, 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)
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)
