Question

In Exercises \(47-52,\) discuss the continuity of the function. \(f(x, y, z)=\frac{z}{x^{2}+y^{2}-9}\)

Short answer

The function \(f(x, y, z)=\frac{z}{x^{2}+y^{2}-9}\) is continuous at all points in the space except for the points lying on the circle defined by the equation \(x^{2} + y^{2} = 9\) in the \(xy\)-plane. At these points, the function is not defined due to division by zero.

Step-by-step solution

Identify critical points

Firstly, solve for the values of \(x\) and \(y\) when the denominator of the function is equal to zero.\n So, we set \(x^{2} + y^{2} - 9 = 0\). This will give us the values of \(x\) and \(y\) where the function might not be continuous.

Interpret the critical Points

We see that the values of \(x\) and \(y\) satisfy the equation of a circle in the \(xy\)-plane with radius 3 centered at the origin. This means that the function is not defined and hence not continuous at any point on this circle in the \(xy\)-plane since the function would include division by zero at these points.

Analyze continuity outside the identified points

For any other points where the denominator of the function does not equal zero, we can use the sum, difference, product, quotient, and composition rules of continuity to determine that the function is continuous. Provided that the denominator is non-zero, all the aforementioned mathematical operations would imply continuity for the given function \(f(x, y, z)=\frac{z}{x^{2}+y^{2}-9}\).

Key concepts

Critical Points

When we dive into the realm of multivariable calculus, one of the fundamental concepts students must understand is that of critical points. Critical points are essentially the coordinates of a function where the behavior of that function changes in some meaningful way. Specifically, they are the points where the gradient of the function (the vector of partial derivatives) is zero or undefined.

In the context of the exercise, identifying critical points involves finding where the function's denominator is equal to zero, thereby pinpointing the locations where the function may not be defined. For the given function \(f(x, y, z)=\frac{z}{x^{2}+y^{2}-9}\), the critical points are those that make the denominator \(x^{2} + y^{2} - 9\) equal to zero. As revealed through the solution, these points form a circle with radius 3 in the \(xy\)-plane, centered at the origin. Understanding critical points is not just about determining where a function might misbehave; it also sets the stage for analyzing features like local maxima, minima, and saddle points, which are critical for understanding the function's bigger picture.

Continuity of a Function

A cornerstone of calculus, the continuity of a function, is a property that ensures smooth transitions of the function values as inputs change incrementally. For a function to be continuous at a point, it must satisfy three conditions: the function must be defined at the point, the limit of the function as it approaches the point must exist, and the value of the function at that point must equal the value of the limit.

In the multivariable scenario, such as with our function \(f(x, y, z)\), continuity requires that small changes in the variables \(x, y,\) and \(z\) do not lead to drastic jumps in the value of \(f\). Following the step-by-step solution, by utilizing the rules of continuity (sum, difference, product, quotient, and composition), we can infer that \(f\) is continuous anywhere except where the denominator is zero. This hinges on the denominator being a non-zero value, which ensures that we don't end up with undefined situations, such as dividing by zero. It's through this meticulous process that we navigate the function's landscape, sifting out the points of continuity from the discontinuities.

Multivariable Functions

Exploring multivariable functions is akin to venturing into a multi-dimensional universe where each input variable adds a layer of complexity. A function of several variables, like the one in our exercise \(f(x, y, z)=\frac{z}{x^{2}+y^{2}-9}\), maps ordered triples \( (x, y, z) \) to a real number. These types of functions can represent real-world phenomena where multiple factors interact to produce an outcome.

What sets multivariable functions apart from their single-variable counterparts is the way in which they change. The methods for analyzing them include partial derivatives, gradients, and level curves, each providing insight into the function's behavior in the vicinity of a given point. Understanding multivariable functions is not just about recognizing their graphical representations, such as surfaces for functions of two variables, but also about grasping how they respond to changes in any of the variables and how these changes are interconnected. These functions demand a concerted effort to dissect and comprehend each variable's influence on the function's value—a multidimensional dance of calculus that is both intricate and fascinating.