Problem 17 - Calculus for Biology and Medicine

Question

Find the range of each function \(f(x, y)\), when defined on the specified domain \(D\). \(f(x, y)=x^{2}+y^{2} ; D=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq 1\\}\)

Step-by-Step Solution

Answer

The range of the function is [0, 2].

Step 1: Understand the Function and Domain

The function given is \(f(x, y) = x^2 + y^2\). The domain \(D\) restricts \(x\) and \(y\) to values between 0 and 1, inclusive. This forms a square within the first quadrant of the xy-plane.

Step 2: Determine Function Values at Domain Boundaries

Evaluate the function at the boundary points within \(D\): - At \((0,0)\), \(f(0,0) = 0^2 + 0^2 = 0\), - At \((1,0)\) and \((0,1)\), \(f(1,0) = 1^2 + 0^2 = 1\) and \(f(0,1) = 0^2 + 1^2 = 1\), - Finally, at \((1,1)\), \(f(1,1) = 1^2 + 1^2 = 2\). The function reaches 2 only at this point.

Step 3: Explore Function Values within the Domain

Because \(f(x, y) = x^2 + y^2\) is a sum of squares, the smallest value within the domain is 0 (at (0,0)) and the sum increases as either \(x\) or \(y\) increases. Intermediate values are combinations like \(f(0.5, 0.5) = 0.5^2 + 0.5^2 = 0.5\), showing the continuous increase from 0 to 2.

Step 4: Conclude the Range

Given the explorations, \(f(x, y)\) smoothly increases from its minimum at the point (0,0) to its maximum at the point (1,1). Thus, the range of \(f(x, y)\) on domain \(D\) is \([0, 2]\).

Key Concepts

Multivariable Functions

When you encounter a multivariable function like \(f(x, y) = x^2 + y^2\), you are dealing with a function that has two input variables, \(x\) and \(y\). Unlike single-variable functions, these functions map points in a two-dimensional space to a single output value. Think of it as creating a surface: each point \((x, y)\) on the xy-plane corresponds to a height or value of the function. This height is given by the expression \(x^2 + y^2\).
When exploring multivariable functions, it's crucial to understand how changes in \(x\) and \(y\) affect the output. In our example, increasing either \(x\) or \(y\) will increase the function's value because we are adding more positive areas to the output. Since these functions operate in multiple dimensions, visualizing them can often involve plotting surfaces or using contour maps to describe levels of function values.

Domain Analysis

Domain analysis is a critical step when working with any function, especially multivariable ones. Here, the domain \(D\) for \(f(x, y) = x^2 + y^2\) is defined as \({(x, y): 0 \leq x \leq 1, 0 \leq y \leq 1}\). This means \(x\) and \(y\) can take any value between 0 and 1, inclusive.
This domain specification forms a square in the first quadrant of the xy-plane, bounded by the lines \(x=0\), \(x=1\), \(y=0\), and \(y=1\).

• To properly understand the range of the function within this domain, it's essential to examine how the function behaves along the edges of this square and within its interior.
• This involves determining the function's values at significant points, like corners, and understanding the behavior in between these points.
• Restricting the domain can limit the range of possible output values of the function, which is why analyzing the domain's constraints is so important in determining the range.

Boundary Evaluation

Evaluating the boundary of the domain is about understanding what happens at the edges of the area where the function is defined. For \(f(x, y)=x^2+y^2\), the critical boundary points to consider are where \(x\) or \(y\) meets its maximum and minimum allowed values within the domain.

• At the corner points, for instance, (0,0), (1,0), (0,1), and (1,1), the function reaches crucial values: 0, 1, and 2.
• Checking these boundary points helps confirm that the function does indeed cover all potential values from its minimum to its maximum as the input variables vary across their allowed range.
• Beyond just checking corners, it's important to understand how the function behaves along these edges (like where \(x\) or \(y\) holds constant), confirming that no potential function values are missed when we consider the values in between.

Thus, through boundary evaluation, we ensure that every possible value the function can take on this domain is accounted for, closing those possible endpoints into a complete range of \([0, 2]\).