Problem 19 - The Calculus with Analytic Geometry

Question

\(f(x, y, z)=x-2 y+z^{2} ; P(3,1,-2), Q(10,7,4)\)

Step-by-Step Solution

Answer

f(3, 1, -2) = 5 and f(10, 7, 4) = 12.

Step 1 - Evaluate f at Point P

Substitute the coordinates of point P into the function. For point P(3,1,-2), the function becomes:t(3, 1, -2)=3-2(1)+(-2)^{2}Simplify the expression:3 - 2 + 4 = 5So, f(3, 1, -2) = 5.

Step 2 - Evaluate f at Point Q

Substitute the coordinates of point Q into the function. For point Q(10,7,4), the function becomes:f(10, 7, 4)=10-2(7)+4^{2}Simplify the expression:10 - 14 + 16 = 12So, f(10, 7, 4) = 12.

Step 3 - Finalize the Answer

Now that both points are evaluated, summarize the results:f(3, 1, -2) = 5f(10, 7, 4) = 12

Key Concepts

Evaluating Functions

Evaluating a function means finding the value of the function at specific points. In our original exercise, we need to evaluate the function \(f(x, y, z)=x-2y+z^2\) at two different points: P(3,1,-2) and Q(10,7,4). To do this, we follow these steps:

• Substitute the coordinates of the point into the function.
• Simplify the resulting expression.

For point P(3,1,-2), we substitute as follows:
\(f(3, 1, -2) = 3 - 2(1) + (-2)^2\)
Next, simplify:
\(3 - 2 + 4 = 5\)
So, \(f(3, 1, -2) = 5\).
For point Q(10,7,4), we substitute as follows:
\(f(10, 7, 4) = 10 - 2(7) + 4^2\)
Next, simplify:
\(10 - 14 + 16 = 12\)
So, \(f(10, 7, 4) = 12\).
Evaluating functions is crucial in multivariable calculus, as it allows you to understand how the function behaves at different points in space.

Coordinate Substitution

Coordinate substitution is a technique used to evaluate functions at specific points. In our case, we simply replace the variables \(x\), \(y\), and \(z\) with the coordinates of the points provided.
For example, if we have the point P(3,1,-2) and the function \(f(x, y, z)=x-2y+z^2\), we substitute \(x = 3\), \(y = 1\), and \(z = -2\) into the function.

So, the function becomes:
\(f(3, 1, -2) = 3 - 2(1) + (-2)^2\)
This technique is fundamental because it converts an abstract function into a concrete numerical value. This makes it easier to understand and analyze the function's behavior at different points. Once you perform the substitution, the next step is simplification.

Simplification

After substituting coordinates into the function, the next step is simplification. Simplification makes complex expressions easier to understand and helps in obtaining the final value of the function at a given point.

Let's look at the simplification process for point P(3,1,-2) in our example:
First, we substitute the coordinates:
\(f(3, 1, -2) = 3 - 2(1) + (-2)^2\)
Next, we perform the multiplications and powers:
\(-2 \times 1 = -2\) and \((-2)^2 = 4\)
Now, combine everything:
\(3 - 2 + 4 = 5\)
So, \(f(3, 1, -2) = 5\).
Simplification ensures that you get a clear, concise numerical value, making further analysis easier. It's a vital step in solving multivariable calculus problems and helps in verifying the correctness of your solution.