HKAL Pure Mathematics Past Paper Topic: Integration
71.
(00II13)
Let n be a positive integer. Define f n
(a) (i)
( x)
x 0 1 0
(1 t 4 ) n dt (1 t 4 ) n dt
.
Show that fn(x) is an odd function.
(ii)
Find f n' ( x) and f n'' ( x) .
(iii)
Sketch the graph of fn(x) for 1 x 1 . (7 marks)
(b) Using the facts A.
t 3 (1 t 4 )n (1 t 4 )n for 0 t 1 and
B.
(1 t 4 )n
t3 (1 t 4 )n for 0 < x t 1 , 3 x
or otherwise, show that 0 1 fn(x)
(1 x 4 ) n 1 for 0 x 1 . x3 (5 marks)
(c) For each x [1, 1] , let g( x) lim f n ( x) . Evaluate g(x) when n
0 x 1 and when x = 0 respectively. Sketch the graph of g(x) for 1 x 1 . (3 marks)
P. 232