91Ó°ÊÓ

Function evaluation is a fundamental process in mathematics where you find the value of a function for specific input values. When a function is given, such as \( f(x, y) = x^2 \sqrt{1 + xy} \), you can plug in different values for \( x \) and \( y \) to determine output values.
In our exercise, we substitute the pairs \((x, y)\) into the function to evaluate it. The pairs given are (3, 1), (1, 15), (0.3, 0.5), and (56, 4).
This involves calculating \( x^2 \) and then multiplying by the square root of \( (1 + xy) \).:

• For \( (3, 1) \), calculate \( 9\sqrt{4} \).
• For \( (1, 15) \), it becomes \( 1\sqrt{16} \).
• For \( (0.3, 0.5) \), we get approximately \( 0.0900\sqrt{1.15} \).
• For \( (56, 4) \), compute \( 3136\sqrt{225} \).

Using tools like graphing calculators or spreadsheets can help simplify these calculations by automating the arithmetic and algebraic manipulations.

Rounding decimals is an essential skill in both mathematics and everyday life. It is the process of trimming down a number to a certain number of significant digits for simplicity or clarity. For example, when mathematical results have long decimal expansions, we often round them to a desired precision.
In statistical and scientific applications, decimals are commonly rounded to four places for a balance of precision and simplicity. When rounding, identify the digit at the requested decimal place, look at the digit immediately after it, and apply rules for whether to round up or stay.
This is important in our table, where all function evaluation results must be rounded to four decimal places:

• In \( f(0.3,0.5) \), after computing, the result is approximately \( 0.102207 \). We round this down to \( 0.1022 \) since the next digit is less than 5.

Rounding helps to present the data concisely and ensures that information is easily interpretable.

Mathematical tables are structured presentations of data, often used to list and summarize large sets of numbers. They allow easy comparison and visualization of results for different input values. In mathematics, they often show values of functions or statistics.
In the context of the exercise, the table is used to display calculated values of \( f(x, y) \) for given pairs of \( x \) and \( y \). This permits a quick check of results and facilitates pattern recognition, if any:

• For each \( x, y \) pair, we fill the table entry after calculating \( f(x, y) \).
• Organization in a table simplifies finding errors or areas needing re-calculation.

Overall, using tables in mathematical computations aids in organizing complex computations and interpreting the results effectively.