a) Describe a graph for which the average rate of change is equal to instantaneous rate of change for its entire domain. Describe a real life situation that this graph could represent.
b) Describe a graph which the average rate of change between two points is equal to the instantaneous rate of change at:
i) one of the two points
ii) the midpoint between two points
c) Describe a real life situation that could be represented by each of the graphs in part b)
The equivalent statement is that:
since the instantaneous rate of change is equal to the average rate of change throughout the domain, the instantaneous rate of change does not vary.
How would you describe a function for which the instantaneous rate of change does not vary?
The mid-point theorem in mathematics says that the average rate of change of a function between two points is equal to the instantaneous rate of change of at least one point between the two end-points. Therefore the graph of any curve would satisfy condition (b)(i).
For part b)(ii), you need to draw a graph in which the tangent to the curve at the mid-point is equal the chord joining the two end-points.
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