Math 2111 Advanced Calculus (I)
Homework 1 Hand in Problems:
1(b)(d), 2(a)(c), 3(b), 4, Lecture Note: Problem 0.2, 0.3, 0.4
1. Use “Second Derivative Test” to find the local maximum and minimum values and saddle point(s) of the function.
(a) f (x, y) = xe−2x2−2y2 (b) f (x, y) = xy(1 − x − y)
(c) f (x, y) = excos y (d) f (x, y) = ey(y2− x2)
2. Find the absolute maximum and minimum values of f on the set D.
(a) f (x, y) = x + y − xy, D is the closed triangular region with vertices (0, 0),(0, 2) and (4, 0)
(b) f (x, y) = 4x + 6y − x2− y2, D = {(x, y)| |x| ≤ 1, |y| ≤ 1}
(c) f (x, y) = xy2, D = {(x, y)| x ≥ 0, y ≥ 0, x2+ y2 ≤ 1}
3. (a) Find the point on the surface y2 = 9 + xz that are closest to the origin.
(b) Find the dimensions of the rectangular box with largest volume if the total surface area is given as 64 cm2.
4. Use the below definition to prove that the following binary relations “∼” are equivalence relation on the given sets.
Definition: Let S be a set. A binary relation “∼” on S is said to be an “equivalence relation” if for all a, b, c ∈ S, then
(a) a ∼ a.
(b) a ∼ b if and only if b ∼ a.
(c) if a ∼ b and b ∼ c then a ∼ c.
Example 1: Check that “=” is an equivalence relation on R.
Proof: For any a, b, c ∈ R, all of the statements that (i) a = a (ii) a = b ⇔ b = a (iii) if a = b and b = c then a = c are trivial. Hence, (a),(b) and (c) in the definition hold and
“=” is an equivalence relation on R.
Example 2: Let S = n I
I is an interval of Ro
be the set of all intervals in R. Check that “⊆” is not an equivalence relation on S.
Proof: For I, J, K ∈ S, I ⊆ I as well as if I ⊆ J and J ⊆ K then I ⊆ K. But the statement that if I ⊆ J then J ⊆ I is false. Hence, (a) and (c) hold for “⊆” but (b) doesn’t. The binary relation “⊆” is not an equivalence relation.
(a) For x, y ∈ N, we say that x ∼ y if the remainders of the division of both x and y by 7 are the same. (ex: 26 = 3 × 7 + 5 and 145 = 20 × 7 + 5. Hence, 26 and 145 have the same remainder “5” in the division by 7. We have 26 ∼ 145.)
Check that “∼” is an equivalence relation on N.
(b) Let S = n
{xn}
{xn} is a convergent sequenceo
be the set of all convergent se- quences. We say that {xn} ∼ {yn} if lim
n→∞xn= lim
n→∞yn. Check that “∼” is an equivalence relation on S.
Lecture Note:
(Page xi) 1. Problem 0.1 2. Problem 0.2 3. Problem 0.3 4. Problem 0.4
