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Exam-Style Questions on Periodic FunctionsProblems on Periodic Functions adapted from questions set in previous Mathematics exams. |
1. | IB Analysis and Approaches |
Consider a function \(f\), such that \(f(x)=7.2\sin(\frac{\pi}{6}x + 2) + b\) where \( 0\le x \le 12\)
(a) Find the period of \(f\).
The function \(f\) has a local maximum at the point (11.18,10.3) , and a local minimum at (5.18.-4.1).
(b) Find the value of b.
(c) Hence, find the value of \(f(7)\).
A second function \(g\) is given by \(g(x)=a\sin(\frac{2\pi}{7}(x -4)) + c\) where \(0 \le x \le 10\)
The function \(g\) passes through the points (2.25,-3) and (5.75,7).
(d) Find the value of \(a\) and the value of \(c\).
(e) Find the value of x for which the functions have the greatest difference.
2. | IB Analysis and Approaches |
A carriage attached to a tall vertical pole in an amusement park ride whisks customers up and down. The height, \(H\) metres, of the base of the carriage above the ground can be modelled by the function \(H(t) = a\cos(0.4t) + b\), for \(a, b \in \mathbb{R}\) and \(t\) is the time in seconds after the ride starts.
(a) Find the period of the function.
The ride begins when its base is at a minimum height of 1 metre above the ground, and it reaches a maximum height of 31 metres above the ground.
(b) Find the values of a and b.
(c) Find the number of times that the carriage reaches its maximum height in the first minute of its motion.
(d) Find the first time that the base of the carriage reaches a height of 15 metres.
A camera is set to take a picture of the ride at a random time during the first 15 seconds of its motion.
(e) Find the probability that the height of the base of the carriage is greater than 10 metres at the time the picture is taken.
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