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)

a) To describe a graph for which the average rate of change is equal to the instantaneous rate of change for its entire domain, we need to visualize a graph that represents a constant function. For example, consider the graph of a horizontal line where the y-value does not change with respect to the x-value. This results in a constant slope or rate of change throughout the entire domain.

A real-life situation that this graph could represent is a car traveling at a constant speed on a straight road. Since the car maintains a constant speed, the distance it travels is proportional to the time elapsed. Therefore, the average rate of change (distance/time) is equal to the instantaneous rate of change (speed) at every point along the journey.

b) i) To describe a graph where the average rate of change between two points is equal to the instantaneous rate of change at one of the points, we can picture a linear function with a constant slope. For instance, consider a line that is not horizontal or vertical, but instead has a fixed slope. In this case, the average rate of change between any two points will be equal to the instantaneous rate of change at either of those points.

ii) To describe a graph where the average rate of change between two points is equal to the instantaneous rate of change at the midpoint between the two points, we can envision a quadratic function. Specifically, a parabola that is symmetrical with respect to its vertex. The vertex lies at the midpoint between the two chosen points. In this situation, the average rate of change between the two points will be equivalent to the instantaneous rate of change at the midpoint.

c) For part b):

i) A real-life situation that could be represented by a graph with a constant slope could be a car traveling at a steady speed on a straight road. The average rate of change between any two points on the graph would correspond to the car's constant velocity, which represents the instantaneous rate of change at any point.

ii) A real-life situation that could be represented by a graph with a quadratic function and a symmetrical parabolic shape could be a projectile motion scenario. For example, if a ball is thrown into the air and follows a parabolic trajectory under the influence of gravity, the average rate of change between two points would be equal to the instantaneous rate of change at the midpoint, representing the ball's instantaneous velocity at that moment.

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