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BUSI3119-E1

A LEVEL 3 MODULE, AUTUMN SEMESTER 2019-2020

ADVANCED CALCULUS FOR BUSINESS, ECONOMICS AND FINANCE

1.  (a) Use L’Hopital rule to find the lix( − )

(8 Marks)

(b) Show that det(adj(A))  =  (det(A))n- 1.

(12 Marks)

2.  (a) Compute the rank of the following matrix:                                                  (6 Marks) |   1            1     |

(b) Find the solution of the following system:                                                    (6 Marks)

−  x + 3yz = 0

w +  4xy + 2z= 3

3w + 7x+  y +  z = 6.

(c) Given that A = [], compute A100.                                                      (8 Marks)

3.  Find the general solution of each of the following differential equations, then, find the solution that satisfies y(1) = 1:

(a)                                         y.= ty                                                   (8 Marks)

(b)                                       ty.+ (1 − t)y = e2t                                                       (12 Marks)

 

4.  Find the general solution of y…− y.− 2y = 6t+ 4e- t .                                   (20 Marks)

5.  Solve the following control problem using the current value formulation            (20 Marks)

max j0T (2u  u2 + 2x)e- 2tdt,          x.= x u,         x(0) = 0,         x(T) free,   u ∈ ( −∞, ∞)