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Chapter 11: Problem 4

Let \(g(x, y, z)=x^{3} y^{2} z \sqrt{10-x-y-z}\) (a) Evaluate \(g(1,2,3)\) . (b) Find and describe the domain of \(g .\)

Short Answer

Expert verified
(a) 24; (b) Domain is all real numbers (x, y, z) such that \(x+y+z \leq 10\).

Step by step solution

01

Evaluate Specific Values

First, we need to substitute the values of \(x=1\), \(y=2\), and \(z=3\) into the function \(g(x, y, z)\). This gives us:\[g(1, 2, 3) = 1^{3} \cdot 2^{2} \cdot 3 \cdot \sqrt{10-1-2-3}\]
02

Simplify Exponents and Multiply

Calculate each part separately. Compute \(1^3 = 1\), \(2^2 = 4\), and multiply with \(3\):\[1 \cdot 4 \cdot 3 = 12\]
03

Calculate Inside the Square Root

Now, calculate the expression inside the square root:\[10 - 1 - 2 - 3 = 4\]
04

Square Root Calculation

Next, take the square root of the result:\[\sqrt{4} = 2\]
05

Final Multiplication

Finally, multiply the results obtained in previous steps:\[g(1, 2, 3) = 12 \cdot 2 = 24\]
06

Define Domains

Identify the constraints for the domain of the function given by the square root:The inside of the square root must be non-negative.\[10 - x - y - z \geq 0\]
07

Express Domain with Inequality

Rearrange to find the allowed set of values for \(x, y, z\):\[x + y + z \leq 10\]
08

Consider Variables Restrictions

Additionally, since the variables appear in the expression \(x^3y^2z\), all variables should be real numbers. There are inherently no restrictions beyond the interval mentioned except to ensure they remain real (excluding negatives in this context).

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Key Concepts

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

Domain of a Function
Understanding the domain of a multivariable function is crucial for identifying the set of all possible input values for which the function is defined. In the function \(g(x, y, z) = x^3 y^2 z \sqrt{10-x-y-z}\), the domain is determined by the condition inside the square root. This condition must be such that the expression inside remains non-negative:
  • This means \(10 - x - y - z \geq 0\), ensuring the square root is real and non-negative.
  • Therefore, we rearrange the inequality to \(x + y + z \leq 10\).
Additionally, all input variables \(x, y, z\) must be real numbers so that the operations are valid, especially given terms like \(x^3y^2z\). Keep these factors in mind to properly navigate the domain of this function.
Function Evaluation
Evaluating a multivariable function means finding its output by substituting specific values for its variables. We're given the function \(g(x, y, z)\) to evaluate at \((1, 2, 3)\). Here鈥檚 a simple walkthrough:
  • First, substitute the values: \(g(1, 2, 3) = 1^3 \cdot 2^2 \cdot 3 \cdot \sqrt{10-1-2-3}\).
  • Calculate each part step by step. \(1^3 = 1\) and \(2^2 = 4\), leading to \(1 \times 4 \times 3 = 12\).
  • Evaluate the square root expression by calculating \(10 - 1 - 2 - 3 = 4\), thus \(\sqrt{4} = 2\).
  • Finally, multiply the computed values: \(12 \times 2 = 24\).
Through this process, we find that the function's output at \((1, 2, 3)\) is 24, illustrating the power of systematic substitution and calculation.
Understanding the Square Root
The square root operation is fundamental in mathematics, especially in functions involving radicals. In the function \(g(x, y, z)\), the square root \(\sqrt{10-x-y-z}\) imposes a significant constraint. Here's why it matters:
  • The expression inside the square root, \(10-x-y-z\), must be non-negative for the square root to be defined in the real number system.
  • A negative value under the square root would yield an imaginary number, which is typically outside the scope for real-valued functions unless specifically stated.
  • Understanding this constraint helps in determining the domain of the function, ensuring all operations remain valid for real numbers.
Such intricacies highlight the importance of properly handling square roots to avert undefined or non-real results in calculations.

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Most popular questions from this chapter

(a) Two surfaces are called orthogonal at a point of inter- section if their normal lines are perpendicular at that point. Show that surfaces with equations \(F(x, y, z)=0\) and \(G(x, y, z)=0\) are orthogonal at a point \(P\) where \(\nabla F \neq 0\) and \(\nabla G \neq 0\) if and only if $$F_{x} G_{x}+F_{y} G_{y}+F_{z} G_{z}=0$$ at \(P\) (b) Use part (a) to show that the surfaces $$z^{2}=x^{2}+y^{2}$$ and $$x^{2}+y^{2}+z^{2}=r^{2}$$ are orthogonal at every point of intersection. Can you see why this is true without using calculus?

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. $$f(x, y)=x e^{-2 x^{2}-2 y^{2}}$$

Find the extreme values of \(f\) on the region described by the inequality. $$ f(x, y)=e^{-x y}, \quad x^{2}+4 y^{2} \leq 1 $$

Find the point on the plane \(x-2 y+3 z=6\) that is closest to the point \((0,1,1) .\)

Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. $$ f(x, y, z)=x^{2}+y^{2}+z^{2} ; \quad x+y+z=12 $$
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