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midterm2 review sheet_sol

# midterm2 review sheet_sol - MATH 20C MIDTERM#2 REVIEW SHEET...

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MATH 20C — MIDTERM #2 REVIEW SHEET SOLUTIONS You are strongly encouraged to solve the problems yourself before reading the solutions here. You will learn much more in that way and prepare better for the test. This solutions sheet is not guaranteed to be free of errors; please let me know if you find one. 1. Sketch the contour map of f ( x, y ) = 3 x + 2 y with contour interval m = 2 . For any c , the level curve at level c has equation 3 x + 2 y = c , or y = - 3 2 x + 1 2 c . This is a line with slope - 3 2 and intercept 1 2 c . Thus the contour curves with interval m = 2 will be lines with slope - 3 2 and intercepts 0, 1, 2, - 1, etc. 2. What do the vertical traces of the graph of the function f ( x, y ) = sin( x/y ) look like? The trace in the plane y = b is z = sin( x/b ), which is a sine curve scaled by a factor of b . The trace in the place x = a is z = sin( a/y ), which is a sine curve that stretches out towards infinity and becomes infinitely oscillatory towards zero, as shown below. 3. Sketch the contour maps of f ( x, y ) = sin( x 2 + y 2 ) with contour intervals m = 1 and 1 2 . With contour interval 1 the only contour curves will be at levels - 1, 0, and 1, since f ( x, y ) is always between - 1 and 1. Similarly, for contour interval 1 2 we have curves at levels - 1, - 1 2 , 0, 1 2 , and 1. Every level curve consists of a sequence of concentric circles, since sin is periodic. The countour line at level 0, for instance, consists of the circles of radius , for all integers n , since on those circles we have x 2 + y 2 = which are the places where sin vanishes. 4. True or false? (a) If the partial derivatives f x and f y both exist at a point ( x, y ) = ( a, b ) , then the function f is differ- entiable at ( a, b ) . False. (b) If lim ( x,y ) (0 , 0) f ( x, y ) exists, and the one-variable limit lim x 0 f ( x, 0) = 4 , then lim ( x,y ) (0 , 0) f ( x, y ) = 4 . True. If a two-variable limit exists, then it must also be the limit approaching that point along any direction. (c) If for some ( a, b ) , the one-variable limits lim x a f ( x, b ) and lim y b f ( a, y ) both exist but are not equal, then lim ( x,y ) ( a,b ) f ( x, y ) does not exist. True, for the same reason. (d) If f is differentiable at ( a, b ) and f x ( a, b ) = f y ( a, b ) = 0 , then the tangent plane to the graph of f at ( a, b ) is horizontal (parallel to the xy -plane). True, from the equation of the tangent plane to the graph of a differentiable function in terms of the partial derivatives. (e) If f is differentiable and f x ( a, b ) = f y ( a, b ) = 0 , while f xx ( a, b ) = f yy ( a, b ) = 3 , then f has a local minimum at ( a, b ) . False, or at least not necessarily true. If f xy ( a, b ) is sufficiently negative, then f might have a saddle point. (f) A continuous function of two variables on a closed and bounded domain achieves a maximum value. True. (g) A differentiable function of two variables is necessarily also continuous. True. 1

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(h) If f is continuous at ( a, b ) , then lim ( x,y ) ( a,b ) f ( x, y ) = f ( a, b ) .
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