Car Depreciation example

 

a.      Find a real world application OR design your own experimental application relating to rates of change. (in the blog folder you will find plenty of examples to get you started)

 

-        Car losing value over the years.

 

b.     Write a narrative or synopsis explaining your application/experiment and include a question. (For example, what is the velocity of the snowball at exactly 2 seconds? Or how can I find the velocity of the baseball at exactly 3 seconds?)

 

- Cars lose value over time, and the rate of loss varies according to the car’s age, make and model. For this example I examine how much a new car depreciate in value over 5 years.

 

- What is the deprecation value of the car at the third year or t=3?

c.            Create a table of values for the data that you have recorded from your application/experiment.

              

Time (years)	Value ($)
0	31252
1	23523
2	20195
3	16867
4	14248
5	11629

 

d.     Graph the points using the data from your table of values (connect the dots). 

 

 

 

 

e.            Calculate the slope (ARC) of at least three secant lines originating from the same point on your graph to three different points on your graph (i.e. maybe you want to know what happens exactly at x = 20, so your points might be (20, 62), (20, 56), (20, 50)).  Explain what you notice about the ARC of these secant lines and what the calculations mean/represent in terms of your experiment/application.

               -      23,523-31,252/1-0= -7,729

               -      20,195-23523/2-1= -3,328

               -      14,248-16867/4-3= -2619

               -      So at x=1 the points are (1, -7,729), (1, -3,328), (1, -2619)

As the car is more used it loses most of its value on the first year and then it loses less each year. 

f.            Sketch an approximation of a tangent line that passes though the same point (P) from part e to which you connected your secant lines (i.e. you would draw a tangent line through the point 20, since that is the same point that you used to calculate your three different secant lines)

 

Choose a second point (Q) on the tangent line, and calculate the slope of the line (PQ). This calculation will be the instantaneous rate of change ((IRC or derivative at a point)…be sure to identify the units correctly).  Explain what this calculation means mathematically and in terms of your experiment/application.

               -     IRC= (-7,729) + (-3,328) + (-2619) /3= - 4,558.7

               -     - 4,558.7$/year*4 years= $18,234.8

The more the years the car is used the more value it will lose

h.           Explain in detail how you know that the value from part g is the IRC. (i.e. since the values of calculations from part d are getting smaller and smaller, this shows that the slope of the secant is getting closer and closer to the tangent line … or some explanation similar to this). BE DETAILED!!!

               -   The smaller the difference in years the more instantaneous it gets. Since the value of calculations from the graph in part d are getting smaller and smaller, it shows that the slope of the secant line is getting closer and closer to the tangent line.