Science:Math Exam Resources/Courses/MATH200/December 2011/Question 01 (a)

MATH200 December 2011

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q2 (a)  •  Q2 (b)  •  Q3 (a)  •  Q3 (b)  •  Q4  •  Q5 (a)  •  Q5 (b)  •  Q6  •  Q7  •  Q8 (a)  •  Q8 (b) i  •  Q8 (b) ii  •  Q8 (b) iii  •  

   Other MATH200 Exams

  •  April 2012  •  December 2011  •  

Question 01 (a)
Consider the function $\displaystyle f(x,y)=e^{-x^{2}+4y^{2}}$ a) Draw a "contour map" of $f$, showing all types of level curves that occur.

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.

Hint 1
Remember that level curves satisfy $f(x,y)=C$ where $C$ is constant. Varying the value of $C$ will give different level curves.

Hint 2
Don't let the appearance of the exponential function trick into thinking this problem is more difficult than it is. Consider that if $f(x,y)=C$, then $\displaystyle \ln(f(x,y))=\ln(C)$ provided $C>0$.

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.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The level curves of $f(x,y)$ satisfy $f(x,y)=C$ for arbitrary values of $C$ such that a solution to $f(x,y)=C$ exists. Starting with that equation we have $\displaystyle e^{-x^{2}+4y^{2}}=C.$ We notice that we can take the natural logarithm of both sides giving us $\displaystyle -x^{2}+4y^{2}=\ln(C),$ where $\ln(C)$ is just an arbitrary constant as well! In other words, the level curves of $e^{-x^{2}+4y^{2}}$ look the same as those of $-x^{2}+4y^{2}=g(x,y)$. The only difference between the level curves in each situation is the values of $C$ corresponding to each specific curve. The level curves of $g(x,y)$ are hyperbolae. If we choose five different values of $C$: $C=e^{-2},e^{-1},1,e^{1},e^{2}$ and draw the resulting level curves, we get the figure below. (Hint: If you begin drawing the level curves starting with $C=1$, the remaining curves should be easier to draw) Some level curves of f(x,y)

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