Directions:
(When creating your problem, be sure to take note of these directions as well as the criteria in the rubric) here is a worked example of what you will do on your own using the instructions below.
Part a:
You are at a bowling party with a group of your friends. One of your friends is a champion at bowling and you guys want to tracks the speed at the bowling ball that he just throw.
So you guys want to find out the instantaneous rate of change of the bowling ball at 2 seconds, approximately 1 seconds before it hits the end of the alley. Thus, the experiment question is, What is the instantaneous rate of change of the bowling ball after 2 seconds?
Create a table of values for the data that you have recorded from your application/experiment.
Graph the points using the data from your table of values (connect the dots).
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.
(2, 16.4) to (2.5, 17.5)
(17.5-16.4)/(2.5-2)=2.2
(2, 16.4) to (3, 17.7)
(17.7-16.4)/(3-2)= 1.3
(1.5, 15.2) to (2, 16.4)
(16.4-15.2)/(2/1.5)= .9
The formula that was used was just the slope formula. The ARC formula shows the calculation of slope of the two points. As we can see, the speed is faster as the bowling ball is going towards the alley. The average rate of change increases when the speed increase at each seconds. For example, the ARC between 2 second and 2.5 is 2.2 and the its only 0.9 for 1.5-2 sec. This is causes because the speed was increased during these .5 sec.
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.
The tangent began at t= 0 sec and ends at t= 2 sec. The points that was used for the tangent line was (0,0) and (2, 16.4). So to calculate the slope, (16.4-0)/(2-0). The slope between these points is 8.2. So from the 0 sec to 2 sec, the IRC of the bowling ball is 8.2 mph.
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).
I know this is right because the speeds from 0 sec to 3 sec are getting faster and faster, until the ball enter the alley.The numbers are getting closer and closer to 8.2 as the seconds are increasing. For example, 2.2 ARC (2-2.5 sec) is a lot closer compare to only .9 (1.5-2).
I remember you telling me about this in class! This problem reminds me a lot about things in physics--in terms of bowling balls and bowling alleys, it makes sense for the bowling ball to be going at a faster velocity as it gets closer to the end. I think that your tangent line's a little off, though. What you have on the graph is what's called a secant line--tangent lines are found on the "side" of a graph. Does that make sense? Overall, though, this is a neat problem! 8 mph for a bowling ball sounds pretty extreme (or maybe that's just me). Good job, though!
ReplyDeletegood observations, yan, about the tangent line.
Deleteprof little
Fun example! It's interesting how the derivative can show up at a lot of situation including billowing. I never thought of it when I used to play bowling :) The tables of the values are a bit not clear, a good idea to make them on Word or Excel and then copy them to the Blog entry box. However, it looks like you did a good job overall calculating and explaining the derivative in the relationship between time and velocity.
ReplyDeleteHi! I think you've done a good job explaining and applying math into real world problem! I couldn't really read the table, they were a little small. But other than that, great!
ReplyDeleteyanny,
ReplyDeletethis is a fun example. your graphs are pretty clear and your ROC calculations are spot on. i agree with yan shi, that your tangent line looks more like a secant line and this is evidenced in the fact that your IRC calculation is a much larger value than that of your ROC secant line values. the tangent line should be (next to) the point you want to evaluate.
also, what would the units be? miles per hour per second? i believe maybe converting everything to seconds or everything to hour would have made interpreting the units a bit more clear.
prof little