Explanation Given: The given pair of set: “Attorneys and men” 1. The set A can be a subset of the set B. In this Case, all the members of the set A are also the members of the set B. Hence, the circle for set A is drawn inside the circle for the set B 2. The set B can be a subset of the set A. In this case, all the members of the set B are also the members of the set A. Hence, the circle for the set B is drawn inside the circle for the set A 3. The set A is disjoint from the set B. In this case, the members of the set A are not the members of the set B or vice versa. That is, the two sets A and B do not have any common members. Hence, there are two separated circles for the set A and the set B which do not touch each other. 4. Set A and set B are overlapping sets. In this case, the two sets A and B have some common members...

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ISBN: 8220100802713

Author: Briggs

Publisher: PEARSON

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Chapter 1.C, Problem 37E

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Draw Venn diagrams with two circles showing the relationship between the following pair of set. Provide an explanation of the diagram you drew.

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Chapter 1.C, Problem 36E

Chapter 1.C, Problem 38E

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Total marks 15 4. : Let f R2 R be defined by f(x1, x2) = 2x²- 8x1x2+4x+2. Find all local minima of f on R². [10 Marks] (ii) Give an example of a function f R2 R which is neither bounded below nor bounded above, and has no critical point. Justify briefly your answer. [5 Marks]

4. Let F RNR be a mapping. (i) x ЄRN ? (ii) : What does it mean to say that F is differentiable at a point [1 Mark] In Theorem 5.4 in the Lecture Notes we proved that if F is differentiable at a point x E RN then F is continuous at x. Proof. Let (n) CRN be a sequence such that xn → x ЄERN as n → ∞. We want to show that F(xn) F(x), which means F is continuous at x. Denote hnxn - x, so that ||hn|| 0. Thus we find ||F(xn) − F(x)|| = ||F(x + hn) − F(x)|| * ||DF (x)hn + R(hn) || (**) ||DF(x)hn||+||R(hn)||| → 0, because the linear mapping DF(x) is continuous and for all large nЄ N, (***) ||R(hn) || ||R(hn) || ≤ → 0. ||hn|| (a) Explain in details why ||hn|| → 0. [3 Marks] (b) Explain the steps labelled (*), (**), (***). [6 Marks]

4. In Theorem 5.4 in the Lecture Notes we proved that if F: RN → Rm is differentiable at x = RN then F is continuous at x. Proof. Let (xn) CRN be a sequence such that x → x Є RN as n → ∞. We want F(x), which means F is continuous at x. to show that F(xn) Denote hn xnx, so that ||hn||| 0. Thus we find ||F (xn) − F(x) || (*) ||F(x + hn) − F(x)|| = ||DF(x)hn + R(hn)|| (**) ||DF(x)hn|| + ||R(hn) || → 0, because the linear mapping DF(x) is continuous and for all large n = N, |||R(hn) || ≤ (***) ||R(hn)|| ||hn|| → 0. Explain the steps labelled (*), (**), (***) [6 Marks] (ii) Give an example of a function F: RR such that F is contin- Total marks 10 uous at x=0 but F is not differentiable at at x = 0. [4 Marks]

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EBK USING AND UNDERSTANDING MATHEMATICS

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