Science:Math Exam Resources/Courses/MATH105/April 2012/Question 08 (d)

MATH105 April 2012

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Question 08 (d)

Short-answer questions. No credit will be given for the answer (even if it is correct) without the accompanying work. Identify and sketch the level curve corresponding to z = e of the function $\displaystyle z=e^{x^{2}+4y^{2}}$ Label the axes of your graph and plot the coordinates of at least four points on the level curve. Also, answer the following: The curve is a parabola ellipse circle hyperbola

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Try finding the level curve when z = e by setting the equation equal to e and simplifying it to classify the curve. To find points on the curve, choose simple values for x (and y) and calculate the corresponding value for y (and x).

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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. If $z=e$ then $e=e^{x^{2}+4y^{2}}$ and by taking the logarithm of both sides: $\displaystyle 1=x^{2}+4y^{2}$ This is an ellipse. The four easiest points are the intercepts. If $x=0$ then $4y^{2}=1$ and $y=\pm 1/2$ are the y-intercepts. If $y=0$ then $x^{2}=1$ and $x=\pm 1$ are the x-intercepts. The sketch is below:

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