Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(g(x, y, z)=\sqrt{x \cos y}+z^{2} .\) Find each value. (a) \(g(4,0,2)\) (b) \(g(-9, \pi, 3)\) (c) \(g(2, \pi / 3,-1)\) (d) \(g(3,6,1.2)\)

Step-by-Step Solution

Answer

(a) 6, (b) 12, (c) 2, (d) 3.137

Step 1: Understand the Function

The function given is \(g(x, y, z)=\sqrt{x \cos y}+z^{2}\). We need to evaluate this function for different sets of \(x\), \(y\), and \(z\). The function has two parts: the square root of the product of \(x\) and the cosine of \(y\), plus the square of \(z\).

Step 2: Evaluate \(g(4,0,2)\)

For part (a), substitute \(x = 4\), \(y = 0\), and \(z = 2\) into the function:\[g(4,0,2) = \sqrt{4 \cos 0} + 2^2\]Since \(\cos 0 = 1\),\[g(4,0,2) = \sqrt{4 \times 1} + 4 = \sqrt{4} + 4 = 2 + 4 = 6\].

Step 3: Evaluate \(g(-9, \pi, 3)\)

For part (b), substitute \(x = -9\), \(y = \pi\), and \(z = 3\):\[g(-9, \pi, 3) = \sqrt{-9 \cos \pi} + 3^2\]Since \(\cos \pi = -1\),\[g(-9, \pi, 3) = \sqrt{-9 \times -1} + 9 = \sqrt{9} + 9 = 3 + 9 = 12\].

Step 4: Evaluate \(g(2, \pi / 3,-1)\)

For part (c), substitute \(x = 2\), \(y = \pi/3\), and \(z = -1\):\[g(2, \pi/3, -1) = \sqrt{2 \cos(\pi/3)} + (-1)^2\]Since \(\cos(\pi/3) = 1/2\),\[g(2, \pi/3, -1) = \sqrt{2 \times \frac{1}{2}} + 1 = \sqrt{1} + 1 = 1 + 1 = 2\].

Step 5: Evaluate \(g(3,6,1.2)\)

For part (d), substitute \(x = 3\), \(y = 6\), and \(z = 1.2\):\[g(3,6,1.2) = \sqrt{3 \cos 6} + (1.2)^2\]First, find \(\cos 6\) using a calculator, approximately \(\cos 6 \approx 0.96017\):\[\sqrt{3 \times 0.96017} + 1.44 = \sqrt{2.88051} + 1.44 \approx 1.697 + 1.44 \approx 3.137\].

Key Concepts

Function Evaluation

Function evaluation is an essential process in mathematics where you substitute specific values into a function to find a result. In our function, \(g(x, y, z) = \sqrt{x \, \cos y} + z^2\), by plugging in different values of \(x\), \(y\), and \(z\), we can determine specific outputs. This exercise involves evaluating the function at different points:

• For \(g(4, 0, 2)\), we substitute \(x = 4\), \(y = 0\), and \(z = 2\).
• For \(g(-9, \pi, 3)\), we substitute \(x = -9\), \(y = \pi\), and \(z = 3\).
• Steps like these help us understand the behavior of the function across various inputs, which is indispensable for further mathematical analysis and applications.

Function evaluation provides clarity on how changes in variables affect the output, giving insight into the function's form and behavior.

Trigonometric Functions

Trigonometric functions like cosine are crucial in multivariable calculus, providing connections between angles and ratios in right triangles. In this exercise, \(\cos y\) plays a key role. Here’s why it matters:

• 
  For \(y = 0\), \(\cos 0 = 1\), which simplifies calculations.
• For \(y = \pi\), \(\cos \pi = -1\), introducing a negative aspect in operations.
• The cosine function is periodic and oscillates between -1 and 1, affecting how the function \(g(x, y, z)\) behaves depending on \(y\).

Understanding trigonometric functions equips you to analyze complex systems involving waves, cycles, and rotations. This exercise shines a light on how cosine's value influences the results, demonstrating its utility in transforming angular measures into linear forms.

Square Roots

Square roots are a mathematical operation that finds the value that, when multiplied by itself, yields the original number. In our function \(g(x, y, z)\), the square root \(\sqrt{x \, \cos y}\) offers insights into how the function scales:

• 
  The square root smooths the function’s rate of increase since it grows slower than a linear or quadratic function.
• Values of \(x\) and \(y\) determine the input to the square root, influencing the stability and magnitude of \(g(x, y, z)\).

By understanding square roots, you can better grasp how they modulate the results and stabilize erratic behavior. They are particularly useful in situations where you want to manage growth, spread, or potential over time or space.

Mathematical Analysis

Mathematical analysis involves in-depth study of functions, their properties, and behaviors. This process helps in evaluating not just isolated values but understanding the continuous nature of the function. In analyzing \(g(x, y, z) = \sqrt{x \, \cos y} + z^2\), you learn:

• How each component of the function contributes to overall changes in the output.
• The effects of nonlinear combinations, such as square roots and cosine, on variable transformations.
• Methods to breakdown multifaceted problems into comprehensible parts, making them easier to solve and interpret.

Multivariable calculus, along with mathematical analysis, provides the tools needed to explore and predict behavior in complex systems, crucial for fields like physics, engineering, and economics.