course Mth 174 If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: `aThe first period (4 months) was shorter than the second (5 months), and the value changed by more during the shorter first period (increase of $300) than during the longer second perios (increase of $200). A greater increase in a shorter period implies a greater rate of change. So the rate was greater during the first period. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating #$&* OK ********************************************* Question: `q002. What were the precise average rates of change during these two periods? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To find the average rate of change, you subtract final from the initial value divided by the difference of the amount of months. So, for the first set, it would be 300 divided by 4 months, which is 75 and for the second set, it would be 200 divided by 5, which is 40 a month. confidence rating #$&* 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aFrom mid-March thru mid-July is 4 months. A change of $300 in four months gives an average rate of change of $300 / (4 months) = $75/month. From mid-July through mid-December is 5 months, during which the value changes by $200, giving an average rate of $200 / (5 months) = $40 / month. Thus the rate was greater during the first period than during the second. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating #$&* OK ********************************************* Question: `q003. If the water in a uniform cylinder is leaking from a hole in the bottom, and if the water depths at clock times t = 10, 40 and 90 seconds are respectively 80 cm, 40 cm and 20 cm, is the depth of the water changing more quickly or less quickly, on the average, between t = 10 sec and t = 40 sec, or between t = 40 sec and t = 90 sec? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Since the water went from 80 to 40 during the 30 second interval between 10 and 40 seconds, you can use the rate of change formula. 40 minus 80, which is -40 divided by 30 seconds. You can do the same for the second set of numbers. 20 minus 40, which is -20 divided by 50. These give you the rate of change. confidence rating #$&* 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aBetween clock times t = 10 sec and t = 40 sec the depth changes by -40 cm, from 80 cm to 40 cm, in 30 seconds. The average rate is therefore -40 cm / (30 sec) = -1.33 cm/s. Between t = 40 sec and t = 90 sec the change is -20 cm and the time interval is 50 sec so the average rate of change is -20 cm/ 50 sec = -.4 cm/s, approx. The depth is changing more quickly during the first time interval. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique rating #$&* OK ********************************************* Question: `q004. How are the two preceding questions actually different versions of the same question? How it is the mathematical reasoning the same in both cases? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Both of these questions deal with the rate of change. It shows that some variable started at one value and went to a different value. The rate of change also depends on another variable, which the difference would be divided by. confidence rating #$&*3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aIn each case we were given a quantity that changed with time. We were given the quantities at three different clock times and we were asked to compare the rates of change over the corresponding intervals. We did this by determining the changes in the quantities and the changes in the clock times, and for each interval dividing change in quantity by change in clock time. We could symbolize this process by representing the change in clock time by `dt and the change in the quantity by `dQ. The rate is thus rate = `dQ /`dt. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I think that my answer is a little different than the actual solution and could have been a little more in depth. ------------------------------------------------ Self-critique rating #$&* OK The remainder of this initial q_a_ assignment concerns the Modeling Project assignments that are part of your course. Question `q005: Different courses are taught with different goals and different emphasis, and no course covers everything. This is certainly true of your prerequisite precalculus (or equivalent) course. This doesn't mean that your precalculus or equivalent course was not a good course (from the preparation of students coming into these calculus courses it appears that some are excellent and some are inadequate, with most somewhere in between). Part of your present calculus course consists of Modeling Project Assignments. These assignments include ideas and techniques related to quadratic, power and exponential functions, and the concept of proportionality. They are presented in a real-world context (which is not fairly common but not universal among good precalculus courses). Ideally you will have developed these skills in your prerequisite Precalculus or equivalent course. However different courses are taught to different standards, and students frequently enter this course without a good understanding of these functions, techniques, and concepts. These topics are therefore part of your review of prerequisite material, which also includes the first chapter in your text. (You are very likely to find things in that chapter that you haven't covered in the prerequisite courses, which is of course why those chapters are there). These Modeling Projects are assigned on your Assignments page, in the appropriate assignments. Please summarize below your understanding of the reason for these modeling projects: Your response: confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Question `q006: The modeling projects are found by clicking on the obvious link on your assignments page, part of the title of which is 'Modeling Project # '. You are advised to look on your Assignments page to locate at least one such link, click on it and see that the link leads you to a worksheet or series of worksheets. Please summarize below your understanding of how to access the modeling projects. Your response: confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Question `q007: The modeling projects include a number of exercises. You are expected to do these exercises as you work through the Modeling Project. You are not expected to submit your work on the exercises, but you may certainly submit questions about them. The Open Query for the corresponding assignment will then ask selected questions about your work on the exercises in the Modeling Project, as well as other material that is part of the assignments. Please summarize you understanding of the fact that there are exercises in the Modeling Projects, which you are expected to do, and which will appear in the Open Query for the corresponding assignments: Your response: confidence rating #$&* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ "