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MAT2615 Assignment 1 (ANSWERS) 2025 - DISTINCTION GUARANTEED

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Well-structured MAT2615 Assignment 1 (ANSWERS) 2025 - DISTINCTION GUARANTEED. (DETAILED ANSWERS - DISTINCTION GUARANTEED!)... 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]

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MAT2615 Assignment 1 2025
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Due Date:15 May 2025




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SECTION A

1. I scored 20 out of 20 on the Self-Assessment Task for Units 1–3.


2. Yes, I found the self-assessment task useful.

3. I found the self-assessment task useful because it allowed me to test my
understanding of key concepts from the first three units of the course. The
task helped reinforce my learning and highlighted areas where I was confident
as well as areas where I needed further clarification. By working through each
question and comparing my answers to the correct ones, I was able to confirm
my understanding and identify gaps in my knowledge. Additionally, the
explanations provided for each correct answer helped me deepen my
conceptual grasp of the material. Overall, it was a helpful exercise to ensure I
am on track with my studies before moving on to more advanced topics.




SECTION B

QUESTION 1

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