Problem 14 - Calculus for Biology and Medicine

Question

Find the largest possible domain and the corresponding range of each function. Determine the equation of the level curves \(f(x, y)=c\), together with the possible values of \(c .\) \(f(x, y)=\sqrt{9-x^{2}-y^{2}}\)

Step-by-Step Solution

Answer

The domain of \( f(x, y) \) is the closed disk \( x^2 + y^2 \leq 9 \), range is \([0, 3]\), level curves are circles \( x^2 + y^2 = 9 - c^2 \) for \( 0 \leq c \leq 3 \).

Step 1: Understand the Function

The function given is \( f(x, y) = \sqrt{9 - x^2 - y^2} \). This is a two-variable function involving a square root. For the function to be real-valued, the expression inside the square root must be non-negative, i.e., \( 9 - x^2 - y^2 \geq 0 \).

Step 2: Find the Domain

The domain consists of all \( (x, y) \) points that satisfy \( 9 - x^2 - y^2 \geq 0 \). Simplifying this inequality gives \( x^2 + y^2 \leq 9 \). This represents a circle centered at the origin with a radius of 3. Thus, the domain is all \( (x, y) \) within and including this circle.

Step 3: Determine the Range

The expression \( \sqrt{9 - x^2 - y^2} \) represents the distance from the boundary of the circle to any point inside it, ranging from \( \sqrt{0} = 0 \) on the boundary to \( \sqrt{9} = 3 \) at the center of the circle. Therefore, the range of the function is \([0, 3]\).

Step 4: Identify Level Curves

A level curve of \( f(x, y) = c \) is given by setting \( \sqrt{9 - x^2 - y^2} = c \), which implies \( 9 - x^2 - y^2 = c^2 \). Rearranging gives \( x^2 + y^2 = 9 - c^2 \). This is another circle centered at the origin with radius \( \sqrt{9 - c^2} \).

Step 5: Determine Possible Values of \(c\)

For \( x^2 + y^2 = 9 - c^2 \) to describe real circles, \( 9 - c^2 \geq 0 \) must hold. Simplifying, \( c^2 \leq 9 \), which gives \( |c| \leq 3 \), or equivalently \( 0 \leq c \leq 3 \).

Key Concepts

Domain and Range

In multivariable calculus, understanding the domain and range of a function is crucial. The **domain** of a function is the set of all possible input values (typically represented as coordinates in two-variable functions) that result in real output values. For the function \( f(x, y) = \sqrt{9 - x^2 - y^2} \), the expression inside the square root must be non-negative to ensure real numbers. Thus, for \( 9 - x^2 - y^2 \geq 0 \), the domain is defined by the inequality \( x^2 + y^2 \leq 9 \). This inequality represents the interior of a circle centered at the origin with a radius of 3. In simpler terms, all points \((x, y)\) lying inside or on the perimeter of this circle are part of the domain. Now, turning to the **range**, it refers to the set of all possible output values. Given the function, the result \( \sqrt{9 - x^2 - y^2} \) varies from \(0\) to \(3\). The maximum value occurs at the center, where \( x^2 + y^2 = 0 \), resulting in \( \sqrt{9} = 3 \), while the minimum value, \( \sqrt{0} = 0 \), occurs on the circle's boundary.Therefore, the function's range is \([0, 3]\), capturing all possible heights the square root reaches as it moves within the domain.

Level Curves

Level curves, an essential concept in multivariable calculus, offer a way to visualize functions of two variables. They represent the set of input points where the output value is constant. For \( f(x, y) = \sqrt{9 - x^2 - y^2} \), finding level curves involves setting \( f(x, y) = c \) for various constants \(c\). This results in the equation \( \sqrt{9 - x^2 - y^2} = c \), leading to \( 9 - x^2 - y^2 = c^2 \). Simplifying further, we see \( x^2 + y^2 = 9 - c^2 \), defining a circle around the origin. Each level curve corresponds to a different \(c\), indicating a different circle radius. The radius is \( \sqrt{9 - c^2} \), demonstrating how the circle's size changes based on \(c\). These curves help visualize how the output varies around different parts of the plane. Understanding this concept allows students to better grasp how the function behaves overall.

Square Root Function

The square root function is fundamental in mathematics, appearing in diverse contexts. In \( f(x, y) = \sqrt{9 - x^2 - y^2} \), the square root dictates the function's behavior by limiting it to non-negative values. The requirement for non-negativity ensures the input to the square root is zero or positive, preventing complex numbers. This is a key consideration when determining the domain, as seen previously.Squares and square roots introduce unique characteristics to functions, such as smoothing transitions and emphasizing symmetry. Within the context of the given function, the square root impacts its range, resulting in limitations, such as \([0, 3]\). The graphical interpretation also showcases specific attributes like radial symmetry around the origin, due to the even powers of \(x\) and \(y\). Recognizing these features and understanding their implications paves the way for deeper comprehension of multivariable calculus concepts.