MAT2615 Assignment 2 2026 Due June 2026

MAT2615
Assignment 2
Due June 2026

,Question 1

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),

• 𝐿 be the tangent line to 𝐶 at the point (1,1),

• 𝑉 be the tangent plane to the graph of 𝑓 at (1,1).



(a) Find the equation of the curve 𝐶

Problem statement

Find the equation of the contour curve of 𝑓 passing through the point (1, −1).

Step 1: Understand what a contour curve is

A contour curve (or level curve) is defined by fixing the value of the function:

𝑓(𝑥, 𝑦) = 𝑘,

where 𝑘 is a constant.

Step 2: Find the level value

Since the curve passes through (1, −1), compute

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

Step 3: Write the contour equation

Set

𝑓(𝑥, 𝑦) = −1.

, That gives

1 − 𝑥 2 − 𝑦 2 = −1.

Step 4: Simplify

𝑥 2 + 𝑦 2 = 2.

Final Answer (a)

𝑥2 + 𝑦2 = 2




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

Problem statement

Find a vector perpendicular to the contour curve 𝐶 at (1,1).

Step 1: Key idea

For a contour curve 𝑓(𝑥, 𝑦) = constant, the gradient vector

∇𝑓(𝑥, 𝑦)

is perpendicular to the curve at that point.

Step 2: Compute the gradient

∂𝑓 ∂𝑓
∇𝑓(𝑥, 𝑦) = ( , ).
∂𝑥 ∂𝑦

Compute the partial derivatives:

∂𝑓 ∂𝑓
= −2𝑥, = −2𝑦.
∂𝑥 ∂𝑦

So,

∇𝑓(𝑥, 𝑦) = (−2𝑥, −2𝑦).