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Summary: Math 2200 Homework 5 (Finish by Friday Oct. 22)
Problem 1 Find the domain and range of f(x, y) = xy+1
(y-2x+2)
.
Solution: Since this is a ratio of polynomials the only possible problem is a divide by zero so the
domain is the plane less the line where the denominator is zero: D = {(x, y) : y = 2x + 2}. The
function shoots of to both ± as we approach the divide-by-zero line. By plugging in values we
see the [c, ) and (-infty, d] ranges on either side of the singularity overlap and so the range is
(-, ).
Problem 2 Find the tangent plane to
f(x, y) =
2x + 3y
x2 + y2
at x = 1, y = 1 and also at x = 3, y = 2.
Solution:
fx =
(x2
+ y2
)(2) - 2x(2x + 3y)
(x2 + y2)2
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