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3 cs309 G W Cox – Spring 10 The University Of Alabama in Hunt sville Computer Science Proving the Theorems Example Theorem 1 x x = x;Find all the zeroes of the polynomial 2xcube xsquare 6x 3 if 2 of its zeroes are √3 and √3 IF one of the zeros of quadratic polynomial is f(x)=14x²42k²x9 is negative of the other, find the value of k

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(e) Below is the graph of z = x2 y2 On the graph of the surface, sketch the traces that you found in parts (a) and (c) For problems 1213, nd an equation of the trace of the surface in the indicated plane Describe the graph of the trace 12 Surface 8x 2 y z2 = 9;X¡2 1 = y ¡4 ¡5 = 03 4 We solve this system to obtain x =2 3 4 = 11 4 y =4¡5 µWe know that x = 5 and y = 12, so we can use the original equation to determine that z = 13 We can plug in all the values that we now know 2(13) dz dt = 2(5)2 2(12)3 dz dt = 46 13 Note that z could also equal −13, but I am assuming that the physical restrictions behind this problem would not allow such an answer

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Z 1 0 2πrhdy = Z 1 0 2π(y 2/5)(12(y2 −y3))dy = 24π Z 1 0 2y2 5 3y3 5 −y4 dy = 24π 2y3 15 3y4 − y5 5 1 0 = 24π 2 15 3 − 1 5 −0 = 2π 38 The region shown here (bounded by y = x2, y = −x4 and x = 1) is to be revolved about the yaxis to generate a solid Which of the methods (disk, washer, shell) could you use toP xis an increasing function, the previous inequality implies that d(x;y) = p jx 1 y 1j p jx 2 y 2j p jx 1 z 1j jz 1 y 1j p jx 2 z 2j jz 2 y 2j p jx 1 z 1j p jy 1 z 1j p jx 2 z 2j p jy 2 z 2j d(x;z) d(z;y) The metric is not derived from a norm on R2 since d( x;

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