Problem 10

Question

Let $$g(x, y, z)=\ln \left(25-x^{2}-y^{2}-z^{2}\right)$$ (a) Evaluate $g(2,-2,4)$. (b) Find the domain of $g$. (c) Find the range of $g$.

Step-by-Step Solution

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Answer

(a) 0; (b) $\{(x,y,z): x^2 + y^2 + z^2 < 25\}$; (c) $(-\infty, \ln(25))$.

1Step 1: Evaluate the Function at Given Values

To evaluate $g(2,-2,4)$, we first substitute $x=2$, $y=-2$, and $z=4$ into the function. Thus, we have: \[g(2,-2,4) = \ln (25 - 2^2 - (-2)^2 - 4^2)\]This simplifies to: \[g(2,-2,4) = \ln (25 - 4 - 4 - 16)\]\[g(2,-2,4) = \ln (1)\]Since the natural logarithm of 1 is 0, \[g(2,-2,4) = 0\].

2Step 2: Determine the Domain of g

The domain of the function $g(x, y, z)$ is determined by the expression inside the logarithm being positive:\[25 - x^2 - y^2 - z^2 > 0\]This implies:\[x^2 + y^2 + z^2 < 25\].Therefore, the domain of $g$ is all $x, y, z$ such that the sum of their squares is less than 25. This can be described as the interior of a sphere centered at the origin with radius 5.

3Step 3: Determine the Range of g

The function $g(x, y, z) = \ln(25 - x^2 - y^2 - z^2)$ takes values depending on the value inside the log function.- The minimum value inside the logarithm occurs when $x^2 + y^2 + z^2$ approaches but is less than 25. In this case, the logarithm approaches $\ln(0^+)$, which tends to $-\infty$.- The maximum value occurs when $x^2 + y^2 + z^2 = 0$, leading to $\ln(25)$.Thus, the range of $g$ is the interval $( -\infty, \ln(25) )$.

Key Concepts

Natural LogarithmDomain of a FunctionRange of a FunctionSphere in Three Dimensions

Natural Logarithm

The natural logarithm, often denoted as $ \ln(x) $, represents the logarithm to the base $ e $, where $ e $ is an irrational constant approximately equal to 2.71828. In calculus, the natural logarithm is essential due to its unique properties with respect to differentiation and integration. The function $ e^x $ and its inverse, $ \ln(x) $, are deeply linked, making them a staple in many areas of mathematics and science.
This logarithmic function is only defined for positive values of $ x $. Consequently, when working with $ \ln(x) $, it's crucial to ensure that the input or argument is greater than zero. This property stems from the fact that raising a positive number to any exponent will result in a positive number, mirroring the domain of the natural logarithm function.

The graph of $ y = \ln(x) $ is increasing, continuous, and passes through the point $ (1,0) $ since $ \ln(1) = 0 $.
As $ x $ approaches 0 from the positive side, $ \ln(x) $ tends to $ -\infty $.

In multivariable calculus contexts, like the function $ g(x, y, z) $, understanding the domain of the logarithm is integral when involving expressions like $ \ln(25 - x^2 - y^2 - z^2) $.

Domain of a Function

When determining the domain of a function, we identify all the possible input values, often referred to as the set of all feasible $ x $ values for which the function is defined. For the function $ g(x, y, z) = \ln(25-x^2-y^2-z^2) $, we must scrutinize the expression inside the logarithm.
This expression, $ 25 - x^2 - y^2 - z^2 $, must remain positive for the natural logarithm to be defined. Hence, it must satisfy:

$ 25 - x^2 - y^2 - z^2 > 0 $
or equivalently, $ x^2 + y^2 + z^2 < 25 $.

This condition signifies that the point $ (x, y, z) $ lies within the interior of a sphere centered at the origin with a radius of 5 in three-dimensional space. Thus, the domain of $ g $ includes all points strictly inside this sphere, excluding points on its surface or beyond.

Range of a Function

The range of a function comprises all the potential output values that the function can produce. For $ g(x, y, z) = \ln(25-x^2-y^2-z^2) $, the range is determined by how the expression within the logarithm fluctuates for permissible $ x, y, $ and $ z $.

The maximum value for $ 25 - x^2 - y^2 - z^2 $ occurs when $ x^2 + y^2 + z^2 = 0 $, meaning $ x, y, z $ are all zero. This gives $ g(0, 0, 0) = \ln(25) $.
The minimum occurs as $ x^2 + y^2 + z^2 \to 25 $, leading $ 25 - x^2 - y^2 - z^2 \to 0^+ $, which makes $ g \to -\infty $.

Hence, the range of $ g $ is $( -\infty, \ln(25) )$. This reflects the output values as $ (x, y, z) $ vary within the domain.

Sphere in Three Dimensions

A sphere in three-dimensional space is a set of all points equidistant from a central point. The formula $ x^2 + y^2 + z^2 = r^2 $ describes a sphere of radius $ r $ centered at the origin $ (0,0,0) $.
In the context of the function $ g(x, y, z) = \ln(25-x^2-y^2-z^2) $, the condition $ x^2 + y^2 + z^2 < 25 $ suggests that $ (x, y, z) $ lies inside such a sphere with radius 5. This sphere's surface marks the boundary of the domain of $ g $, but the domain itself includes all internal points.

The radius determines the sphere's size, with all points $ 5 $ units away from the center forming its boundary.
These criteria are useful when exploring geometrical properties relating to volume and surface area in calculus or physics.

Understanding spheres assists in visualizing and solving problems in multivariable calculus that involve domains restricted to spherical regions.

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