y) = f(g(x,y)) \) where \( g(x, y) = y - x \) and \( f(u) = \cos(u) \).

Step 4 - Determine continuity

The cosine function, \( f(u) = \cos(u) \), is continuous for all real numbers \( u \), while the function \( g(x, y) = y - x \) is also continuous for all \( x \) and \( y \) in \( \mathbb{R} \). Since the composition of continuous functions is continuous, \( h(x, y) = \cos(y-x) \) is continuous for all real numbers \( (x, y) \).

Step 5 - State conclusion about continuity

Thus, the function \( h(x, y) = \cos(y-x) \) is continuous for all values \( (x, y) \in \mathbb{R}^2 \).

Key Concepts

Function Composition

Function composition is a fundamental concept in calculus, which involves creating a new function by combining two or more functions. It allows us to build complex operations from simpler ones. In the given exercise, the function \( h(x, y) = \cos(y-x) \) is expressed as a composition of two functions.
First, we identify the inner function, which takes the variables \( x \) and \( y \) and combines them. Here, the inner function \( g(x, y) = y-x \) simply subtracts \( x \) from \( y \). This output acts as the input for the outer function.

• The inner function \( g(x, y) = y-x \) represents a basic arithmetic operation, which is continuous across all real numbers.

Next, the outer function \( f(u) = \cos(u) \) accepts the result of the inner function. By writing \( h(x, y) = f(g(x, y)) \), we denote that \( h \) is the composition of \( f \) and \( g \). This notation shows how we input the result of \( g(x, y) \) into \( f \), revealing the dependency and flow of data.

• The outer function utilizes the result of the inner function, forming a dependency chain essential to achieve the desired calculation: \( \cos(y-x) \).

Understanding function composition is crucial in multivariable calculus as it breaks down complex processes into manageable parts.

Continuity

Continuity is a key concept in calculus and analysis, ensuring there are no sudden breaks or jumps in the values of a function. To determine where a function is continuous, one must examine the individual functions involved.
In this exercise, both the inner function \( g(x, y) = y-x \) and the outer \( f(u) = \cos(u) \) must be continuous for their composition \( h(x, y) = f(g(x, y)) \) to be continuous.

• The function \( g(x, y) = y-x \) is a basic linear function. Linear operations such as addition and subtraction are continuous over all real values of \( x \) and \( y \).

Meanwhile,

• the cosine function \( f(u) = \cos(u) \) is well-known to be continuous for all real numbers \( u \). This is due to its nature as a periodic trigonometric function that smoothly oscillates between -1 and 1 without breaks.

When both functions are continuous individually, their composition \( h(x, y) = \cos(y-x) \) is also continuous. This continuity holds for all pairs \( (x, y) \) in \( \mathbb{R}^2 \), offering a smooth calculation process for the entire input space.

Cosine Function

The cosine function is a critical part of trigonometry, often appearing in various calculus problems due to its periodic nature and applications in oscillations and waves. Defined as \( \cos(u) \), where \( u \) is any real number, it describes a continuous wave.
One of the unique characteristics of the cosine function is its periodicity with a period of \( 2\pi \). This means that \( \cos(u + 2\pi) = \cos(u) \) for any value of \( u \). This periodicity makes it particularly useful in fields like physics and engineering, which commonly deal with repeating cycles.

• The range of the cosine function is between -1 and 1, inclusive. No matter the input \( u \), the output will always remain within this interval, ensuring predictability.

Additionally, the cosine function is not only continuous but also differentiable everywhere, providing both a smooth graph and the capacity to ascertain rates of change effectively. When used in multivariable calculus, especially in compositions like \( h(x,y) = \cos(y-x) \), it operates predictably across the entire real number line.

• Thus, the function serves as a robust component in both theoretical explorations and practical calculations.