MAT2615 Assignment 2 | 100% COMPLETLY SOLVED| 2026

MAT2615
Assignment 2

2026

, MAT2615 Assignment 2 - 2026

Calculus in Higher Dimensions

University of South Africa (UNISA)



Question 1: Contour Curves and Tangent Planes

Given: f(x, y) = 1 − x² − y², with contour curve C through point (1, −1), and we analyze at point
(1, 1).


(a) Equation of curve C [2 marks]

Solution:

First, find the level by evaluating f at (1, 1):

f(1, 1) = 1 − 1² − 1² = 1 − 1 − 1 = −1

The contour curve C is where f(x, y) = −1:

1 − x² − y² = −1

x² + y² = 2

Answer: The equation of curve C is x² + y² = 2


(b) Vector perpendicular to C at (1, 1) [2 marks]

Solution:

By Theorem 7.9.1, the gradient ∇f is perpendicular to the contour curve at any point.

Calculate the gradient:

∇f(x, y) = (∂f/∂x, ∂f/∂y) = (−2x, −2y)

At point (1, 1):