182E2-S2003 - MA 182 TEST II SPRING 2003 (11 pts) 1. If...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

This note was uploaded on 09/15/2011 for the course MATH 182 taught by Professor staff during the Fall '09 term at Purdue University-West Lafayette.

Page1 / 4

182E2-S2003 - MA 182 TEST II SPRING 2003 (11 pts) 1. If...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online