Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot the graphs of each of the following functions on \(-2 \leq x \leq 2,-2 \leq y \leq 2\), and determine where on this set they are discontinuous. (a) \(f(x, y)=x^{2} /\left(x^{2}+y^{2}\right), f(0,0)=0\) (b) \(f(x, y)=\tan \left(x^{2}+y^{2}\right) /\left(x^{2}+y^{2}\right), f(0,0)=0\)

Step-by-Step Solution

Answer

(a) Continuous; (b) Discontinuous at circles centered at origin with radii \( \sqrt{\frac{\pi}{2}}, \sqrt{\frac{3\pi}{2}}, \ldots \).

Step 1: Understand the Function Domains

First, interpret the given functions and recognize where potential discontinuities may arise. The function (a) \( f(x, y) = \frac{x^2}{x^2 + y^2} \) becomes undefined at \( (0,0) \) since the denominator becomes zero, but it's defined there as 0. The function (b) \( f(x, y) = \frac{\tan(x^2 + y^2)}{x^2 + y^2} \) becomes problematic at \( (0,0) \) and when \( x^2 + y^2 = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \) due to the tangent function.

Step 2: Analyze Continuity of Function (a)

Evaluate the continuity of \( f(x, y) \) at \( (0,0) \) and over the given intervals. For points \((x, y) eq (0, 0)\), \( f(x, y) = \frac{x^2}{x^2 + y^2} \) is continuous since both numerator and denominator are polynomials. At \((0,0)\), calculate the limit \( \lim_{(x,y)\to(0,0)} \frac{x^2}{x^2+y^2} = 0 \), which matches \( f(0,0) \). Hence, \((0,0)\) is not a point of discontinuity.

Step 3: Locate Discontinuities for Function (b)

For \( f(x, y) = \frac{\tan(x^2 + y^2)}{x^2 + y^2} \), check the limit at \((0,0)\): \( \lim_{(x,y)\to(0,0)} \frac{\tan(x^2+y^2)}{x^2+y^2} = 1 \) (using \( \lim_{z \to 0} \frac{\tan z}{z} = 1 \)). So, it is continuous at \((0,0)\). Check for other discontinuities where \( x^2 + y^2\) equals \( \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \), as these cause the tangent function to become undefined.

Step 4: Graphical Analysis

Graph both functions over the domain \(-2 \leq x \leq 2, -2 \leq y \leq 2 \) and observe the behavior. For function (a), the graph is smooth and continuous over the entire range with no disruptions, confirming the analytical finding. For function (b), observe spikes or asymptotes at radial distances \( \sqrt{\frac{\pi}{2}} \), \( \sqrt{\frac{3\pi}{2}} \), etc., indicating discontinuities due to undefined tangents.

Key Concepts

Discontinuity Analysis

When dealing with multivariable functions, understanding where they are discontinuous is crucial. Discontinuity occurs when the function is not well-behaved at certain points in its domain. For function (a), \( f(x, y) = \frac{x^2}{x^2 + y^2} \), potential discontinuity at \((0,0)\) arises because both the numerator and denominator are zero. However, the function is pre-defined to be zero at \((0,0)\). By calculating the limit \( \lim_{(x,y)\to(0,0)} \frac{x^2}{x^2+y^2} = 0 \), we find it continuous at this point.
Function (b), \( f(x, y) = \frac{\tan(x^2 + y^2)}{x^2 + y^2} \), is more complex. It is undefined when \( x^2 + y^2 = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \), since the tangent function becomes infinite. However, at \((0,0)\), an analysis using limits, specifically \( \lim_{(x,y)\to(0,0)} \frac{\tan(x^2+y^2)}{x^2+y^2} = 1 \), shows that the function is continuous at the origin. This shows the importance of calculating limits at points with potential issues to determine continuity.

Function Graphing

Graphing multivariable functions provides a visual understanding of their behavior over a specific domain. For the mathematic region given as \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\), it's beneficial to visualize how these functions operate.
By graphing function (a), \( f(x, y) = \frac{x^2}{x^2 + y^2} \), we see a smooth and continuous surface confirming its analytical continuity. There's no disruption or sudden change, showing it behaves well over this range.
On the contrary, function (b), \( f(x, y) = \frac{\tan(x^2 + y^2)}{x^2 + y^2} \), presents spikes and asymptotes where \( x^2 + y^2 \) meets the problematic values \( \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \). These visual elements highlight discontinuities, indicating regions where the tangent function is undefined. Thus, graphical analysis serves as a powerful tool to accompany algebraic investigations of continuity.

Limits and Continuity

Limits help determine whether a function remains stable as inputs approach certain values. For function (a), \( f(x, y) = \frac{x^2}{x^2 + y^2} \), the limit at \((0,0)\) was essential in confirming its continuity there. By finding that the limit equaled the function’s value at the origin, continuity was established.
Function (b), with its more intricate tangent composition, further illustrates limits’ significance. Although directly at \((0,0)\), the function seemed suspicious, careful use of limits helped verify that \( \lim_{(x,y)\to(0,0)} \frac{\tan(x^2+y^2)}{x^2+y^2} = 1 \), ensuring continuity despite the undefined behavior at other radial distances.
This process shows how limits act as a powerful method to carefully navigate potential discontinuities by thoroughly assessing nearby function behavior, and ensuring smooth transitions in multivariable calculus.