91Ó°ÊÓ

In multivariable calculus, a function of several variables is a function that depends on two or more variables. For example, the function provided in this exercise is defined as follows:
\[ f(x, y) = x^2 - 3xy - y^2. \]
Here, the function depends on both the variable \(x\) and the variable \(y\). This is different from a single-variable function which depends on only one variable. It’s important to understand that each input pair \((x, y)\) maps to a single output, a real number.
This kind of function is typical in real-world scenarios where multiple factors (variables) affect the outcome. For instance:

• Height above sea level might depend on both the latitude and longitude.
• The temperature at a given point in a room can depend on both the point’s height and distance from a heat source.

The substitution method involves replacing variables in a function with specific values to evaluate the function. Let's apply this to the given function:
1. We identified the function:
\[ f(x, y) = x^2 - 3xy - y^2. \]
2. To find \( f(5,0) \), we substitute 5 for \( x \) and 0 for \( y \):
\[ f(5, 0) = 5^2 - 3(5)(0) - 0^2 = 25. \]
3. Similarly, to find \( f(5, -2) \), we substitute 5 for \( x \) and -2 for \( y \):
\[ f(5, -2) = 5^2 - 3(5)(-2) - (-2)^2 = 25 + 30 - 4 = 51. \]
4. For \( f(a, b) \), we simply substitute \( a \) for \( x \) and \( b \) for \( y \):
\[ f(a, b) = a^2 - 3ab - b^2. \]
This method helps to compute the values of the function for any given pairs of \(x\) and \(y\) quickly and accurately.

Evaluating a multivariable function means finding the output of the function for specific values of the input variables. Here’s a step-by-step understanding of this concept using our example:
1. **Determine the function:**
We start with the function \( f(x, y) = x^2 - 3xy - y^2. \) This tells us how each pair \((x, y)\) impacts the output.
2. **Choose values for the variables:**
To evaluate the function, we need specific values for \( x \) and \( y \). For instance, \( (5, 0) \), \( (5, -2) \), and \( (a, b) \).
3. **Substitute and simplify:**
After choosing the specific values, we plug them into the function and simplify to find the result:

• For \( f(5, 0) \):
  \[ f(5, 0) = 5^2 - 3(5)(0) - 0^2 = 25. \]
• For \( f(5, -2) \):
  \[ f(5, -2) = 5^2 - 3(5)(-2) - (-2)^2 = 25 + 30 - 4 = 51. \]
• For \( f(a, b) \):
  \[ f(a, b) = a^2 - 3ab - b^2. \]

This process ensures that we correctly evaluate the function for any given inputs, thereby understanding the behavior and result of the function for various values.