Problem 6
Question
Find the domain of the function. \(g(x, y)=\sqrt{25-x^{2}-y^{2}}\)
Step-by-Step Solution
Verified Answer
The domain is a circle with radius 5 centered at the origin: \( x^2 + y^2 \leq 25 \).
1Step 1: Understand the Function
The given function is a square root function, \( g(x, y) = \sqrt{25 - x^2 - y^2} \). The expression under the square root must be non-negative for the square root to be defined.
2Step 2: Set Condition for Domain
Identify the condition for the expression to be valid under the square root. We need to have:\[ 25 - x^2 - y^2 \geq 0 \]This condition must hold true for real function values to exist.
3Step 3: Rearrange the Inequality
Rearrange the inequality to find the region of acceptability in the \(xy\)-plane:\[ x^2 + y^2 \leq 25 \]
4Step 4: Interpret the Inequality Geometrically
The inequality \( x^2 + y^2 \leq 25 \) represents a circle in the \(xy\)-plane with a radius of 5, centered at the origin \((0,0)\). The domain will thus include all points inside and on this circle.
5Step 5: Conclusion on the Domain
The domain of the function \( g(x, y) \) includes all the \((x, y)\) coordinate pairs such that they lie on or inside the circle of radius 5 centered at the origin.
Key Concepts
Square root functionInequalityCircle in the xy-plane
Square root function
Square root functions involve expressions where the square root of some variable or expression is taken. In the function \( g(x, y) = \sqrt{25 - x^2 - y^2} \), the variable components \( x \) and \( y \) are under the square root sign.
The critical aspect of a square root function is determining the values for which the expression under the square root is defined. This requirement demands that any expression under the square root must be non-negative.
For example:
The critical aspect of a square root function is determining the values for which the expression under the square root is defined. This requirement demands that any expression under the square root must be non-negative.
For example:
- In \( g(x, y) = \sqrt{25 - x^2 - y^2} \), the term \( 25 - x^2 - y^2 \) must be greater than or equal to zero for the square root result to be a real number.
- If the expression under the root were negative, the result would not be real but rather complex, which is often outside the scope of the problem.
Inequality
Inequalities are mathematical expressions that show the relationship of one expression to another, signified by symbols such as \( \geq, \leq, >, \text{and} < \).
For the function \( g(x, y) = \sqrt{25 - x^2 - y^2} \), an inequality forms when we set the requirement for the expression under the square root to be non-negative.
Let's see how it works:
These skills allow you to characterize and visualize domains in graphical form, as seen in the circle represented within the context of this problem.
For the function \( g(x, y) = \sqrt{25 - x^2 - y^2} \), an inequality forms when we set the requirement for the expression under the square root to be non-negative.
Let's see how it works:
- The inequality \( 25 - x^2 - y^2 \geq 0 \) must hold for the domain of \( g(x, y) \) to be valid.
- This simplifies to \( x^2 + y^2 \leq 25 \), indicating an area where the function maintains real values.
These skills allow you to characterize and visualize domains in graphical form, as seen in the circle represented within the context of this problem.
Circle in the xy-plane
A circle in the \( xy \)-plane is a set of points equidistant from a common point called the center. Its general equation is \( x^2 + y^2 = r^2 \) where \( r \) is the radius.
In the function \( g(x, y) = \sqrt{25 - x^2 - y^2} \), the inequality \( x^2 + y^2 \leq 25 \) describes a circle with a radius of 5 centered at the origin \((0,0)\).
This can be broken down as follows:
Visual representation aids in comprehension, providing a clear and concrete interpretation of potentially complex algebraic expressions.
In the function \( g(x, y) = \sqrt{25 - x^2 - y^2} \), the inequality \( x^2 + y^2 \leq 25 \) describes a circle with a radius of 5 centered at the origin \((0,0)\).
This can be broken down as follows:
- All points \((x, y)\) satisfying \( x^2 + y^2 \leq 25 \) lie inside or on the boundary of the circle.
- The circle equation \( x^2 + y^2 = 25 \) defines the boundary or edge of the circle.
Visual representation aids in comprehension, providing a clear and concrete interpretation of potentially complex algebraic expressions.
Other exercises in this chapter
Problem 6
Find the first partial derivatives of the function. $$ f(x, y)=\sqrt{x^{2}+y^{2}} $$
View solution Problem 6
Evaluate the limit. $$ \lim _{x, y) \rightarrow(-1,1)} \frac{x^{2}+2 x y^{2}+y^{4}}{1+y^{2}} $$
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Find the minimum value of \(f\) subject to the given constraint. In each case assume that the minimum value exists. $$ f(x, y)=4 x^{2}+y^{3}+3 y+7 ; 2 x^{2}+\fr
View solution Problem 7
Find all critical points. Determine whether each critical point yields a relative maximum value, a relative minimum value, or a saddle point. $$ f(x, y)=x^{2} y
View solution