MATH237 - MIDTERM EXAM - WINTER 20071. Consider the functionf(x,y) = 2 +x2+y2.a). Sketch the level curves off(x,y),f(x,y) =CforC= 2,3,4and indi-cate how the level curves change asCis increased.b). Sketch the cross-section wherey= 0, i.e.z=f(x,0), in thexz-plane andsketch the graphz=f(x,y)in three dimensions.2. Consider the functionf(x,y) = 2−√x2+ 9y2.a). Find∂ f∂x(0,0)if it exists or verify that it does not exist. Isfdifferentiableat(0,0)?b). Show thatfis differentiable for all(x,y)negationslash= (0,0). Justify your answer.3. Consider the functionf(x,y) =1 + sin(xy)y.a). Find the linear approximation tofat(0,1).b). Use the linear approximation to approximatef(0.1,0.8).4. Consider the functiong(x,y) = 5−x2−y2.a). From the point(1,1)in which direction does the function value increasemost rapidly? Specify the direction as a unit vector.b). From the point(1,1), indicate any and all directions in which the rate ofchange is equal to zero. Specify directions as unit vectors.c). From the point(1,1), indicate any and all directions in which the rate of
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