Problem 403
Question
For the following exercises, find the largest interval of continuity for the function. \(g(x, y)=\ln \left(4-x^{2}-y^{2}\right)\)
Step-by-Step Solution
Verified Answer
The largest interval of continuity is the interior of the circle: \( x^2 + y^2 < 4 \).
1Step 1: Understanding the Function
We are given the function \( g(x, y) = \ln(4 - x^2 - y^2) \). This is a natural logarithmic function. For the logarithm to be defined, its argument must be greater than zero.
2Step 2: Setting the Inequality Constraint
Since the argument of the logarithm is \( 4 - x^2 - y^2 \), we need \( 4 - x^2 - y^2 > 0 \). Rearranging this inequality yields \( x^2 + y^2 < 4 \). This represents a circular region.
3Step 3: Geometrical Interpretation
The inequality \( x^2 + y^2 < 4 \) describes the interior of a circle with center at the origin (0,0) in the \(xy\)-plane and a radius of 2. The boundary, where \( x^2 + y^2 = 4 \), is not included because the logarithm is undefined at zero.
4Step 4: Determine the Interval of Continuity
Since the natural logarithm is continuous wherever it is defined, and given \( x^2 + y^2 < 4 \) defines the region of continuity, the largest interval of continuity for \( g(x, y) \) is the interior of the circle with radius 2, centered at the origin.
Key Concepts
Natural Logarithmic FunctionInequality ConstraintCircular RegionInterval of Continuity
Natural Logarithmic Function
The natural logarithmic function is expressed using the symbol \( \ln(x) \) which refers to the logarithm to the base \( e \), where \( e \approx 2.71828 \), a mathematical constant. Natural logs are prevalent in math because of their interesting properties and applications.
For a function like \( g(x, y) = \ln(4 - x^2 - y^2) \), the role of the logarithm is to transform the expression \( 4 - x^2 - y^2 \). Functions involving natural logs are only defined for input values that are strictly greater than zero.
Whenever we see expressions with natural logs, they have undefined regions where the inside of the log becomes zero or negative. By solving the inequality \( 4 - x^2 - y^2 > 0 \), we ensure that the function remains within a defined region, which is crucial for continuity and practical applications of such functions.
Using logs can simplify expressions and solve problems involving exponential growth or decay, making them invaluable in various scientific fields.
For a function like \( g(x, y) = \ln(4 - x^2 - y^2) \), the role of the logarithm is to transform the expression \( 4 - x^2 - y^2 \). Functions involving natural logs are only defined for input values that are strictly greater than zero.
Whenever we see expressions with natural logs, they have undefined regions where the inside of the log becomes zero or negative. By solving the inequality \( 4 - x^2 - y^2 > 0 \), we ensure that the function remains within a defined region, which is crucial for continuity and practical applications of such functions.
Using logs can simplify expressions and solve problems involving exponential growth or decay, making them invaluable in various scientific fields.
Inequality Constraint
Inequality constraints are used in mathematics to limit the values that variables can take on. When we see an inequality like \( 4 - x^2 - y^2 > 0 \), it means we're looking for \( x \) and \( y \) values that satisfy that condition.
Such constraints are necessary for ensuring that functions like \( \ln(4 - x^2 - y^2) \) are defined. Without them, we might try to take the log of zero or a negative number, which is not possible in real number mathematics.
By rearranging \( 4 - x^2 - y^2 > 0 \) to \( x^2 + y^2 < 4 \), we clearly see the valid region of \( (x, y) \) points. Inequality constraints form the basis of defining feasible regions in mathematics, essential for optimization problems, ensuring solutions are grounded in reality.
Such constraints are necessary for ensuring that functions like \( \ln(4 - x^2 - y^2) \) are defined. Without them, we might try to take the log of zero or a negative number, which is not possible in real number mathematics.
By rearranging \( 4 - x^2 - y^2 > 0 \) to \( x^2 + y^2 < 4 \), we clearly see the valid region of \( (x, y) \) points. Inequality constraints form the basis of defining feasible regions in mathematics, essential for optimization problems, ensuring solutions are grounded in reality.
Circular Region
The concept of a circular region arises when dealing with inequalities such as \( x^2 + y^2 < 4 \). This inequality represents a circle on a Cartesian plane with a few specific characteristics:
This circle helps visualize the set of all points \( (x, y) \) that fall within it, each satisfying \( x^2 + y^2 < 4 \).
Graphically, the entire shaded region inside the circle boundary marks where \( g(x, y) = \ln(4 - x^2 - y^2) \) is valid. The edge of the circle isn't included because \( x^2 + y^2 = 4 \) makes \( 4 - x^2 - y^2 = 0 \), where the logarithm becomes undefined.
Overall, understanding circular regions enriches our grasp of spatial areas in math, valuable for solving geometric and physical problems alike.
- Center: Origin \((0,0)\)
- Radius: 2
- Type: Open circle (excludes boundary)
This circle helps visualize the set of all points \( (x, y) \) that fall within it, each satisfying \( x^2 + y^2 < 4 \).
Graphically, the entire shaded region inside the circle boundary marks where \( g(x, y) = \ln(4 - x^2 - y^2) \) is valid. The edge of the circle isn't included because \( x^2 + y^2 = 4 \) makes \( 4 - x^2 - y^2 = 0 \), where the logarithm becomes undefined.
Overall, understanding circular regions enriches our grasp of spatial areas in math, valuable for solving geometric and physical problems alike.
Interval of Continuity
The interval of continuity for a function is the set of all inputs where the function is continuous and well-defined. For \( g(x, y) = \ln(4 - x^2 - y^2) \), this is dictated by the inequality \( x^2 + y^2 < 4 \).
An important fact about the natural logarithm is that it is continuous wherever it is defined, providing the function is neither zero nor negative inside the log. Thus, the region \( x^2 + y^2 < 4 \) becomes our domain of continuity.
In practical terms, the interval of continuity is the two-dimensional region within the circle of radius 2 on the plane. It excludes the circle’s boundary to ensure \( 4 - x^2 - y^2 > 0 \) at all times.
This concept is crucial in calculus, real analysis, and applied problems, ensuring that mathematical operations within such an interval are smooth and predictable, allowing for accurate modeling and solution derivation.
An important fact about the natural logarithm is that it is continuous wherever it is defined, providing the function is neither zero nor negative inside the log. Thus, the region \( x^2 + y^2 < 4 \) becomes our domain of continuity.
In practical terms, the interval of continuity is the two-dimensional region within the circle of radius 2 on the plane. It excludes the circle’s boundary to ensure \( 4 - x^2 - y^2 > 0 \) at all times.
This concept is crucial in calculus, real analysis, and applied problems, ensuring that mathematical operations within such an interval are smooth and predictable, allowing for accurate modeling and solution derivation.
Other exercises in this chapter
Problem 401
Evaluate the following limits, if they exist. If they do not exist, prove it. \(\lim _{(x, y) \rightarrow(0,0)} \frac{4 x y}{x-2 y^{2}}\)
View solution Problem 402
For the following exercises, find the largest interval of continuity for the function. \(f(x, y)=x^{3} \sin ^{-1}(y)\)
View solution Problem 404
For the following exercises, find all first partial derivatives. \(f(x, y)=\sqrt{x^{2}-y^{2}}\)
View solution Problem 405
For the following exercises, find all first partial derivatives. \(u(x, y)=x^{4}-3 x y+1, x=2 t, y=t^{3}\)
View solution