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## ENGINEERING MATH III

by: Evert Christiansen

14

0

2

# ENGINEERING MATH III MATH 251

Evert Christiansen
Texas A&M
GPA 3.92

Jianxin Zhou

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COURSE
PROF.
Jianxin Zhou
TYPE
Class Notes
PAGES
2
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 2 page Class Notes was uploaded by Evert Christiansen on Wednesday October 21, 2015. The Class Notes belongs to MATH 251 at Texas A&M University taught by Jianxin Zhou in Fall. Since its upload, it has received 14 views. For similar materials see /class/226029/math-251-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15
Math 251JZh0u H D 03 F 0 00 CD Form Keys to Exam I Fall 2008 FormA0vlvg2xx21 Sox71 FormB0vlvg72zx21 Sox1 Form A 1311 711P271 1 72P34 03132731 gtlt P3131 075 78gt So P7z75y7824 Form B P1111P271 1 72P34 03132131 gtlt P3731 937521 So P73z75y2z76 FormA l 2y2171 hm 2y2171 2y211l 39 39 magt700 x2 y magt700 x2 02 2 y 1 1 239 FormB g y1172 g hm hm 7 my700 xzy 1 71 my700 my 17 1 my 11 1 FormAfm3y73fy2y3z4 ThusT734y7z75N273ty714t2757t FormAfm23y4fy3x73 ThusT4x73y7z75N714ty273t2757t Form A 2 7 y 7 22 z 0 h paraboloid Form B 2 y 22 z 7 8 0 e paraboloid 0 A yo All flt07y0 fwlt07110gtA fy07yoAy Form A 0 2Az 7005y0 1Ay 008 ay mfoyo 3 f1 7fy 7 Thus f0 Azy0 Ay 2847 Form B 0 2 Ax 7004y0 1 Ag 005fxy m f0y0 3 f1 7fy 7 Thus f0 Azy0 Ay 291 ii ii am 14273417 F39 FormA Fzyz 227y272z27yFm4Fy8Fz710 Thus 7 7 3 4 3f 3 a 722 u lt 7 gt7 ilt 7igt Uig FormB Fyz 2y272y27xF18Fy4Fz710 Thus 37 g 3 u7 lt 7gt77 lt7 gtu A f1 326y 0fy 6x3y2 0gt 2y 7x2x 7y2gt 2y 7iy4 GP 07 07 727 67 fyy 6y7 fmy 67D 7 At 07 07 D 736 lt 07 a saddle At 72 7 2 D gt 0 fm lt 0 a local max Form B f1 3x276y 0fy 76x3y2 0gt 2y 2 y2gt 2y 1y GP 07 6x7fyy 6y7fmy 67D 7 saddle At 22 D gt 0fm gt 0 a local min Let C 40000 32000 dwh Oh g A 2hd2hwdw dw Aw 72Cw2d 0Ad 72Cd2w 0gt 2C dwz dZw Thus d w 2040h 1020 Math 251JZhou 1x 9 03 F 0 7 OD ForrnsAampBAgC go 1 m Form A m z2y2dydx 0 ix Keys to Exam II Fall 2008 Forrns AampB Straightforward evaluation 32 Switch order 4 x5 Form A 4 ysin 2 dy dx 17 cos 16 04 o Form B 4 y cos x2 dy dx sin 16 0 0 05 39 rdrdt 1 12 Straightforward evaluation Form A 27 Form B 9 Forrns AampB 07 since E is symmetric about xzplane and the integrand is odd in y Skipped a7 since most students did not do well For b and C7 1 27r 39rcos 92 Form A 7 rzrdzdr d0 7W 4 0 1 0 4 27r 2 39rcos 92 Form B rzrdz dr d0 157T 0 1 0 g l m Form A 3 pzpz sin bdp dab d0 0 0 0 g t m Form B pzpz sin dpd d 0 0 0 1 m 8 My zx2y2dydzE7Mm0 So 0 ix WHO 9279 370 y 2 2 2 1 y 2 2 8 Form Bm x ydzdy7Mw yz yddyi7My0 So 4 0 7y 3 0 7y 15 i71707g 1 uxiy7 z 2uv7 6x7yil ltagtv2y gtTyyiul b6u7173 12 l l Cux7y17vz2y 2 ForrnA v idvdu ForrnB Lidvdu 0 0 0 0 cosu3 0 0 cosy

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