Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(f(x, y)=\left\\{\begin{array}{cl}\frac{\sin (x y)}{x y}, & \text { if } x y \neq 0 \\ 1, & \text { if } x y=0\end{array}\right.\)

Step-by-Step Solution

Answer

The function is piecewise and continuous, defined as 1 at \(xy = 0\) and \(\frac{\sin(xy)}{xy}\) when \(xy \neq 0\).

Step 1: Understand the Function Definition

The function given is a piecewise function. It is defined differently depending on whether the product of \(x\) and \(y\) is zero or not. We have: - \( f(x,y) = \frac{\sin(xy)}{xy} \) when \( xy eq 0 \).- \( f(x,y) = 1 \) when \( xy = 0 \).

Step 2: Evaluate for Conditions When \(xy = 0\)

The value of \( f(x, y) \) is directly given as 1 when \( xy = 0 \). This occurs when \( x = 0 \) or \( y = 0 \). So, for any inputs where either variable is zero, \( f(x, y) = 1 \) is the output.

Step 3: Simplify the Expression for \(xy \neq 0\)

For \( xy eq 0 \), the function is \( f(x,y) = \frac{\sin(xy)}{xy} \). Recall that as \( u \to 0 \), \( \frac{\sin(u)}{u} \to 1 \). Hence, for non-zero values of \(xy\), we directly use this expression. However, if \( xy \) approaches zero, the expression approaches 1, due to the small angle approximation for sine.

Step 4: Verify Continuity at \(xy = 0\)

For the function to be continuous at the boundary where \( xy = 0 \), the limit of \( \frac{\sin(xy)}{xy} \) as \( xy \to 0 \) must equal the function value at \( xy = 0 \), which is 1. As shown, the limit \( \lim_{xy \to 0} \frac{\sin(xy)}{xy} = 1 \), so the function is indeed continuous across this boundary.

Key Concepts

Piecewise Functions

Piecewise functions are functions that have different expressions based on specific conditions or intervals. They are essential when a single formula isn't sufficient to define a function for all cases. For example, in our exercise, the function is defined as

• \( f(x, y) = \frac{\sin(xy)}{xy} \)
• if \( xy eq 0 \)
• and \( f(x, y) = 1 \)
• if \( xy = 0 \)

This means the function behaves differently depending on the value of the product \(xy\). If \(xy\) equals zero, we use the second part of the definition, making the function value equal to 1, ensuring proper handling of undefined expressions in the first part. Using piecewise functions helps simplify complex function behaviors into manageable pieces.

Continuity

Continuity in functions means that you can draw the graph without lifting your pencil. In more technical terms, a function is continuous at a point if the limit as you approach the point from all directions equals the function's value at that point. For piecewise functions like ours, it is crucial to verify continuity at the boundaries where there is a change in the expression.
For our function, we specifically check the boundary condition at \(xy = 0\). We need the limit of \( \frac{\sin(xy)}{xy} \) as \(xy\) approaches zero to be equal to 1, which is the defined function value at \(xy = 0\). Since the limit does approach 1, the function is continuous across this boundary.

Limit Evaluation

Limit evaluation helps determine the behavior of functions as inputs approach a particular value. For piecewise functions, it's crucial to assess the limit at transition points to ensure continuity. In our case, the expression \( \frac{\sin(xy)}{xy} \) is evaluated as \(xy\) approaches zero.Using the fact that \( \lim_{u \to 0} \frac{\sin(u)}{u} = 1 \), we can find that as \(xy\) gets closer to zero, \( \lim_{xy \to 0} \frac{\sin(xy)}{xy} = 1 \). This is important because it tells us the function's behavior at the boundary is predictable and controlled, confirming the function's continuity.

Sine Function

The sine function, \( \sin(x) \), is a fundamental trigonometric function that describes the y-coordinate of a point on a unit circle as it rotates counterclockwise from the positive x-axis. It's essential in various fields, from engineering to physics, because of its periodic nature.
In calculus, a crucial property is how \( \sin(x) \) behaves as \(x\) approaches zero. For small angles, \( \sin(x) \) is approximately equal to \(x\). Hence, \( \frac{\sin(x)}{x} \) approaches 1 as \(x\) approaches zero. This property is used in our function to determine the behavior of \( \frac{\sin(xy)}{xy} \) when \(xy\) nears zero, ensuring continuity.

Function Evaluation

Function evaluation is the process of determining the output of a function for given input values. For our piecewise function, it involves determining which piece of the function to use based on the input condition.- If \( xy = 0 \), we directly use \( f(x, y) = 1 \).- If \( xy eq 0 \), we use the rational expression \( \frac{\sin(xy)}{xy} \).Evaluating functions accurately ensures proper function operation and understanding. It also helps in identifying points of discontinuity and transitional behavior in more complex functions. Function evaluation is fundamental to verifying the correctness of piecewise definitions and ensuring seamless transitions at boundaries.