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Unformatted text preview: we get u(x,y) = x^2 y^2/2 + f(y) where f(y) is some undetermined function of y . Putting this back into the other equation we get x^2 y = x^2 y + f'(y) so f is actually a constant, i.e. u(x,y) = x^2 y^2/2 + C. Substituting x=0,y=0 we get C=1 , so u(1,1) = 3/2 . Question 5: Since ln|x| is an anti-derivative of 1/x for all x other than 0, we have from the Fundamental theorem of Calculus that the integral is ln|-2| - ln|-4| = ln 2 - ln 4 = ln (1/2) = - ln 2. Question 6: Since x ranges between -1 and 1, x^3 ranges between -1 and 1 and so 100 + x^3 ranges between 99 and 101. Thus 1/(100+x^3) <= 1/99. Since the integral of 1/99 from -1 to 1 is clearly equal to 2/99, we are done. Solutions to the quiz 1 of 1 9/19/2011 3:46 PM...
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This note was uploaded on 09/19/2011 for the course MATH 132 taught by Professor Grossman during the Spring '08 term at UCLA.
- Spring '08