Excursions in Mathematics, Loose-Leaf Edition Plus MyLab Math with Pearson eText -- 18 Week Access Card Package
9th Edition
ISBN: 9780136208754
Author: Tannenbaum, Peter
Publisher: PEARSON
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Chapter 3, Problem 57E
To determine
1.
To explain:
The placement of each player’s markers.
To determine
b.
The allocation of comic books to each player and the comic books that are left over.
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Total marks 15
3.
(i)
Let FRN Rm be a mapping and x = RN is a given
point. Which of the following statements are true? Construct counterex-
amples for any that are false.
(a)
If F is continuous at x then F is differentiable at x.
(b)
If F is differentiable at x then F is continuous at x.
If F is differentiable at x then F has all 1st order partial
(c)
derivatives at x.
(d) If all 1st order partial derivatives of F exist and are con-
tinuous on RN then F is differentiable at x.
[5 Marks]
(ii) Let mappings
F= (F1, F2) R³ → R² and
G=(G1, G2) R² → R²
:
be defined by
F₁ (x1, x2, x3) = x1 + x²,
G1(1, 2) = 31,
F2(x1, x2, x3) = x² + x3,
G2(1, 2)=sin(1+ y2).
By using the chain rule, calculate the Jacobian matrix of the mapping
GoF R3 R²,
i.e., JGoF(x1, x2, x3). What is JGOF(0, 0, 0)?
(iii)
[7 Marks]
Give reasons why the mapping Go F is differentiable at
(0, 0, 0) R³ and determine the derivative matrix D(GF)(0, 0, 0).
[3 Marks]
Chapter 3 Solutions
Excursions in Mathematics, Loose-Leaf Edition Plus MyLab Math with Pearson eText -- 18 Week Access Card Package
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- 5. (i) Let f R2 R be defined by f(x1, x2) = x² - 4x1x2 + 2x3. Find all local minima of f on R². (ii) [10 Marks] Give an example of a function f: R2 R which is not bounded above and has exactly one critical point, which is a minimum. Justify briefly Total marks 15 your answer. [5 Marks]arrow_forwardTotal 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]arrow_forward4. 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]arrow_forward
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