Let's solve each part step by step, with explanations after each step: --- ### **Step 1: (a) Level-one contour of \( z = y^2 - x^2 \)** **Set the level value to 1:** \[ z = y^2 - x^2 = 1 \] **Rearrange:** \[ y^2 - x^2 = 1 \] This is the equation of a **hyperbola** in the \(xy\)-plane. **Explanation:** The level-one contour is a hyperbola opening upwards and downwards (along the y-axis), centered at the origin. --- ### **Step 2: (b) Level-one contour of \( z = 1 - x + 2y \)** **Set the level value to 1:** \[ z = 1 - x + 2y = 1 \] **Simplify:** \[ 1 - x + 2y = 1 \implies -x + 2y = 0 \implies x = 2y \] This is a **straight line** in the \(xy\)-plane. **Explanation:** The level-one contour is a straight line passing through the origin with a slope of \( \frac{1}{2} \). --- ### **Step 3: (c) Contour of \( z = x^2 + y^2 \) through (1, 1), and label its level** **Plug in the point (1,1):** \[ z = 1^2 + 1^2 = 2 \] So the contour is: \[ x^2 + y^2 = 2 \] This is a **circle** centered at the origin with radius \( \sqrt{2} \). **Explanation:** The contour passing through (1,1) is a circle of radius \( \sqrt{2} \) and the level is **2**. --- ## **Final Answers:** **(a)** The level-one contour is \( y^2 - x^2 = 1 \) (a hyperbola). **(b)** The level-one contour is \( x = 2y \) (a straight line). **(c)** The contour through (1,1) is \( x^2 + y^2 = 2 \) (a circle), and its level is **2**.