Asked Oct 30, 2019

Let f(x,y)= x2-y2

a) On the set of xy-axes, draw the level curves L(x,y) = k for k= 1,2,3. On another set of axes, draw the level curves f(x,y)= k for k=1,2,3. (Do this part by hand)

b) How do the contour maps in part (a) reflect the fact that L is the linear approximation to f at the point (2,1)? Explain briefly in words.



Expert Answer

Step 1

The level curves f (x, y) = k are just the traces of the graph of f in the horizontal plane z = k projected down to the xy-plane.



Let f (x, y) = x2 – y2, now the level curve k = x2 – y2


For k = 1


Consider the curve k = x2 – y2


Image Transcriptionclose

=1 k k this is the equation of hyperbola where k 0 Ifk is positive, the hyperbolas open to the left and right. If k is negative, the hyperbolas open to up and dowm.

Step 2

Now suppose for k =1


The obtained equation is equation of hyperbola and open up to the left and right, shown in the below diagram:



Image Transcriptionclose


Step 3

For k = 1, 2, 3, …. The obtained curve is sh...


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