Exam 2 Fall 2013 - Math 318 Test 2 1 2 3 Total Name Student...

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Unformatted text preview: Math 318 Test 2 Nov 11, 2813 1. 2. 3. Total: Name: Student No. 45 minutes. No aids are aliowed. There are 3 problems and 6 pages in totai, each problem is worth 10 points, for a total of 30 points. Problem 1. {3+7} (a) Let U C R” be open, and let a E U. For a. function f : U —> R”, state What it means for f to be difik—ax‘entiable at a. (1)) Define the function : R2 + ER “a /"‘x \‘rié-E \___/ H {35%;}: ¥(o)”1c{§) —>a f). :L {figflfl “ tad We {7 6’9 Qfi a “973(0) +4; '6; {j a ‘ ._., 0 +‘% {’ 0&4“ : D<C(Oo): Ca 5") 420:) 45:)" W “(:9 Mag) [3+ [<1— L (1.1+ I?) 3h— ( Li: L1 {£14. kl. £70) {He-+16 :24. (25:5) : o u “’ ° Problem 2. (5+5) (a) Consider the sphere in R3 {(mjyfi) E R312 +312 +32 2 169} Find the equation of the tangent piane to the sphere at the point (3,4,12). V » [M 2 6&3“ a?“ x 2* I (3) m h 4 .. 9 v¥(g)- (:5) , v: ,t a 24. TL 42;?” $1M A: 3% (ea in w»; (9 ) . (Jag. ) -_—.~ 0 2‘? 2-42. 5%» (a? 4. 8'; 9?— +249=~ 293w hum? +IZZ-m/5/20 (‘0) Assume that f : R3 —> R Satisfies fi+fl+fl=u Sh 0w that there exists a, differentiable ful'zction g : 1R2 -—~+ R such that f(§)=g<:::)r Problem 3.(5+5) Consider the curve ’7 : [0, 27?”! % R3 where 3cos(t) fit) 2 (3311103)) . 4t Compute the. al'clength of 7. , 49,16 4 W: {Wm/J r: (/(wawcwmame) + 4 S“ (121? £67) sJ “mangoe- 0 N :2] $1,577“ 0 (1')) Compute the curvature of 7. ...
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