Day 23 Understanding the rate of change when it is not constant

Introduction So far, we have looked at cases where the rate of change is constant. For instance, we have considered a situation where when an equal amount of water is poured into a cylindrical beaker, the increases in height, after every instance, is the same. However, most situations in life involving the rate of change does not have a constant increase. In such a case, we consider a non-uniform rate of change. We may also refer to non-uniform rate of change as a variable rate of change since it keeps on varying.

Vocabulary: Variable rate of change This refers to an increase or decrease that is not constant with respect to a unit change. This can be done in the notebooks or on vocabulary cards. Whatever system you use 

Case 1 Get two beakers, a cylindrical and a conical one Case 1 Get two beakers, a cylindrical and a conical one. Cylindrical beaker Conical Beaker When we take a small container then use it to fill each of the above beakers, we find that in every fill, the height in the cylindrical container increases with a constant height.

For the case of a conical beaker, for every fill, the height of water increases with a smaller height than that of the cylindrical beaker. As the level of the water rises, the height of every fill increases gradually. Considering the increase in height in every fill, we have a rate which we can compare. For the case of the cylindrical beaker, there is a constant increase in height in every fill. For the case of the conical beaker, the increase in height in every fill varies; it is smaller in the beginning and keeps on increasing gradually. This is, therefore, a perfect case of a non-constant rate of change.

Drawing the graphs showing the height in every fill, we have the following. Number of fills Height Number of fills Height For conical beaker For cylindrical beaker

CASE II Consider the motion of a vehicle. Due to stop overs, traffic, downward slopes and upwards slopes, the rate at which the vehicle moves (the speed) varies. Thus there is no constant rate of change of distance but a variable one. Therefore, a vehicle exhibits a non-constant rate of motion.

Homework Describe one situation in your locality that observe a variable rate of change.

answer to the homework The student should describe a situation whose rate of change keeps on changing such as (i). Rate of walking to school (ii). The rate of swimming in the pool (iii). In most cases, the rate of improvement in class performance Among others.

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