Blog 2

Instantaneous results for change in price of vehicles over the years:

Depreciation refers to decrease in value of asset after certain period of use. That is why; the cars or any type of vehicles or machines that we buy today sell for fewer prices in the future. They are said to lose value over the years or depreciate and are sold at lower price than when they were bought.

The price of the vehicle decreases over the years because its value gets depreciated with time. As such, the more number of years the vehicle has been used or years passed since it was bought, the more its depreciation and less its price. There may be plenty of reasons as to why the value of assets such as cars depreciates such as vehicle condition, availability of new model, etc. Nevertheless, the rate of change in price of vehicles depends on time. Thus, time is an independent variable causing changes in the price of the vehicles. Price of the vehicles is dependent variable.

The rate of change in price of vehicles due to time can be expressed in two forms: first the average rate of change which indicates the overall rate of change within certain amount of years and second the instantaneous rate of change that indicates the rate of change in price at particular moment or instant. It is very important to consider instantaneous rate of change because it helps us understand the immediate impact on car’s value due to change in usage period or its value at a particular instant such as exactly after 2 months of use.

Thus, with the help of instantaneous rate of change we may determine what the exact value of the vehicle would be decreased after 2 months of use.

The following table provides the details of the price of a vehicle and its price over the years of its use (extracted from MHR Calculus and Vectors, Chapter 1):

Time (years)	Value ($)
0	22000
1	16200
2	14350
3	11760
4	8980
5	7820
6	6950
7	6270
8	5060
9	4380
10	4050

By plotting the data above in a graph paper and calculating the slope of secants that can be constructed in the curve formed, different rates of changes in price due to change in time or use period cal be determined. This is because the average rate of change is equivalent to the slope of secants of the curve. Secants are the lines joining two points in a curve.

Similarly, the tangent of the same curve at any point shall give the instantaneous rate of change of the dependent variable. Tangent is the straight line that touches only one point of a function. That is why tangent can provide the instantaneous answer to the rate of change.

The following picture shows the graphical representation of the data given in table above. Three secants PA, PB and PC have been drawn in the curve with point P being the common point among them. The slopes of these secants have been calculated alongside. The slopes of each of these secants give the average rate of change in the price of vehicles from the years 2 to 6, 8 and 10. Thus, the slopes of these secants give a overall picture of how the price of vehicles have changed within the period of 4, 6 and 8 years after being bought. In average, the rate of change in the price of vehicles in 4 years is $1850 i.e. the price will have decreased by that amount within that duration. The price of the vehicle will approximately decrease by $1548.3 and $1287.5 within the span of 6 to 8 years respectively. The thing to be noticed in the slope value of secants is that, the less is the distance between two points the less is the value of the slope. For example, the least difference in x-variable time from among all three secants in the given graph is 4 years. The slope value for the secant is also the least. So the closer the secant is from the common point P, the less in the slope.

Likewise a tangent has also been drawn passing through the common point P. By choosing a random second point Q (1.4, 16000), the slope of the tangent line is also calculated. This value of slope is the instantaneous rate of change or the derivative of function (relation between time and price of vehicle). Thus, after 1.4 years of buying the car, its value will decrease by the amount of $2750. The slope of this tangent QP gives the instantaneous rate of change in price of vehicle when at specific point of time that is when time (x) = 1.4 years.

Tangent touches a curve at only one point. At the same time if a secant is moved closer and closer to a common point, at one moment it is going to touch only one point in the curve, i.e. they slowly start behaving like a tangent. A logical extrapolation shows that the slope of tangent indeed gives us the instantaneous rate of change. From our calculations we can see that as we have moved the secants closer to point P, the values of their slopes have also decreased. If you continue decreasing the distance between points in the curve and point P, at an instance when the distance between two points becomes zero, and it is at that moment the secant will behave like a tangent. The slope of this line is going to give the instantaneous rate of change in price. Thus, we can say that the value obtained as a slope of the tangent QP is in fact, the instantaneous rate of change of price of the vehicle with respect to independent variable time.