Problem 15 - Active Calculus Multivariable

Question

Use the properties of continuity to determine the set of points at which each of the following functions is continuous. Justify your answers. a. The function \(f\) defined by \(f(x, y)=\frac{x+2 y}{x-y}\) b. The function \(g\) defined by \(g(x, y)=\frac{\sin (x)}{1+e^{y}}\) c. The function \(h\) defined by $$h(x, y)=\left\\{\begin{array}{ll}\frac{x y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\\0 & \text { if }(x, y)=(0,0)\end{array}\right.$$ d. The function \(k\) defined by $$k(x, y)=\left\\{\begin{array}{ll} \frac{x^{2} y^{4}}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\\0 & \text { if }(x, y)=(0,0)\end{array}\right.$$

Step-by-Step Solution

Answer

a. The function f(x, y) is continuous everywhere except on the line x = y. b. The function g(x, y) is continuous everywhere in the xy-plane. c. The function h(x, y) is continuous for all points (x, y) ≠ (0, 0). d. The function k(x, y) is continuous everywhere in the xy-plane.

Step 1: Consider the function f(x,y)

We have the function f(x, y) = \(\frac{x + 2y}{x - y}\). To identify the points of continuity, we must find where the denominator is not equal to zero. Thus, we have: x - y ≠ 0 x ≠ y The function f(x, y) is continuous everywhere except on the line x = y. b.

Step 2: Consider the function g(x,y)

We have the function g(x, y) = \(\frac{\sin(x)}{1 + e^y}\). Both sine and exponential functions are continuous everywhere. Since the sum of continuous functions is continuous, 1 + e^y is continuous everywhere. The quotient of continuous functions is continuous everywhere the denominator is non-zero. In this case, the denominator is always greater than zero: 1 + e^y > 0 Thus, g(x, y) is continuous everywhere in the xy-plane. c.

Step 3: Consider the function h(x,y)

We have the function h(x, y) defined by: $$h(x, y)=\left\\{\begin{array}{ll}\frac{x y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\\0 & \text { if }(x, y)=(0,0)\end{array}\right.$$ At the point (0, 0), h(x, y) = 0. For all other points, h(x, y) = \(\frac{xy}{x^2 + y^2}\). The function \(\frac{xy}{x^2 + y^2}\) is continuous for all points (x, y) ≠ (0, 0). To determine if h(x, y) is continuous at (0, 0), we should check the limit: \[\lim_{(x,y) \to (0,0)} \frac{xy}{x^2 + y^2}\] We can evaluate this by polar coordinates: \(x = r\cos\theta\), \(y = r\sin\theta\) \[\lim_{r \to 0}\frac{r^2 \cos\theta\sin\theta}{r^2(\cos^2\theta + \sin^2\theta)} = \lim_{r \to 0}\cos\theta\sin\theta\] Since \(\cos\theta\sin\theta\) is independent of r, the limit is equal to \(\cos\theta\sin\theta\), which is not unique for all θ. Therefore, h(x, y) is not continuous at (0, 0). So, h(x, y) is continuous for all points (x, y) ≠ (0, 0). d.

Step 4: Consider the function k(x,y)

We have the function k(x, y) defined by: $$k(x, y)=\left\\{\begin{array}{ll} \frac{x^{2} y^{4}}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\\0 & \text { if }(x, y)=(0,0)\end{array}\right.$$ At the point (0, 0), k(x, y) = 0. For all other points, k(x, y) = \(\frac{x^2 y^4}{x^2 + y^2}\). The function \(\frac{x^2 y^4}{x^2 + y^2}\) is continuous for all points (x, y) ≠ (0, 0). To determine if k(x, y) is continuous at (0, 0), we should check the limit: \[\lim_{(x,y) \to (0,0)} \frac{x^2 y^4}{x^2 + y^2}\] Again, we can evaluate this using polar coordinates: \(x = r\cos\theta\), \(y = r\sin\theta\) \[\lim_{r \to 0}\frac{r^6 \cos^2\theta\sin^4\theta}{r^2(\cos^2\theta + \sin^2\theta)} = \lim_{r \to 0} r^4\cos^2\theta\sin^4\theta\] Since the limit is equal to 0 for all θ, k(x, y) is continuous at (0, 0). So, k(x, y) is continuous everywhere in the xy-plane.

Key Concepts

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions of multiple variables. In this realm, we deal with functions that have inputs and outputs in higher dimensions, such as two or three dimensions.
It encompasses crucial topics like partial derivatives, multiple integrals, and continuity in higher dimensions. When analyzing the continuity of a function in multivariable calculus, we need to consider all possible directions in which we can approach a given point.
This leads to techniques such as evaluating limits using different paths, including straight lines and curves, to ensure that the function behaves consistently from all directions.

Functions

Functions in calculus are ways to express relationships between different quantities. In multivariable calculus, functions often involve two or more variables, like the example functions in the exercise: \(f(x, y)\), \(g(x, y)\), \(h(x, y)\), and \(k(x, y)\).
Understanding these functions means looking at the relationships they dictate and how outputs change as inputs vary.

• For \(f(x, y) = \frac{x+2y}{x-y}\), it is defined everywhere except when \(x = y\) because that makes the denominator zero.
• The function \(g(x, y) = \frac{\sin(x)}{1+e^y}\) is defined everywhere since the denominator cannot be zero.
• Functions \(h\) and \(k\) involve conditionally defined outputs dependent on the point in question, especially at the origin (0,0), where special attention is required.

It's key to note how these functions behave differently around singularities or points where they aren't well-defined in typical forms.

Limit Evaluations

Evaluating limits is a foundational concept when assessing the continuity of multivariable functions. A function is continuous at a point if its limit matches the function's value remains consistent as you approach that point from all directions.
For instance, in the exercise with \(h(x, y)\), checking the limit as \((x, y)\) approaches \((0, 0)\) involves converting to polar coordinates. This technique considers the function's behavior when \(x\) and \(y\) transform into a radius and angle system.Polar substitution allows the conversion:

• \(x = r\cos\theta\)
• \(y = r\sin\theta\)

This simplifies the expressions and focuses on the impact of paths other than straight lines, confirming if the limit is path-independent. If it's not, as is shown for \(h(x, y)\), the function cannot be continuous at that point.

Polar Coordinates

Polar coordinates are an alternative to Cartesian coordinates, useful for evaluating limits in multivariable calculus. They represent points in the plane using a distance from a reference point and an angle from a reference line.
This system is particularly valuable in evaluating the behavior of functions around a singularity, such as the origin (0,0).

• Converting \((x, y)\) to polar coordinates involves the equations \(x = r\cos\theta\) and \(y = r\sin\theta\).
• This substitution helps isolate the radial component \(r\) and examines how functions behave as \(r\) approaches zero.

By doing this, you can scrutinize whether the function's behavior is inherently tied to direction (angle \(\theta\)) or is purely radial. This method is vital when straightforward Cartesian approaches are invalid, especially in proving or disproving continuity at problematic points such as where the denominator might approach zero.