234-seeger-08fafin - .Math 234 Final Exam Do problems 1—8...

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Unformatted text preview: .Math 234 Final Exam Do problems 1—8. Problems 9/ 10 are for extra credit (given only for substantial work). If you leave early then leave quietly. YOUR NAME: No. l: No. 2: No. 3: No. 4: No. 5: N0. 6: No'. 7: No. 8: sum: No. 9 and 10. E.C’.: 1. A curve C in the plane is given in polar coordinates by the equation 7" = 2 cos 6 where 6’ ranges from 0 to 7r / 2. (i) Sketch the curve. (ii) Write down a parametrization t 1—> R(t) of this curve; it is recommended to choose the angle as a parameter. (iii) Find the total length of the curve. (iv) Find the line integral f0 y ds. Formulas: 2 sin6cos 6 = sin(20), 2 0082 6 = 1 + cos(26). 2. Assume the differentiable function of three variables, f (3:, y, 2), satisfies f(P)=-3, fz(P):2)fy(P)=4afz(P):—3. for the given point P = (77 3, 1). ) Compute the value of the directional derivative of f at P in the direction 1 v (a (b) Find the direction of maximal decrease of f at P. . (0) Suppose that the surface (through P) is given by the equation f (ray, z) = 3. Find an equation for the tangent plane through P. (d) A UW professor teaching Math 234 is making up questions for the final exam. Given f as above he wants to find an example of a path of a particle, r(t), which passes through the point P = (7,3,1) at time t = O with speed 2, and he wants to make sure that the t—derivative of the function f (r(t)) at time t = 0 is equal to zero. Find such apath r(t) (and provide the proper justification). 11 3. Consider the surface defined by e:E + yz3 + z4 = 2 Check that the point P = (0,0,1) belongs to the surface. The equation defines 2 as a function of a: and y, so that z 2 2(90, y) (at least for small x, y) and so that 2(0, 0) = 1. Compute the partial derivatives 82 dz 82 2 @(050)) 'a—$(O)O)a and (8y)2 (0, 0). 4. Let ' F(x, y) = (6312 + 3x2y)i + (2a:yey2 + 903)} (i) Show that F is a conservative vector field in the plane (Without first finding a potential). (ii) Now find a potential of F (Le. a function f so that V f = F), with the additional property that f(0, 0) = 1. (iii) Compute the line integral fCF - dr for the curve parametrized by r(t) = cos ti + sin(2t)j, 0 S t S 27r. 5. An iterated double integral is given as /01 { /y1 mgemydx}dy. (i) This can be interpreted as an integral ffD fdA over a domain D. Sketch this domain. (ii) Reverse the order of integration and then compute the integral. 6. Let Q be the region in 3—space given by Q: {(x,y,z):m2+y2+22 S 1, 22 0,0 Sm2+y2 g 1/4}. Make a sketch of Q and compute its volume. 7. Find the points on the curve :32 + my + y2 = 1 which are nearest to and farthest from the origin. 8. Let a curve be parametrized by 1 1 4 r(s) = —E sin(38)i + E COS(35)j + ~5fk, Where 8 E [0, 2%]. (1) Show that the curve is parametrized by arclength and determine the total length of the curve. (ii) Compute the unit tangent vector T, the principal unit normal vector N, the binormal unit vector B, all as functions of s. i ‘ (iii) Show that the curvature and torsion are independent of s and compute them. 10 9. (Extra credit). Let D be a domain which is described as D = {(6%) i a S a: S @9106) S y 3 92m}. Make a sketch. Let bD be the boundary of the curve (with a “counterclockwise” parametrization). Give the proof of the following version of Green’s theorem: if dez —-/ QJ—W—dA. bD 33/ (Find suitable parametrizations of all parts of the boundary and then reduce to the fundamental theorem of calculus). 11 10. (Extra credit). Prove the Cauchy—Schwarz inequality n 2 TL 71 k=1 i=1 j=1 Hint: One way to do this is by starting with the observation that the sum of Squares Zgzlt'mk + tyk)2 is always nonnegative and then Choose an optimal t. ...
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