MATH110 April 2018
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q2 (e) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q7 (c) • Q7 (d) • Q7 (e) • Q8 (a) • Q8 (b) • Q8 (c) • Q8 (d) • Q9 • Q10 •
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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Hint 1
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Find the relation between the length of the silk thread and the -coordinate of the location of the spider.
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Hint 3
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Use either the chain rule (Solution 1) or the implicit differentiation (Solution 2).
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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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Solution 1
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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.
Let be the -coordinate of the location of the spider in time . Since the spider moves along the graph , the location of the spider in time is . Note that the length of the silk thread is the distance between two points and . Using the distance formula, we get
Since is a function of and is the function of , by the chain rule, we have
By the Hint 2, we have .
Since can be written as the composite of two functions , where and , we apply the chain rule with and to get
Plugging and into the formula for , we obtain
At the location of the spider , i.e., , we therefore have
Answer:
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Solution 2
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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.
Let be the -coordinate of the location of the spider in time . Since the spider moves along the graph , the location of the spider in time is . Note that the length of the silk thread is the distance between two points and . Using the distance formula, we get
Now, we find using the implicit differentiation. Since
we take the derivative with respect to to get
On the other hand, we compute the derivative of the function with respect to using chain rule (note that x depends on time t):
Therefore
Noting that and , we have
Finally we isolate the derivative of with respect to to get
Answer:
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