Kyle Vansant's Second Blog Post

Jimmy is speeding down a deserted country road at 70 mph when he encounters a school zone. Jimmy knows that school zones have speed cameras, so he slows down rapidly to the posted speed of 25 mph over five seconds. What was Jimmy's acceleration at the third second after he hit the breaks?

Time (seconds)	Velocity of Jimmy's Car (mph)
0	70
1	65
2	57
3	46
4	32
5	25

Secant #1: (3,46) to (5,25)        ARC of Secant #1: 25-46 = -10.5
                                                                                    5-3

Secant #2: (3,46) to (4,32)        ARC of Secant #2: 32-46 = -14
                                                                                    4-3

Secant #3: (3,46) to (3.5, 39)    ARC of Secant #3: 39-46 = -14
                                                                                  3.5-3

The ARC of these secant lines should be getting closer and closer to the IRC at point (3,46). However, because the derivative of the function that describes the velocity of Jimmy's car is linear between one and four seconds, it is possible that the IRC from when one second after Jimmy hit the breaks to four seconds after he hit the breaks is the same for each point within that interval.

Tangent Line Points: (2.5,53) and (3,46)
Tangent Line Slope: 46-53 = -14
                                  3-2.5

At three seconds after Jimmy hit the breaks, his acceleration was -14 miles per hour per second. This calculation is both possible and accurate because the tangent line, while only touching one point on the function, provides enough information to calculate the IRC, This makes the tangent line a much more accurate method for calculating derivatives than secant lines, because it is not an estimate. The tangent line is a linear function with an easily calculated slope that incorporates the intended point of measure and only the intended point of measure from the original function. The ARC of a tangent line is equal to the IRC of the intended point of measure.

I know that the value of the IRC is -14 because the secant lines between one and four seconds and the tangent line have exactly the same slope. This is possible because I used a piecewise function to model the behavior of the car as it decelerated. Between seconds one and four, the derivative of this function has an ARC of -14.