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題組內容

1. Assume a (real valued) function y = f(x) satisfy y⁵+9^y=x³+x. Suppose g(x) is a (real valued) function satisfying f(g(x))=x.

(b) (10%) The derivative g'(1)=_____

其他申論題

10 Consider the funetion f(x,y) = ax2 + βy2. Find values for a and β so that (a) (0,0) is a local minimum (b) (0,0) is a local maximum, and (c) (0,0) is a saddle.

(a) (5%) The value g(1) must be 2. True or false? Answer:_____

(a) (5%) Let s(t) be the distance (in meters) between the two particles in t minutes. s(t)=_____

(b) (10%) Suppose at t= t0 the two particles are closest to each other. Then t0=_____

3. (10%) Compute the indefinite integral =_____

4. (10%) Let Ω be a regioin (on the xy-plane) enclosed by , x=0 and y=e. Let S be the solid obtained by revolving Ω about the y-axis. Let V be the volume of S. V=_____

(a) (10%) If the unit vector u (in the xy-plane) at p is the direction (among all directions at p) along which the height (i.e. the value of z) of S increases most rapidly, then u =_____

(b) (10%) Write H for the plane x+2y+3z=1 and the curve C for the intersection S∩H. Let L be the tangent line to C at P and N be the plane perpendicular to L at P. Then the equation of N is _____

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