MAT2615 Assignment 1 (DETAILED ANSWERS) 2025 - DISTINCTION GUARANTEED

MAT2615 Assignment 1 (DETAILED ANSWERS) 2025 - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED Answers, guidelines, workings and references ,... 1. (Sections 2.11,2.12) The parametric equations of two lines are given below: ℓ1 : (x, y, z) = (1, 0, 0) + t(1, 0, 1), t ∈ R ℓ2 : (x, y, z) = (1, 0,−1) + t(0, 1, 1), t ∈ R Calculate the equation of the plane containing these two lines. [5] 2. (Sections 2.11,2.12) Given the two planes 3x + 2y − z − 4 = 0 and −x − 2y + 2z = 0. Find a parametric equation for the intersection. [5] 3. (Sections 2.11,2.12) Find the point of intersection of the line ℓ : (x, y, z) = (5, 4,−1)+t(1, 1, 0), t ∈ R and the plane 2x + y − z = 3. [5] 4. (Sections 2.5,2.6,4.3) Consider the R2 − R function defined by f (x, y) = 2x + 2y − 3. Prove from first principles that lim(x,y)→(−1,1) f (x, y) = −3 [5] 5. (Sections 4.3,4.4,4.5) Determine whether the following limits exist. If you suspect that a limit does not exist, try to prove so by using limits along curves. If you suspect that the limit does exist, you must use the ϵ − δ definition, or the limit laws, or a combination of the two. (a) lim (x,y)→(0,0) sin(x + y) x + y (5) (b) lim (x,y)→(1,1) y + 1 x − 1 (5) (c) lim (x,y)→(0,0) x2 + y2 xy (5) 15 (d) lim (x,y)→(π/2,π/2) cos x sin y + y tan x (5) [20] 6. (Sections 4.4,4.7) Consider the R2 − R function given by f (x, y) = ( −2x2+xy+y2 y2+2xy if 2x ̸= −y 3 2 if (x, y) = (1,−2) or (x, y) = (2,−4). (a) Write down the domain Df of f . (2) (b) Determine lim (x,y)→(1,−2) f (x, y) and lim (x,y)→(2,−4) f (x, y). (3) (c) Calculate f (1,−2) and f (2,−4). (4) (d) Is f continuous at (x, y) = (1,−2)? (2) (e) Is f continuous at (x, y) = (2,−4)? (2) (f) Is f a continuous function? (2) Give reasons for your answers to (d), (e) and (f). [15]