How do you evaluate F(0,4,5) given F(x,y,z)=11x^2 y^2 + 5z^2?

Substitute x=0, y=4, z=5 into the function: F(0,4,5) = 11(0)^2(4)^2 + 5(5)^2 = 0 + 525 = (0, 4, 5)

Is the value of F(0,4,5) a vector or a scalar?

F(0,4,5) is a scalar value, specifically 125.

What are the steps to calculate F(0,4,5) for the function F(x,y,z)?

Replace x=0, y=4, z=5 into F: F=110^24^2 + 55^2 = 0 + 125 = 125

What is the importance of substituting specific values into a function like F(x,y,z)?

It allows us to evaluate the function at particular points to find specific values or analyze behavior.

Let F(x,y,z)=11x 2y 2+5z 2. Calculate F(0,4,5). (Write Your Solution Using The Form (,,). Use Symbolic

Understanding the Function F(x, y, z) = 11x²y² + 5z²

In this article, we will explore how to evaluate the function F(x, y, z) = 11x²y² + 5z² at specific coordinate points. This task involves substituting given values for x, y, and z into the function and simplifying the expression step-by-step. Our primary goal is to compute F(0, 4, 5) and express the solution in the symbolic form ( , , ) — that is, as an ordered triple.

Step-by-Step Approach to Calculating F(0, 4, 5)

Understanding the Function and Its Components

The function F(x, y, z) = 11x²y² + 5z² consists of two main terms:

11x²y²: a product of squared variables x and y, scaled by 11

5z²: the square of z, scaled by 5

To evaluate this function at a specific point, we need to substitute the corresponding values of x, y, and z and perform the calculations accordingly.

Substituting the Values (0, 4, 5) into the Function

Step 1: Substitute x = 0, y = 4, z = 5

Replacing variables with their respective values gives:



F(0, 4, 5) = 11  (0)²  (4)² + 5  (5)²

Step 2: Simplify each term separately

First term: 11 0² 4²

Second term: 5 5²

Calculating each component:

0² = 0

4² = 16

5² = 25

Step 3: Plug in the simplified values



F(0, 4, 5) = 11  0  16 + 5  25

Step 4: Perform the multiplications

11 0 16 = 0

5 25 = 125

Step 5: Add the results



F(0, 4, 5) = 0 + 125 = 125

Expressing the Solution as an Ordered Triple

The problem statement requests the solution in the symbolic form ( , , ), which represents the point in space where the function is evaluated. Since the function's input is (x, y, z) and the output is a scalar, the ordered triple notation is used to represent the input point.

Therefore, the answer is:



(0, 4, 5)

Summary of the Calculation Process

Here's a quick summary of the steps involved:

Identify the function and the point of evaluation.

Substitute the given values into the function.

Simplify each term separately.

Calculate each squared term.

Perform the multiplications and addition.

Express the input point as an ordered triple ( , , ).

Additional Tips for Evaluating Multivariable Functions

1. Always double-check substitutions

Errors often occur during substitution, especially with signs or powers. Double-check each step to ensure accuracy.

2. Simplify step-wise

Break down complex expressions into smaller parts, simplifying each step to avoid mistakes.

3. Use parentheses to clarify order of operations

When writing expressions, parentheses help clarify the sequence of calculations, especially in nested operations.

4. Practice with different functions

Evaluating various functions enhances understanding and speed, especially when working with more complex expressions.

Conclusion

Calculating the value of a multivariable function at a given point involves systematic substitution and simplification. In our example, we evaluated F(0, 4, 5) by substituting the point's coordinates into the function and simplifying step-by-step, resulting in the value 125. The input point, expressed as an ordered triple, is (0, 4, 5), which accurately reflects the variables' values used in the calculation.

Mastering this process is essential for students and professionals working in calculus, physics, engineering, and related fields, where functions of multiple variables are commonplace. With practice, evaluating such functions becomes quick and intuitive, allowing for efficient analysis of complex systems and models.