Consider the system of parametric equations defined by x(t) and y(t), where t is the parameter.
Over a given interval a ≤ t ≤ b within the domain, the average rate of change can be computed for x(t) and y(t) independently as shown below.
Average rate of change for x(t) :
Average rate of change for y(t) :
The ratio of the average rate of change of y to the average rate of x gives the slope of the graph between the points on the curve corresponding to t = a and t = b.
Slope of the graph :
Problem 1 :
For the particle whose position is determined by the equations
over the given interval -1 ≤ t ≤ 1, find the following.
(i) The average rate of change for x(t).
(ii) The average rate of change for y(t).
(iii) The slope of the graph between the points on the curve corresponding to t = -1 and t = 1.
Solution :
(i) The average rate of change for x(t) :
(ii) The average rate of change for y(t) :
(iii) The slope of the graph :
Problem 2 :
x = 3t
y = 3t2 - 1
For the above parametric equations, over the given interval 0 ≤ t ≤ 2, find the following.
(i) The average rate of change for x(t).
(ii) The average rate of change for y(t).
(iii) The slope of the graph between the points on the curve corresponding to t = 0 and t = 2.
Solution :
(i) The average rate of change for x(t) :
(ii) The average rate of change for y(t) :
(iii) The slope of the graph :
Problem 3 :
x = 2(t - sin t)
y = 2(1 - cos t)
For the above parametric equations, over the given interval 0≤ t ≤ 2π, find the following.
(i) The average rate of change for x(t).
(ii) The average rate of change for y(t).
(iii) The slope of the graph between the points on the curve corresponding to t = 0 and t = 2π.
Solution :
(i) The average rate of change for x(t) :
(ii) The average rate of change for y(t) :
(iii) The slope of the graph :
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