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Chapter 10: Problem 12

Evaluate each function at the given point. \(g\left(x_{1}, x_{2}, x_{3}, x_{4}\right)=x_{1} x_{4} \sqrt{x_{2} x_{3}}\) at \((1,8,2,-1)\)

Short Answer

Expert verified
The function evaluates to -4.

Step by step solution

01

Understand the function

The function given is a multivariable function: \(g(x_1, x_2, x_3, x_4) = x_1 x_4 \sqrt{x_2 x_3}\). It involves four variables \(x_1, x_2, x_3,\) and \(x_4\). The expression includes a product of these variables and the square root of their product.
02

Identify each variable

Here, you need to substitute the values \(x_1 = 1\), \(x_2 = 8\), \(x_3 = 2\), and \(x_4 = -1\) into the function. These are the specific inputs given for this exercise.
03

Substitute and simplify under the square root

Calculate the expression under the square root: \(\sqrt{x_2 x_3} = \sqrt{8 \times 2}\). First, compute the product inside the root: \(8 \times 2 = 16\). Then, find the square root: \(\sqrt{16} = 4\).
04

Evaluate the entire function

Now substitute all values into the function: \(g(x_1, x_2, x_3, x_4) = 1 \times (-1) \times 4\). Compute the product: \(1 \times (-1) \times 4 = -4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Evaluate Functions
Evaluating a function means determining its output when specific values are assigned to its variables. In this case, we're looking at the function \(g(x_1, x_2, x_3, x_4) = x_1 x_4 \sqrt{x_2 x_3}\). To evaluate this function at a specific point, such as \((1,8,2,-1)\), follow these steps:
  • Understand the structure of the function and what it represents.
  • Identify which values correspond to each variable. Here, \(x_1 = 1\), \(x_2 = 8\), \(x_3 = 2\), and \(x_4 = -1\).
  • Substitute these values into the function and simplify the expression.
  • Perform operations such as multiplication, division, and finding square roots as required.
  • The final outcome is your evaluated function value.
Performing these evaluations helps you understand the behavior of the function with different input values, which is crucial in multivariable calculus.
Multivariable Functions
A multivariable function is where more than one input variable affects the output. Here, we work with the function \(g(x_1, x_2, x_3, x_4)\). Such functions are common in calculus involving several dimensions. They help model real-world issues more accurately as most phenomena depend on multiple factors. Some key points about multivariable functions include:
  • They can have countless applications, from engineering problems to economics and physics.
  • Each variable in the function holds a unique value or dimension, influencing the overall output.
  • Understanding these functions requires handling multiple inputs simultaneously and seeing how they interact in the function.
  • These functions often produce surfaces or curves when graphed, offering visual insights into their behavior.
Studying multivariable functions allows deeper control and understanding of complex systems.
Functions of Several Variables
Functions of several variables, like the one we examined \(g(x_1, x_2, x_3, x_4)\), involve more than one independent variable which influences the dependent outcome. Unlike single-variable functions that involve one-dimensional lines, these functions create more complex surfaces.Being adept in dealing with several variables requires keen skill in
  • Spotting the role of each variable and how it contributes to the function's output.
  • Evaluating the combined effect of these variables, such as using multiplication and operations like square roots in our example.
  • Visualizing how changes in one variable influence the result, offering a better grip on dynamic systems.
  • Often, it's essential to understand the concept of partial derivatives, which show how the function changes as one of the inputs changes, holding others constant.
These concepts are crucial in calculus as they provide details necessary for analyzing systems with multi-faceted interactions.

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