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

Function (a) is discontinuous at (0,0). Function (b) is continuous at (0,0) but discontinuous elsewhere where tan is undefined.

Step 1: Understand the Domain

First, identify the domain of the functions. Both functions are defined on the entire plane except where denominators become zero, meaning potential discontinuity at points where the denominators tend to zero. Additionally, both functions have defined values at \((0,0)\).

Step 2: Examine Function (a) for Discontinuities

For function (a), \(f(x, y)=x^{2}/(x^{2}+y^{2})\), the denominator is zero at \((0, 0)\), yet the function assigns a value of 0 at this point. Investigate continuity at \((0, 0)\) by checking the limit \(\lim_{(x,y)\to(0,0)} x^2/(x^2+y^2)\). In polar coordinates, \(r=\sqrt{x^2+y^2}\), the limit becomes \(\lim_{r\to0} \cos^2(\theta)\), which is dependent on \(\theta\) and thus undefined.

Step 3: Examine Function (b) for Discontinuities

For function (b), \(f(x, y)=\tan(x^2+y^2)/(x^2+y^2)\), similarly examine potential discontinuity at \((0,0)\). The point \((0,0)\) has a value of 0. Check the limit \(\lim_{(x,y)\to(0,0)} \tan(r^2)/(r^2)\). As \(\tan(0)/0\) evaluates to 0, we determine if this tendency continues. Convert to polar coordinates and examine the ratio as \(r\to0\), recognizing that \(\tan(\theta)\approx\theta\) for small angles; thus, the function converges to its defined value at \((0,0)\). Away from \((0,0)\), watch for points at which \(\tan(x^2+y^2)\) is undefined.

Step 4: Plot the Functions

Create plots for both functions within the specified domain, \(-2 \leq x \leq 2\), \(-2 \leq y \leq 2\). Observe graphical changes in the functions, noting the behavior around \((0,0)\) and identify any undefined points for \(f(b)\) where \(x^2 + y^2 = \pi/2, 3\pi/2,...\).

Step 5: Solution Conclusion

Conclude that function (a) is discontinuous at \((0,0)\) as it does not have a defined limit regardless of the path taken toward it. For function (b), the designated value at \((0,0)\) is consistent with its limit and is not discontinuous there. However, \(f(b)\) includes other discontinuous points where \(\tan\) becomes undefined.

Key Concepts

Continuity and Discontinuity

Functions are said to be continuous at a point if the limit of the function as it approaches the point equals the function’s value at that point. Discontinuity occurs when this condition isn't satisfied. Calculating the limit and comparing it to the function's value is key to determining discontinuity.
For example, consider the function given as part of exercise (a):

• Function: \( f(x, y) = \frac{x^{2}}{x^{2} + y^{2}} \)
• At \((0,0)\), the denominator is zero, which is a red flag potentially indicating discontinuity.
• The function assigns a value of 0 at this point, so the check involves verifying whether \( \lim_{(x,y)\to(0,0)} \frac{x^2}{x^2+y^2} = 0 \).

However, because the limit depends on the path taken to approach \((0,0)\), such as using polar coordinates (described further below), the limit becomes undefined, proving the function is discontinuous at this point.

Polar Coordinates

Polar coordinates express a point in terms of its distance from the origin, \(r\), and an angle, \(\theta\), with respect to the positive x-axis. They are particularly useful in multivariable calculus when examining functions' behavior as they approach the origin, simplifying the calculation of limits.
In Cartesian coordinates \((x, y)\), these can be converted to polar coordinates using:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)
• \( r = \sqrt{x^2 + y^2} \)

By converting function (a) into polar coordinates, the expression simplifies to \( \lim_{r\to0} \cos^2(\theta) \), meaning the limit depends on \(\theta\). This path-dependent result is critical for discovering points of discontinuity. For function (b) in the original exercise, converting to polar coordinates allowed determining consistent behavior in the limit at \((0,0)\).

Limit Evaluation

Evaluating limits is a crucial process in understanding the continuity of functions with multiple variables. By determining the limit of a function as it approaches a point, you can assess if the function behaves consistently.
Consider function (a) again:

• To evaluate the limit \( \lim_{(x,y)\to(0,0)} \frac{x^2}{x^2+y^2} \), it helps to express this in polar form, resulting in \( \lim_{r\to0} \cos^2(\theta) \).
• This expression shows that the limit can vary based on \(\theta\), proving the discontinuity since no single, consistent limit exists as \((x,y)\) approaches \((0,0)\).

For function (b), the limit \( \lim_{(x,y)\to(0,0)} \frac{\tan(r^2)}{r^2} \) behaves according to \(\tan(0)/0\) which evaluates to 0. The ratio approaches its assigned value consistently, confirming continuity at that point. Limit evaluation, therefore, is integral to both identifying and understanding points of discontinuity effectively. By prefetching the points where \tan(x^2+y^2)\ turns zero, it helps pinpoint and avoid discontinuity in function (b).