MAT2615 Assignment 2 (Accurate Solutions) Due June 2026

MAT2615
Assignment 2
Due June 2026

,Question 1

Problem Statement

Consider the function 𝑓: ℝ2 → ℝ defined by

𝑓(𝑥, 𝑦) = 1 − 𝑥 2 − 𝑦 2 .

Let 𝐶 be the contour curve of 𝑓 through the point (1, −1).
Let 𝐿 be the tangent line to 𝐶 at the point (1,1).
Let 𝑉 be the tangent plane to the graph of 𝑓 at the point (1,1).

Answer parts (a) to (f).



(a) Equation of the contour curve 𝐶

Step 1: Identify the level value

A contour curve consists of all points where 𝑓(𝑥, 𝑦) equals a constant.

Evaluate 𝑓 at the given point:

𝑓(1, −1) = 1 − 1 2 − (−1)2 = 1 − 1 − 1 = −1

Step 2: Set up the contour equation

1 − 𝑥 2 − 𝑦 2 = −1

Step 3: Simplify

𝑥2 + 𝑦2 = 2

Final Answer (a)

𝑥2 + 𝑦2 = 2

, (b) A vector in ℝ2 perpendicular to 𝐶 at (1,1)

Step 1: Use the gradient

For a contour curve 𝑓(𝑥, 𝑦) = 𝑐, the gradient ∇𝑓 is perpendicular to the curve.

∂𝑓 ∂𝑓
∇𝑓(𝑥, 𝑦) = ( , )
∂𝑥 ∂𝑦

Step 2: Compute partial derivatives

∂𝑓 ∂𝑓
= −2𝑥, = −2𝑦
∂𝑥 ∂𝑦

Step 3: Evaluate at (1,1)

∇𝑓(1,1) = (−2, −2)

Final Answer (b)

⟨−2, −2⟩

(Any nonzero scalar multiple is also correct.)



(c) Cartesian equation of the tangent line 𝐿

Step 1: Use the gradient-based tangent formula

For a contour curve:

∇𝑓(𝑥0 , 𝑦0 ) ⋅ ⟨𝑥 − 𝑥 0 , 𝑦 − 𝑦0 ⟩ = 0

Step 2: Substitute values

(−2, −2) ⋅ (𝑥 − 1, 𝑦 − 1) = 0

Step 3: Compute the dot product

−2(𝑥 − 1) − 2(𝑦 − 1) = 0