MAT2615 Assignment 1 (COMPLETE ANSWERS) 2025 - DUE 15 May 2025

,1. 1. (Sections 2.11,2.12)Calculate the equation for the plane containing the
lines`1and`2, where`1is given by theparametric equation(x, y, z) = (1,0,−1) +
t(1,1,1), t ∈Rand `2is given by the parametric equation(x, y, z) = (2,1,0) +
t(1,−1,0), t ∈R.[5]2. (Sections 2.11,2.12)Given the two planesx−y+ 2z−1 = 0
and 3x+ 2y−6z+ 4 = 0. Find a parametric equationfor the intersection.[4]

(Sections 3.1,3.2)Consider the surfaces in R3defined by the equationsf(x, y)
= 2px2+y2g(x, y) = 1 + x2+y2.(a) What shapes are described by f,gand their
intersection? (2)(b) Give a parametric equation describing the intersection.
(2)[4]4.
(Sections 2.5,2.6,4.3)Consider the R2−Rfunction defined byf(x, y) = 3x+
2y.Prove from first principles thatlim(x,y)→(1,−1) f(x, y) = 1.[5]5.
(Sections 7.2, 7.4, 7.7) Let fbe the R2−Rfunction defined byf(x, y) =
(x−y)3.(a)Determine the rate of increase infat the point(2,1)in the direction
of the vector(1,−1) .(5)(Study Definition 7.7.1 and Remark 7.7.2(1). Then use
Theorem 7.7.3.)(b) What is the rate of increase in fat (2,1) in the direction of
the negative X-axis? (3)25

(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]



𝑷𝒓𝒐𝒃𝒍𝒆𝒎 𝟏: 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒕𝒉𝒆 𝑷𝒍𝒂𝒏𝒆 𝑪𝒐𝒏𝒕𝒂𝒊𝒏𝒊𝒏𝒈 𝑻𝒘𝒐 𝑳𝒊𝒏𝒆𝒔

𝑮𝒊𝒗𝒆𝒏:

 𝐿𝑖𝑛𝑒 ℓ1ℓ1: (𝑥, 𝑦, 𝑧) = (1,0, −1) + 𝑡(1,1,1)(𝑥, 𝑦, 𝑧) = (1,0, −1) + 𝑡(1,1,1)
 𝐿𝑖𝑛𝑒 ℓ2ℓ2: (𝑥, 𝑦, 𝑧) = (2,1,0) + 𝑡(1, −1,0)(𝑥, 𝑦, 𝑧) = (2,1,0) + 𝑡(1, −1,0)
𝑶𝒃𝒋𝒆𝒄𝒕𝒊𝒗𝒆:
𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑛𝑔 𝑏𝑜𝑡ℎ 𝑙𝑖𝑛𝑒𝑠.

𝑨𝒑𝒑𝒓𝒐𝒂𝒄𝒉:

1. 𝑰𝒅𝒆𝒏𝒕𝒊𝒇𝒚 𝑫𝒊𝒓𝒆𝒄𝒕𝒊𝒐𝒏 𝑽𝒆𝒄𝒕𝒐𝒓𝒔:
o 𝐷𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 ℓ1ℓ1: 𝑑1 = (1,1,1)𝒅1 = (1,1,1)
o 𝐷𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 ℓ2ℓ2: 𝑑2 = (1, −1,0)𝒅2 = (1, −1,0)
2. 𝑭𝒊𝒏𝒅 𝒂 𝑷𝒐𝒊𝒏𝒕 𝒐𝒏 𝑬𝒂𝒄𝒉 𝑳𝒊𝒏𝒆:
o 𝑃𝑜𝑖𝑛𝑡 𝑜𝑛 ℓ1ℓ1: 𝑃1 = (1,0, −1)𝑃1 = (1,0, −1)
o 𝑃𝑜𝑖𝑛𝑡 𝑜𝑛 ℓ2ℓ2: 𝑃2 = (2,1,0)𝑃2 = (2,1,0)
3. 𝑽𝒆𝒓𝒊𝒇𝒚 𝒊𝒇 𝑳𝒊𝒏𝒆𝒔 𝒂𝒓𝒆 𝑷𝒂𝒓𝒂𝒍𝒍𝒆𝒍 𝒐𝒓 𝑰𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒊𝒏𝒈:
o 𝐶ℎ𝑒𝑐𝑘 𝑖𝑓 𝑑1𝒅1 𝑎𝑛𝑑 𝑑2𝒅2 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑏𝑦 𝑠𝑒𝑒𝑖𝑛𝑔 𝑖𝑓 𝑜𝑛𝑒 𝑖𝑠 𝑎 𝑠𝑐𝑎𝑙𝑎𝑟
o 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟.
 𝑑1 = (1,1,1)𝒅1 = (1,1,1)
 𝑑2 = (1, −1,0)𝒅2 = (1, −1,0)
 𝑇ℎ𝑒𝑦 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑠𝑖𝑛𝑐𝑒 𝑡ℎ𝑒𝑟𝑒′𝑠 𝑛𝑜 𝑠𝑐𝑎𝑙𝑎𝑟 𝑘𝑘 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑑1 =
𝑘𝑑2𝒅1 = 𝑘𝒅2.
o 𝐶ℎ𝑒𝑐𝑘 𝑖𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡 𝑏𝑦 𝑠𝑒𝑡𝑡𝑖𝑛𝑔 𝑡ℎ𝑒𝑖𝑟 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑟𝑖𝑐 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠