give answer in 2 step with explanation at the end of each step and final answer at the end:1. (2 pts) Find the equations of the cross-sections of z = y? — zy with fixed z = 0,1, and fixed
= 0,1. Sket

Question

give answer in 2 step with explanation at the end of each step and final answer at the end:1. (2 pts) Find the equations of the cross-sections of z = y? — zy with fixed z = 0,1, and fixed
= 0,1. Sketch them in the appropriate coordinate planes.

VIPSolutions · Accepted Answer

Let's solve the problem step by step:

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### **Step 1: Cross-sections with fixed $x = 0, 1$**

Given: $z = y^2 - xy$

- For $x = 0$:  
  Substitute $x = 0$ into the equation:
  $$
  z = y^2 - (0)y = y^2
  $$
  **Explanation:** This is a parabola in the $yz$-plane.

- For $x = 1$:  
  Substitute $x = 1$ into the equation:
  $$
  z = y^2 - (1)y = y^2 - y
  $$
  **Explanation:** This is also a parabola in the $yz$-plane, but shifted.

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### **Step 2: Cross-sections with fixed $y = 0, 1$**

- For $y = 0$:  
  Substitute $y = 0$ into the equation:
  $$
  z = (0)^2 - x(0) = 0
  $$
  **Explanation:** This is the line $z = 0$ in the $xz$-plane.

- For $y = 1$:  
  Substitute $y = 1$ into the equation:
  $$
  z = (1)^2 - x(1) = 1 - x
  $$
  **Explanation:** This is a straight line with a slope of $-1$ in the $xz$-plane.

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### **Final Answer:**

- For fixed $x = 0$: $z = y^2$ (parabola in $yz$-plane)
- For fixed $x = 1$: $z = y^2 - y$ (parabola in $yz$-plane)
- For fixed $y = 0$: $z = 0$ (line in $xz$-plane)
- For fixed $y = 1$: $z = 1 - x$ (line in $xz$-plane)

You can sketch these by plotting the corresponding equations in the $yz$-plane for fixed $x$, and in the $xz$-plane for fixed $y$.

✨ VIPSolutions

give answer in 2 step with explanation at the end of each step and final answer at the end:1. (2 pts) Find the equations of the cross-sections of z = y? — zy with fixed z = 0,1, and fixed = 0,1. Sketch them in the appropriate coordinate planes.

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