Problem 19 - The Calculus with Analytic Geometry

Question

In Exercises 15 through 20 , find the domain and range of the function \(f\) and draw a sketch of the graph. \(f(x, y)=\sqrt{10-x-y^{2}}\)

Step-by-Step Solution

Answer

Domain: \(x + y^{2} \leq 10\), Range: \[0, \sqrt{10}\]

Step 1 - Identify the function

Given the function: $$f(x, y) = \sqrt{10 - x - y^{2}}$$

Step 2 - Determine the domain

The function $$\sqrt{10 - x - y^{2}}$$ must have a non-negative argument under the square root. This means: $$10 - x - y^{2} \geq 0$$ which implies $$ x + y^{2} \leq 10$$. So the domain consists of all \(x, y\) pairs such that $$x + y^{2} \leq 10$$.

Step 3 - Determine the range

The values that $$\sqrt{10 - x - y^{2}}$$ can produce are non-negative and depend on the value inside the square root. Since $$10 - x - y^{2}$$ ranges from 0 to 10, \ $$\sqrt{10 - x - y^{2}}$$ ranges from 0 to \sqrt{10}.

Step 4 - Sketch the graph

1. Draw the region of the domain: This is the set of all points \(x, y\) such that $$x \leq 10 - y^{2}$$.2. Draw the curve \(x + y^{2} = 10\), which represents the boundary of the domain.3. Remember the function's values range from 0 to \(\sqrt{10}\), so you would sketch an upper surface starting at z=0 and rising up to \(\sqrt{10}\).

Key Concepts

Multivariable Functions

A multivariable function is a function that depends on more than one input. In this case, the function is given by \(\f(x, y) = \sqrt{10 - x - y^2}\), and it takes two inputs, x and y. These types of functions are common in calculus and can describe surfaces in three-dimensional space. Understanding multivariable functions is key to visualizing relationships between more than two quantities. When working with such functions, you need to evaluate how each input affects the outcome, or output, of the function. For the given function, both x and y together determine the value under the square root, which then determines the output. The domain and range of a multivariable function take into account all possible pairs of inputs (in this case, (x, y)) and the corresponding output values. This means you need to consider constraints and feasible input regions carefully.

Square Root Function

The square root function \(\sqrt{u}\) is one of the most fundamental functions in mathematics and only produces non-negative outputs for real input values because √u represents the principal (non-negative) square root of u. When dealing with the function \(\f(x, y) = \sqrt{10 - x - y^2} \) the expression inside the square root, \(10 - x - y^2\), must be non-negative. This means \(10 - x - y^2 \geq 0\) which rearranges to \(x + y^2 \leq 10\). Understanding how to work with square roots in the context of another function is essential as it impacts the domain. In this function, anything within the square root must be zero or positive to produce real-number results. Thus, the domain is determined by ensuring the expression under the square root does not go negative. The output or range of the function will be all possible values that the square root can take, here ranging from 0 to \(\sqrt{10}\).

Graphing Functions

Graphing functions, particularly multivariable ones, helps in visualizing complex relationships between variables. To graph the given function, \(f(x, y) = \sqrt{10 - x - y^2}\), follow a few steps:

• Determine the Domain: This is where \(x + y^2 \leq 10\). Plot this inequality to get the region in the x-y plane.
• Sketch the Boundary: Draw the curve representing \(x + y^2 = 10\). This provides the edge of the domain.
• Plot the Function: Since the function outputs values from 0 to \(\sqrt{10}\), visualize the surface rising from z = 0 to z = \(\sqrt{10}\) over the domain.

When graphing, remember to keep in mind the constraints on the variables. A proper graph will help you understand the complete behavior of the function in its domain, and you will see how changes in x and y affect the function's value. Visualizing graphs of multivariable functions aids in comprehending the overall shape and surface defined by the function, enhancing spatial reasoning and analytical skills.