Decide if the following statements must be true, might be true, or could not be

Question

Decide if the following statements must be true, might be true, or could not be true. Explain your reasoning. For a), b), c), assume that the function z=f(x,y) is defined everywhere.

a) The level curves corresponding to f(x,y)=1 and f(x,y)=-1 cross at the origin.

b)The level curve f(x,y)=1 consists of the circle x2+y2=2 and the circle x2+y2=3, but no other points.

c)The level curve f(x,y)=1 consists of two lines that intersect at the origin.

d)For the surface defined by z=e-x^2, there is a level curve thorugh every point (x,y) and these level curves are lines parallel to the y-axis.

Explanation / Answer

Decide if the following statements must be true, might be true, or could not be true. Explain your reasoning. For a), b), c), assume that the function z=f(x,y) is defined everywhere.

a) The level curves corresponding to f(x,y)=1 and f(x,y)=-1 cross at the origin.

FALSE ...PROOF BY COUNTER EXAMPLE ...

CONSIDER F[X,Y]=X+Y ...X+Y=1 AND X+Y = -1 DO NOT CROSS AT ORIGIN .

b)The level curve f(x,y)=1 consists of the circle x2+y2=2 and the circle x2+y2=3, but no other points.QUESTION IS NOT VERY CLEAR ....

DO YOU MEAN ? ....F[X,Y] = X^2+Y^2-2 = 1 ........AND F[X,Y]=X^2+Y^2-3=1

IN SUCH A CASE IT IS FALSE ..WE CAN HAVE F[X,Y]= X^2+Y^2-4=1 ALSO ..

IF YOU MEAN ...F[X,Y] = X^2+Y^2 ONLY , THEN IT CONSISTS OF THE CIRCLE ..

X^2+Y^2=1.....THE UNIT CIRCLE ONLY .....

c)The level curve f(x,y)=1 consists of two lines that intersect at the origin.

FALSE ...PROOF BY COUNTER EXAMPLE ...

CONSIDER F[X,Y]=X+Y ...X+Y=1 AND X+Y = -1 DO NOT CROSS AT ORIGIN .

d)For the surface defined by z=e-x^2, there is a level curve thorugh every point (x,y) and these level curves are lines parallel to the y-axis.

THAT IS THE SURFACE IS INDEPENDENT OF Y ..FOR ALL VALUES OF Y WE SHALL HAVE Z = E^[-X^2] SO ..

WE SHALL HAVE F=Z - [E^-(X^2) ] = C , WHERE C IS ACONSTANT ...SAY C= 0 , C=1,..ETC ...IN ALL THESE CASES WE WILL HAVE X AND Z RELATED AS ABOVE FOR ANY VALUE OF Y ..THAT IS WE SHALL HAVE PARALLEL CURVED SYRFACES WITH THE LEVEL CURVE OF F=Z - [E^-(X^2) ] = C IN 2 D FORMING ITS CONTOUR .

SINCE E^-X^2 IS DEFINED FOR EVEY VALUE OF X , WE HAVE DOMAIN WRT X AS ANY REAL NUMBER.. BUT Z IS RESTRICTED TO C+E^(-X^2)....THAT IS C TO INFINITY

ONLY Z CAN NEVER BE LESS THAN C ..

SO IF YOU ASK FOR LEVL CURVE THROUGH EVERY POINT [X,Y[ ...there is a level curve thorugh every point (x,y) and these level curves are lines parallel to the y-axis..

IT IS TRUE

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