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Function evaluation means finding the value of a function for specific values of its variables. This involves substituting these values into the function and then simplifying the expression. In our example, the function is given as \[f(x, y) = x^2 - 3xy - y^2 \] To evaluate this function at certain points like \(f(5,0)\) and \(f(5,-2)\), follow the steps below:

• Substitute the given values for \(x\) and \(y\) into the function.
• Simplify the resulting expression step-by-step

For instance, for \(f(5,0)\), we substitute \(x = 5\) and \(y = 0\): \[ f(5,0) = 5^2 - 3(5)(0) - 0^2 = 25 \] The same logic applies for \(f(5,-2)\) by substituting \(x = 5\) and \(y = -2\): \[ f(5,-2) = 5^2 - 3(5)(-2) - (-2)^2 = 25 + 30 - 4 = 51 \] Evaluating functions is essential in multivariable calculus as it forms the foundation of more complex computations.

The substitution method involves replacing variables with specific values or expressions to simplify calculations or solve equations. Here’s how it’s done in our function: \[ f(x, y) = x^2 - 3xy - y^2 \] When computing \(f(5,0)\):

• Substitute \(x = 5\) and \(y = 0\) into the function
• \[ f(5, 0) = 5^2 - 3(5)(0) - 0^2 \] = 25

Similarly, for \(f(5, -2)\):

• Substitute \(x = 5\) and \(y = -2\)
• \[ f(5, -2) = 5^2 - 3(5)(-2) - (-2)^2 \] = 51

Lastly, for a general evaluation at \(x = a\) and \(y = b\): \[ f(a, b) = a^2 - 3ab - b^2 \] This method is widely used in solving equations, integrals, and in various mathematical scenarios where direct computation might be complex.

A polynomial function consists of terms that are non-negative integer powers of variables. In this example, \[ f(x, y) = x^2 - 3xy - y^2 \] is a polynomial function because it involves terms that are sums of powers of \(x\) and \(y\). Key characteristics of polynomial functions:

• Polynomials are made up of monomials (terms like \(x^2\), \(-3xy\), and \(-y^2\)) which are added or subtracted.
• They are defined for all real numbers
• They exhibit smooth and continuous curves when graphed

Polynomial functions feature prominently in calculus due to their straightforward derivative and integral properties. They simplify the analysis and provide a simpler framework for understanding complex mathematical concepts. The above function \[ f(x, y) = x^2 - 3xy - y^2 \] showcases how polynomial functions can represent multivariable scenarios efficiently and effectively.