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) Ah, the magical graph where average and instantaneous rate of change hold hands throughout its entire domain! Picture a straight line that never wavers, like a robot doing the robot dance. In this case, the average rate of change between any two points will always be equal to the slope (or rate of change) of that line at any given point. It's as if time freezes and everything is in perfect harmony, like a synchronized swimming routine.
A real-life situation that could be represented by this graph is a car driving uniformly at a constant speed on a straight road, without any stop signs, traffic lights, or other disruptions. The average rate of change between any two points on the graph would simply be the speed of the car, which is also the instantaneous rate of change at any given point.
b) Alright, let's dive into the world of more specific scenarios!
i) Imagine a graph that looks like a roller coaster with a steep rise and a sudden drop. At one specific point, where the roller coaster is at its highest point (let's call it Point A), the average rate of change between any other point and Point A will be equal to the slope (or rate of change) at Point A. It's like being at the peak of excitement and experiencing that adrenaline rush.
A real-life situation that could be represented by this graph is a bungee jumper leaping off a bridge. At the moment when they reach the highest point of their fall, the average rate of change between their position and that point will match the instantaneous rate of change at that highest point.
ii) Now, picture a graph that resembles a see-saw, with two points on either side of a middle point (let's call it Point B). If the slope (or rate of change) at Point B is equal to the average rate of change between the two other points, it's like having perfect balance. Just like a skilled tightrope walker, the rate of change at the midpoint keeps everything in equilibrium.
A real-life situation that could be represented by this graph is a pendulum swinging back and forth. At the midpoint of its swing, where it reaches its highest point, the average rate of change between the two extremes will match the instantaneous rate of change at that midpoint.
c) For the roller coaster graph (i), a real-life situation could be a skydiver free-falling from a plane. At the highest point of their descent, the average rate of change between their position and that point would match the instantaneous rate of change at that highest point.
For the see-saw graph (ii), a real-life situation could be a pendulum in a clock. At the midpoint of its swing, the average rate of change between the extremes of its swing would match the instantaneous rate of change at that midpoint.
a) A graph that represents a constant function, where the value of the function remains the same for all points in its domain, will have the average rate of change equal to the instantaneous rate of change for its entire domain. This means that the slope of the tangent line at any point on the graph will be the same as its average rate of change over any interval.
A real-life situation that this graph could represent is driving a car at a constant speed on a straight road. The distance covered by the car would be the function, and since the car is traveling at a constant speed, the distance covered by the car would change at a constant rate. The average rate of change between any two points on the graph would be equal to the rate at which the car is traveling at any given moment.
b) i) A graph that represents a linear function will have the average rate of change between any two points equal to the instantaneous rate of change at one of the two points. This means that the slope of the tangent line at one of the two points will be equal to the average rate of change between those two points.
ii) A graph that represents a quadratic function will have the average rate of change between two points equal to the instantaneous rate of change at the midpoint between those two points. This means that the slope of the tangent line at the midpoint will be equal to the average rate of change between the two points.
c) i) A real-life situation that could be represented by a linear graph is a car accelerating or decelerating at a constant rate. The function would represent the car's distance covered over time, and the average rate of change between any two points would represent the car's average speed over that interval. The instantaneous rate of change at one of the two points would represent the car's speed at that particular moment.
ii) A real-life situation that could be represented by a quadratic graph is an object being thrown vertically upwards or downwards. The function would represent the object's height above the ground over time, and the average rate of change between two points would represent the average velocity of the object over that interval. The instantaneous rate of change at the midpoint between two points would represent the object's velocity at the peak height.
a) To have a graph where the average rate of change is equal to the instantaneous rate of change for its entire domain, we need a function that is both continuous and linear. A linear function has a constant rate of change throughout its entire domain, which means the instantaneous rate of change remains the same for any point on the graph.
For example, consider a graph of a car's speed over time during a trip where the car maintains a constant velocity. The graph would be a straight line, indicating that the car is traveling at a consistent speed and the instantaneous rate of change (the slope of the line) remains the same for the entire duration of the trip.
b)
i) To have a graph where the average rate of change between two points is equal to the instantaneous rate of change at one of the two points, we need a function that is nonlinear and exhibits symmetry. A parabolic function is a good example of this. The average rate of change between any two points on a parabolic graph will be equal to the instantaneous rate of change at the vertex of the parabola.
ii) To have 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 again need a nonlinear function. A cubic function provides a good example of this. The average rate of change between two points on a cubic graph will be equal to the instantaneous rate of change at the midpoint between the two points.
c) Real-life situations for the graphs described in part b:
i) One situation that could be represented by a parabolic graph is the trajectory of a projectile. When an object is launched into the air and follows a parabolic path, the average rate of change between any two points on the trajectory will be equal to the instantaneous rate of change at the highest point of the trajectory.
ii) A situation that could be represented by a cubic graph is the growth of a population over time, where the growth rate is constantly changing. The average rate of change in population size between any two points will be equal to the instantaneous rate of change at the midpoint between the two points. This could be seen, for example, in the growth of a bacteria colony or the population of a city.
a)
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?
b)
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.