Problem 6 - Calculus an Applied Approach

Question

Find the function values. $$ f(x, y, z)=\sqrt{x+y+z} $$ $$ \text { (a) } f(0,5,4) \quad \text { (b) } f(6,8,-3) $$

Step-by-Step Solution

Answer

(a) f(0,5,4) = 3 , (b) f(6,8,-3) = \( \sqrt{11}\)

Step 1: Calculate (a)

For f(0,5,4) substitute these values into the function to get:\( f(0,5,4)= \sqrt{0+5+4} = \sqrt{9} = 3\)

Step 2: Calculate (b)

For f(6,8,-3), again you substitute these values into the function to get: \( f(6,8,-3) = \sqrt{6+8-3} = \sqrt{11}\)

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics, especially when working with multivariable functions. This involves determining the output of a function based on the given input values. Let's explore this through an example.

Consider a function of three variables:

• \( f(x, y, z) = \sqrt{x+y+z} \)

Here, the inputs are \(x, y, z\), and you're asked to calculate the function's output using the specified input values.
By substituting specific numbers into \(x, y, z\) in the function, you can find the result. For instance, if

• \( f(x, y, z) \) is evaluated at \( (0, 5, 4) \), you substitute these values in to get \( f(0, 5, 4) = \sqrt{0+5+4} \).

Function evaluation helps you see how different inputs affect the outcome, which is particularly useful in analyzing multivariable functions and their behavior.

Square Roots

When you encounter a function involving square roots, it's important to understand the basics of how square roots work. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because \(3 \times 3 = 9\).

Square roots can also deal with non-perfect squares like 11. In these cases, the square root is not a whole number and is often represented as an approximate value or left as a square root symbol. For example:

• The square root of 11 is an irrational number; precisely, \( \sqrt{11} \) approximately equals 3.3166.
• When dealing with functions, especially those with multivariable inputs, understanding the role of square roots helps you interpret the output correctly.

Therefore, the function \( f(x, y, z) = \sqrt{x+y+z} \) simplifies computations using both integer and non-integer results.

Step-by-Step Solutions

Step-by-step solutions are incredibly useful for breaking down complex mathematical problems into manageable parts, ensuring each phase is understood before moving to the next. In evaluating multivariable functions, proceeding stepwise makes it easier to follow the process.

Let's look at how this can be applied effectively:

• Begin by identifying what the problem asks you to find. Here, you're supposed to evaluate the function \( f(x, y, z) \) at specific points.
• Next, perform the substitution of variables. For example, substitute \(x, y, z\) with\( 0, 5, 4 \) respectively to find \( f(0, 5, 4) \).
• Once values are substituted, simplify the expression: \( \sqrt{0 + 5 + 4} = \sqrt{9} \), which equals 3.
• Repeat similar steps for other given sets of variables, such as \( (6, 8, -3) \), simplifying to \( \sqrt{11} \).

Following a step-by-step approach ensures clarity and helps avoid mistakes by focusing on one small part of the problem at a time. This also makes reviewing and checking work much simpler.