Problem 7 - Calculus and its applications

Question

For \(f(x, y, z)=x^{2}-y^{2}+z^{2},\) find \(f(-1,2,3)\) and \(f(2,-1,3)\).

Step-by-Step Solution

Answer

For \((-1, 2, 3)\), \(f(-1, 2, 3) = 6\). For \((2, -1, 3)\), \(f(2, -1, 3) = 12\).

Step 1: Understand the Function

The given function is \( f(x, y, z) = x^2 - y^2 + z^2 \). This means you need to square the value of \(x\), subtract the square of \(y\), and then add the square of \(z\).

Step 2: Substitute the First Set of Values into the Function

First, substitute \((-1, 2, 3)\) into the function: \[f(-1, 2, 3) = (-1)^2 - (2)^2 + (3)^2\]

Step 3: Compute the Squares and Simplify for the First Set

Calculate each term: - \((-1)^2 = 1\)- \((2)^2 = 4\)- \((3)^2 = 9\)Substitute back into the function: \[f(-1, 2, 3) = 1 - 4 + 9 = 6\]

Step 4: Substitute the Second Set of Values into the Function

Now, substitute \((2, -1, 3)\) into the function: \[f(2, -1, 3) = (2)^2 - (-1)^2 + (3)^2\]

Step 5: Compute the Squares and Simplify for the Second Set

Calculate each term: - \((2)^2 = 4\)- \((-1)^2 = 1\)- \((3)^2 = 9\)Substitute back into the function: \[f(2, -1, 3) = 4 - 1 + 9 = 12\]

Key Concepts

Function Evaluation

In the realm of multivariable functions, function evaluation is a critical skill. When given a specific function and a set of input values, your task is to compute the output. In our example, consider the function \( f(x, y, z) = x^2 - y^2 + z^2 \). To evaluate the function at specific inputs, you substitute the given values for each variable.

For instance:

• For the point \((-1, 2, 3)\), replace \(x\) with \(-1\), \(y\) with \(2\), and \(z\) with \(3\) resulting in the expression \((-1)^2 - (2)^2 + (3)^2\).
• For the point \((2, -1, 3)\), substitute \(x = 2\), \(y = -1\), and \(z = 3\) resulting in \((2)^2 - (-1)^2 + (3)^2\).

By substituting these values, we transform the general function notation into specific numerical calculations. This process is vital for both understanding the function's behavior and for practical applications in engineering, economics, and science.

Algebraic Manipulation

Once the specific values are substituted into the function, the next step involves algebraic manipulation. This process requires basic arithmetic operations—primarily addition, subtraction, and squaring numbers in this case.

Let’s consider the first evaluation:

• Calculate each square: \((-1)^2\) becomes \(1\).
• \((2)^2\) leads to \(4\).
• \((3)^2\) equals \(9\).

Taking these results, the function simplification follows:

• Combine as \(1 - 4 + 9\), which computes to \(6\).

In the second evaluation, you perform similar operations:

• For \(2^2\), you get \(4\).
• \((-1)^2\) gives \(1\).
• \(3^2\) results in \(9\).

Now, combine the outcomes: \(4 - 1 + 9 = 12\).

This manipulation not only reinforces basic arithmetic skills but also enables deeper understanding of how different parts of a function interact.

Coordinate Substitution

Coordinate substitution is a straightforward concept that is crucial when dealing with functions of multiple variables. It involves taking coordinates from a multi-dimensional space and applying them to a function’s variables.

Understanding the origin of these coordinates is essential. Typically, they represent values from a graph, data from an experiment, or specific points of interest in a study. For any given function like \( f(x, y, z) \), each point defined by coordinates \((x, y, z)\) translates to real numbers substituted in place of the variables.

• In our example, coordinates \((-1, 2, 3)\) are selected to represent one point in a three-dimensional space.
• Coordinates \((2, -1, 3)\) present another distinct point.

This substitution helps in evaluating how changes in certain dimensions or variables affect the overall output of the function. It allows us to identify behavior patterns and draw conclusions about the function's performance under varying conditions. Appreciating this concept provides insight into the versatile applications of multivariable calculus in real-life scenarios.