RATES of CHANGE
(in Trigonometric Functions)
PROBLEM
A motion detector is used to measure the horizontal distance between a
person and a child on a swing. The data is shown on the graph below:
The maximum distance away from the
motion detector is _______________.
The minimum distance away from the
motion detector is _______________.
The time required to complete one
swinging cycle is ________________.
The child is moving ________________
the motion detector between 0 and 0.8s
and ___________________ the motion
detector between 0.8 and 1.6s.
The instantaneous rate of change (velocity) can be analyzed by drawing tangent
lines at various points over one swing cycle.
At the maximum and minimum
distances from the motion
detector, the tangent line is
horizontal.
 the IRC is ____________.
Between 0s and 0.4s, the tangent lines become steeper as time increases.
the child’s speed is ____________________.
Between 0.4s and 0.8s, the tangent lines are getting less steep as time increases.
the child’s speed is ____________________.
When is the child’s speed the fastest? ______________
Where is this point located? ______________________________________________
The slopes of the tangent lines are negative.
the child is moving __________________________ the motion detector.
What happens at 0.8 seconds? ____________________________________________
A similar pattern of tangent lines can be seen between 0.8s and 1.6s. The child’s speed
increases at first (tangents become steeper) and then begins to decrease again
(tangents become less steep). The child’s speed is the fastest at t = __________.
The slopes of the tangents are positive on this interval, indicating that the child is
moving _______________________ the motion detector.
SUMMARY
The rate of change will be the greatest at the point that
lies halfway between the maximum and minimum values.
The equation that models the child’s
distance from the motion detector
over time (in seconds) is
  
d(t)  1.6 cos
t   2.2 , where d(t)
 0.8 
is the distance from the child to the
motion detector, in metres.
max =
period =
min =
A.
Determine the time and speed at which the child was moving the fastest.
B.
Determine the child’s average speed over the interval of time where the child
swung toward the motion detector.
Homework:
p.370–373 #2, 7, 12