Problem 88 - Calculus Early Transcendentals

Question

a. Consider the function \(w=f(x, y, z)\). List all possible second partial derivatives that could be computed. b. Let \(f(x, y, z)=x^{2} y+2 x z^{2}-3 y^{2} z\) and determine which second partial derivatives are equal. c. How many second partial derivatives does \(p=g(w, x, y, z)\) have?

Step-by-Step Solution

Answer

Based on the step by step solution, list the equal second partial derivatives for the function \(f(x, y, z) = x^2y + 2xz^2 - 3y^2z\).

Step 1: Identify first partial derivatives

We have three variables, \(x\), \(y\), and \(z\). Compute the first partial derivatives with respect to each: 1. \(\frac{\partial f}{\partial x}\) 2. \(\frac{\partial f}{\partial y}\) 3. \(\frac{\partial f}{\partial z}\)

Step 2: Calculate second partial derivatives

Compute the second partial derivatives for each first partial derivative obtained in step 1: 1. \(\frac{\partial^2 f}{\partial x^2}\) 2. \(\frac{\partial^2 f}{\partial y^2}\) 3. \(\frac{\partial^2 f}{\partial z^2}\) 4. \(\frac{\partial^2 f}{\partial x \partial y}\) 5. \(\frac{\partial^2 f}{\partial x \partial z}\) 6. \(\frac{\partial^2 f}{\partial y \partial x}\) 7. \(\frac{\partial^2 f}{\partial y \partial z}\) 8. \(\frac{\partial^2 f}{\partial z \partial x}\) 9. \(\frac{\partial^2 f}{\partial z \partial y}\) b. Determine which second partial derivatives are equal for a given function \(f(x, y, z) = x^2y + 2xz^2 - 3y^2z\):

Step 1: Compute second partial derivatives

Calculate each of the second partial derivatives from part (a), keeping in mind the function \(f(x, y, z) = x^2y + 2xz^2 - 3y^2z\). 1. \(\frac{\partial^2 f}{\partial x^2} = 2y\) 2. \(\frac{\partial^2 f}{\partial y^2} = -6z\) 3. \(\frac{\partial^2 f}{\partial z^2} = 4x\) 4. \(\frac{\partial^2 f}{\partial x \partial y} = 2x\) 5. \(\frac{\partial^2 f}{\partial x \partial z} = 4z\) 6. \(\frac{\partial^2 f}{\partial y \partial x} = 2x\) 7. \(\frac{\partial^2 f}{\partial y \partial z} = -3(2y)\) 8. \(\frac{\partial^2 f}{\partial z \partial x} = 4z\) 9. \(\frac{\partial^2 f}{\partial z \partial y} = -3(2y)\)

Step 2: Identify equal second partial derivatives

Compare the second partial derivatives obtained in the previous step, and identify the equal ones: 1. \(\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}\). 2. \(\frac{\partial^2 f}{\partial x \partial z} = \frac{\partial^2 f}{\partial z \partial x}\). 3. \(\frac{\partial^2 f}{\partial y \partial z} = \frac{\partial^2 f}{\partial z \partial y}\). c. How many second partial derivatives does \(p = g(w, x, y, z)\) have?

Step 1: Calculate the number of second partial derivatives

With four variables \(w\), \(x\), \(y\), and \(z\), we need to compute the second partial derivatives with respect to these variables. For each variable, compute the derivative with respect to the other three variables, including itself: In total, we have \(4 \times 4 = 16\) possible second partial derivatives.