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182E2-S2003 - MA 182 TEST II SPRING 2003(11 pts 1 If f:c,y...

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Unformatted text preview: MA 182 TEST II SPRING 2003 (11 pts) 1. If f(:c,y, z) = ln(:z:2 + y2 + 1) + y + 62:2 find a) Vf(1,1,0), b) the direction of maximum change of f at (1,1,0) df , z' 23' 2k . . — 1 1 h — — — —— . (3) d8 at ( , ,0) mt e 3 + 3 3 directlon (11 pts) 2. Find the volume of the solid over the triangle bounded by y = 0, y = :v, and a: = 1 under z=3—$—y. (11 pts) 3. Find all maxima, minima, and points of inflection for f (x, y) : 4x3; — x3 — y3. (11 Pts) 4' If f($,y,z) = 3623/ +yz — 2 subject to 11:2 + 3,2 + 2:2 = 6 find at 6x (m, y7 = (27 17 1). (11 pts) 5. Find the tangent plane and normal line to x2 + y2 + z2 = 30 at (1,2,5). 7rr2h , . If the volume 1s computed for 'r = 4 and h = 4, (11 pts) 6. The volume of a cone is estimate the error if r, infact, is 4.2 and h, in fact, equals 3.9. (11 pts) 7. Find the maximum and minimum values of f (m, y, z) = x — 23/ + 5z on the surface $2+y2+22 = 25. 1 . f = — (12 pts) 8 I my) 1 _x_y a) find the linear approximation, [(37, y) near (0, 0), b) find the quadratic approximation, q(x, 3;) near (0,0), 0) estimate the error |f(m,y) — “x, if < 10—2 and < 10—2. HINT: < 10—2 and < 10“2 implies 1 — m — y 2 .98. Do EITHER 9) or 10) or 11). INDICATE YOUR CHOICE: (11 pts) 9. If f(:c,y,z) has a maximum at P, a point on the surface g(x,y,z) = 10, show Vf(p) = W900)- (11 pts) 10. If a particle moves 10—2 units along the helm at = 3cos t, y = sint, z : 4t from (Pix/i 3\/§ 7r 71' T, Tm) towards (0,3, 27r), t goes from Z towards —2—, and “50, y, Z) = $2 + 1/2 + 2 estimate AF. (11 pts) 11. a) If f, fw, fy, fm, f fmy, fyy are continuous state the Taylor Formula of order 2 with ($0,310) as a starting point. 1'9“ 10(53): f(x0a b) Use the formula of part a) to ShOW |f(m,y) -— Z($,y)l S — xol + ly — yoll2 where M 2 maximum of Ifml, lfwyl, lfyyl- MATHEMATICS 182 TEST 3 (15 pts) 1) Set up integrals but do not evaluate them for the mass of the solid between z = «$2 + y2 and 1:2 +3;2 +z2 = 9 ifthe density 6 = x in A) Rectangular coordinates, B) Spherical coordinates, C) Cylindrical coordinates. (14 pts) 2) Change the integral 2 W / / dydzc 0 —\_/4-—a:2 into polar coordinates and evaluate it. (14 pts) 3) Evaluate f fds if f = xyz and C' is the line connecting (0,0,0) to (1,2, —1). 0 (14 pts) 4) Find the mass of the volume above z = y2, below 2 = 4, and between at = O and a: = 1 if the density 6 = x. (12 pts) 5) Find the average distance from (0,0,0) to a point (x,y, z) belonging to the set R = {($,y,z)|x2 +3;2 + z2 s 4). (11 pts) 6) Find the work done by the force R = yi + 503' + x214; over the curve m 2 cost, y 2 sint, zzt, Ogtg27r. (6 pts) 7) Let R = {(u,v)1 S u S 1.01, 1 S v S 1.01} be a set in the u — 1) plane_. Let :c = 11122 y = uzv + m; be a map from the u —- 1) plane into the a: — y_ plane. If R is the image of R under the given map what is the approximate area of R. (14 pts) 8) Let be the region in the :1: — y plane bounded by y = O, y = 3:, and x + 2y = 2 use the following steps to evaluate / (:1: + 2y)ey_"’dA using R the substitution u = a: + 2y , v = a: - y. A) Sketch R and its image in the u — 1) plane. 306,11) 8(u, v). C) Evaluate the integral. B) Find ...
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