MATH200 December 2012
• Q1 (a) • Q1 (b) • Q2 • Q3 (a) • Q3 (b) • Q3 (c) • Q4 • Q5 • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q7 (c) • Q8 (a) • Q8 (b) • Q9 • Q10 •
Assume that the function satisfies the equation and the mixed partial derivatives and are equal. Let be some constant and let . Find the value of such that
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?
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.
Remember the Chain Rule for function depending on several variables:
Use the Chain Rule to calculate ,
and then the second derivatives applying the chain rule twice.
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We apply the Chain rule to calculate
For the second derivatives, we apply the Chain rule again to the already calculated derivatives
Now we set together
where the last step is already given by the problem.
we conclude that is the value that
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