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MAT2615 Assignment 2 Memo | Due June 2026. All questions fully answered. 1. (Sections 3.2, 7.5 and 7.9) Consider the R2 − R function f defined by
f (x, y) = 1 − x2 − y2.
Let C be the contour curve of f through the point (1,−1), let L be the tangent to C at (x, y) =
(1, 1) and let V be the tangent plane to f at (x, y) = (1, 1).
(a) Find the equation of the curve C. (2)
(b) Find a vector in R2 that is perpendicular to C at (x, y) = (1, 1). (2)
(c) Find the Cartesian equation of the line L. (3)
(d) Find a vector in R3 that is perpendicular to the graph of f at the point (x, y, z) = (1, 1, 3).(3)
(e) Find the Cartesian equation of the plane V. (3)
(f) Draw a sketch to visualize the graph of f , together with appropriate sections of the line L
and the plane V. Also show the vectors that you obtained in (b) and (d) on your sketch.
(3)
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