91Ó°ÊÓ

When you're working with functions of multiple variables, understanding the domain and range is crucial. For our function, \( f(x, y) = \exp \left[-(x^2 + y^2)\right] \), the domain and range are relatively straightforward.

The "domain" of a function refers to all the possible input values. In this case, since the squares \( x^2 \) and \( y^2 \) are defined for all real numbers, the domain includes every pair of real numbers \((x, y)\). That's why we say the domain is \( \mathbb{R}^2 \), representing the entirety of the xy-plane.

Next, the "range" describes all possible outputs of the function. The exponential part always gives a positive result, since an exponential function never outputs a negative value. In this case, the range is \((0, 1]\). Why not exactly 0? Because no real value of \( x \) and \( y \) will make the sum \( x^2 + y^2 \to \infty \) in a finite manner. At the same time, the function can output its maximum value of 1 when \( x = 0 \) and \( y = 0 \), giving \( \exp(0) \).

So, think of the domain as all the spots where you can "plug in" values, and the range as all the results you could get.

Exponential functions are essential in both single variable and multivariable calculus. They're known for their rapid growth and decay characteristics. In our function \( f(x, y) = \exp \left[-(x^2 + y^2)\right] \), the exponential decay is particularly interesting.

Let's break down the negative sign before the square terms: \(-(x^2 + y^2)\). Here, the larger the sum of \( x^2 + y^2 \), the smaller your output since \( \exp(-u) \rightarrow 0 \) as \( u \rightarrow \infty \). Conversely, smaller values of \( x^2 + y^2 \) yield larger function values, maximizing at 1 when \( (x, y) = (0, 0) \).

Exponential functions in multivariable calculus can model real-world phenomena that involve areas like growth processes or decay models. Their derivatives are straightforward, often leading back to the original exponential function, but in our scenario, we're more interested in how the function values spread over 2D space.

In summary, exponential functions cover many everyday applications and interactions. When you see them as part of a multivariable function, envision how they grow outwards from the center, diminishing in intensity as you move further away.

Level curves are fascinating visual aids in multivariable calculus, offering insights into the shape and behavior of a function across a plane. For \( f(x, y) = \exp \left[-(x^2 + y^2)\right] \), these level curves are circles centered at the origin.

To find a level curve, you set the function equal to a constant \( c \):
\( \exp \left[-(x^2 + y^2)\right] = c \).
Taking the natural logarithm, \(-(x^2 + y^2) = \ln(c)\), leading to \( x^2 + y^2 = -\ln(c) \). This is a familiar equation of a circle with a radius of \( \sqrt{-\ln(c)} \).

What's intriguing is how these circles vary depending on \( c \):

• When \( c \) is close to 0, the logarithm is very negative, resulting in a large circle.
• As \( c \) approaches 1, the circles shrink, reflecting the declining function output as you approach the limit.

Only values \( 0 < c \leq 1 \) are valid for the range; \( c = 0 \) isn't defined since it implies an infinitely large radius. Visualize these curves as records of the function's behavior at different heights, akin to topographic lines on a map indicating elevation. They serve as a handy illustration of how function values distribute over space.