For Problems 1-8, if necessary, review Section 1.2. Problems 1-4 refer to the following system of linear inequalities: 4 x + y ≤ 20 3 x + 5 y ≤ 37 x ≥ 0 y ≥ 0 Is the point 4 , 5 in the solution region?
For Problems 1-8, if necessary, review Section 1.2. Problems 1-4 refer to the following system of linear inequalities: 4 x + y ≤ 20 3 x + 5 y ≤ 37 x ≥ 0 y ≥ 0 Is the point 4 , 5 in the solution region?
Solution Summary: The author explains that the point (4,5) does not lie in the solution region of the provided system of linear inequalities.
2.
if
limit.
Recall that a sequence (x(n)) CR2 converges to the limit x = R²
lim ||x(n)x|| = 0.
818
-
(i) Prove that a convergent sequence (x(n)) has at most one
[4 Marks]
(ii)
Give an example of a bounded sequence (x(n)) CR2 that
has no limit and has accumulation points (1, 0) and (0, 1) [3 Marks]
(iii) Give an example of a sequence (x(n))neN CR2 which is
located on the hyperbola x2 1/x1, contains infinitely many different
Total marks 10 points and converges to the limit x = (2, 1/2).
[3 Marks]
3. (i) Consider a mapping F: RN
Rm. Explain in your own words
the relationship between the existence of all partial derivatives of F and dif-
ferentiability of F at a point x = RN.
(ii)
[3 Marks]
Calculate the gradient of the following function f: R2 → R,
f(x) = ||x||3,
Total marks 10
where ||x|| = √√√x² + x/2.
[7 Marks]
1.
(i)
(ii)
which are not.
What does it mean to say that a set ECR2 is closed?
[1 Mark]
Identify which of the following subsets of R2 are closed and
(a)
A = [-1, 1] × (1, 3)
(b)
B = [-1, 1] x {1,3}
(c)
C = {(1/n², 1/n2) ER2 | n EN}
Provide a sketch and a brief explanation to each of your answers.
[6 Marks]
(iii) Give an example of a closed set which does not have interior
points.
[3 Marks]
Chapter 5 Solutions
Finite Mathematics for Business, Economics, Life Sciences and Social Sciences
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