MATH102 December 2012
• QA 1 • QA 2 • QA 3 • QB 1 • QB 2 • QB 3 • QB 4 • QB 5 • QB 6 • QC 1 • QC 2 • QC 3(a) • QC 3(b) • QC 3(c) • QC 4(a) • QC 4(b) • QC 4(c) • QC 51 • QC 52 •
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
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If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
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[show]Hint 1
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To read slope field graphs, one chooses a t and x value and goes to that point of the graph. The slope of the line that appears there is the value of dx/dt at those particular x and t values (for example, flat horizontal lines have slope 0 and so dx/dt=0 there).
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Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
- If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
- If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
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[show]Solution
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Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies.
Field A:
- The ODE must be independent of
, because restricted to every vertical line (restricted to any ), the slope field looks equal.
- We have equilibriums at
and , because the slopes get more and more horizontal, the closer they are at .
- For
the slope is positive.
This fits exactly to the ODE (d).
Field B:
- The ODE must be independent of
, because restricted to every vertical line (restricted to any ), the slope field looks equal.
- We have equilibriums at
and , because the slopes get more and more horizontal, the closer they are at .
- For
the slope is negative.
This fits exactly to the ODE (c).
Field C:
- The ODE must be independent of
, because restricted to every horizontal line (restricted to any ), the slope field looks equal.
- We have equilibriums at
and , because the slopes get more and more horizontal, the closer they are at .
- For
the slope is positive, for and the slope is negative.
This fits exactly to the ODE (a).
ODE (b) and Field D:
This ODE causes a slope field where the slopes are horizontal for every and . This is not the case for the Field D. Hence, ODE (b) and Field D cannot fit.
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