#$&* course Mth 173 9/14 11:50
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Given Solution: `aThe three points are (10, 80), (40, 40) and (90, 20). From the first point to the second the rise is from 80 to 40, or -40, and the run is from 10 to 40, or 30. So the slope is -40 / 30 = -1.33. From the second point to the third the rise is from 40 to 20, or -20, and the run is from 40 to 90, or 50, so the slope is -20 / 50 = -.4. Click on 'Next Picture' to see graph. `routine graph3 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q002. Look at your results for the slopes, and look the results for the average rates of change. What do you notice? In what way then does the graph represent the average rate of change? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We can surely notice that the average rate of change and the slopes are same. Let us consider the first 2 points that are ( 10 , 80 ) and ( 40 , 40 ). A rise of -40 as indicated in the slope is nothing but an increase of -40 cm in depth. Now a run of 30 is nothing but the time interval in seconds in which the change in depth occurred. Thus the average rate at which the depth changes can be given by dividing the increase in depth by the time duration in which the change occurred that is -40 cm / 30 sec, which thus says that the depth changes at an average rate of -1.33 cm/sec. The slope of the line is nothing but rise / run which is also -1.33. Thus it can be said that the average rate of change is nothing but the slope. Thus the graph represents the average rate change by its slope. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe slopes and the rates of change are numerically equal. For example between the second and third points the rise of -20 represents the -20 cm change in depth and the run of 50 represents the 50 seconds required to make this change, so the slope represents the -20 cm / (50 sec) average rate of change over the second time interval. We therefore see that slope represents average rate of change. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q003. To what extent do you think your graph with three points and straight line segments between them accurately depicts the detailed behavior of the water over the 80-second period of observation? How do you think the actual behavior of the system differs from that of the graph? How do you think the graph of the actual behavior of the system would differ from that of the graph you made? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The graph with the 3 points and straight segments between them just depict the average rate at which the depth changes over the time interval and not the actual rate at which it changes. The actual graph would instead be a curve and not a straight line passing through the 3 points. This can be fairly said as the rate at which depth decreases decrease as time passes by, thus the graph would be one with decreasing steepness, whereas a straight line indicates one with a constant slope and a constant rate of change of depth which is the average rate of change of depth. Thus concluding the though the graph I made could calculate the average rate of change of depth with time only whereas the actual graph would be a curve with the slope at each point indicating the rate of change of depth at that particular time. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe straight line segments would indicate a constant rate of change of depth. It is fairly clear that as depth decreases, the rate of change of depth will decrease, so that the rate of change of depth will not be constant. The graph will therefore never be straight, but will be a curve which is decreasing at a decreasing rate. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `q004. From the given information, do you think you can accurately infer the detailed behavior of the water depth over the 80-second period? Do you think you can infer the detailed behavior better than you could the values of the stocks? Why or why not? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 3 points given would almost accurately infer the detailed behavior of the water depth over the 80-second period. As far as we know that the graph is a curvilinear graph and the 3 points are accurate we definitely can obtain the 3 parameters using the 3 points and obtain a graph passing through them. The graph would be almost the same as the actual graph, accuracy will increase if the points are as far as possible. The detailed behavior of the change in water depth would be inferred much better than that of the stock values. This is because stock market has no particular rate to change, someday it may increase drastically and the next day it will decrease slowly, there is no such fixed change or rate, whereas in case of change in depth we definitely know that when we pin a hole at the bottom the depth in going to decrease and would decrease at an decreasing rate, which make data much more detailed. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aIt will turn out that three data points will be sufficient to infer the detailed behavior, provided the data are accurate. However you might or might not be aware of that at this point, so you could draw either conclusion. However it should be clear that the behavior of the water depth is much more predictable than the behavior of the stock market. We don't know on a given day whether the market will go up or down, but we do know that if we shoot a hole in the bottom of a full bucket the water level will decrease, and we expect that identical holes in identical buckets should result in the same depth vs. clock time behavior. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!