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

Consider the function a) Draw a "contour map" of , 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 where is constant. Varying the value of 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 , then provided . 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The level curves of satisfy for arbitrary values of such that a solution to exists. Starting with that equation we have We notice that we can take the natural logarithm of both sides giving us where is just an arbitrary constant as well! In other words, the level curves of look the same as those of . The only difference between the level curves in each situation is the values of corresponding to each specific curve. The level curves of are hyperbolae. If we choose five different values of : and draw the resulting level curves, we get the figure below. (Hint: If you begin drawing the level curves starting with , the remaining curves should be easier to draw) 