Periodic functions > Periodicity
123456Periodicity

Solutions to the exercises

Exercise 1
a

f ( 25 ) = f ( 1 ) = 5 .

b

x = 2 + k 3 .

c

x = 1 x = 4 x = 7 x = 2,5 x = 5,5 x = 8,5 .

Exercise 2
a

h ( 4,5 ) = -1 , h ( 10,5 ) = -1 en h ( 16,5 ) = -1 .

b

h ( 0,75 ) = 1 2 2 because you get a rectangular triangle with a 45 o angle that is isosceles with legs 1 2 = 1 2 2 .

c

They are all 1 2 2 since the times differ exactly one period with t = 0,75 .

d

t = 0,75 + k 6 t = 2,25 + k 6 .

Exercise 3
a

h = 45 + 1,5 cos ( 60 ° ) = 45,75 m.

b

Smooth graph through h ( 0 ) = 46,5 , h ( 60 ) = 45,75 , h ( 90 ) = 45 , h ( 120 ) = 44,25 , h ( 180 ) = 43,5 , h ( 240 ) = 44,25 , h ( 270 ) = 45 , h ( 300 ) = 45,75 and h ( 360 ) = 46,5 , etc...

c

h = 45 + 1,5 cos ( α ) = 46 gives α 48,2 ° . The asked for distance is 2 sin ( 48,2 ° ) 1,49 m.

Exercise 4
a

h ( 0 ) = 5 and h ( 0,5 ) = 3,75 .

b

The period is 2 seconds.

c

h ( 6 ) = h ( 0 ) = 5 and h ( 6,5 ) = h ( 0,5 ) = 3,75 .

d

h ( 15 ) = h ( -1 ) = 0 and h ( 15,5 ) = h ( -0,5 ) = 3,75 .

e

formulate your own answer.

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