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Level curves

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Describe the level curves of the function. Sketch the level curves for the given values of c.

f(x,y) = x^2 + 2y^2, c = 0,1,2,3,4

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The expert describes the level curves of the function. Sketch the level curves for the given values of c.

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Solution. We analyze it as follows.

(1) If c=0, then f(x,y)=c gives rise to
x^2 + 2y^2=0 <==>x=y=0
So, in this case, the curve of the function is just a point
O(0,0).

(2) If c=1, then f(x,y)=c gives rise to
x^2 + 2y^2=1 <==> x^2/1^2+y^2/[(1/2)^(1/2)]^2=1
which is an ellipse with x-intercept 1 and y-intercept (1/2)^(1/2), in other words, the intersection points with x-axis is (-1,0) and (1,0) and the ...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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