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### Solutions to Exam I (Blue), Ma123,F04

Course: MATH 123, Fall 2004
School: SDSMT
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Word Count: 695

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123 Math Calculus I Exam I FaD2004 RWJ Instructions : Readquestions carefully. Name: 5ol..J..'1)o,-J'J Please show your work/reasoning asto help me awardyou (partial) credit. so You will necd to compute a dmvative of a function &quot;using the definition&quot; only in question5; in all otherproblemsusethe variousrules/shortcuts we've discussed that Note point values- 105points are listed.the maximumyou...

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123 Math Calculus I Exam I FaD2004 RWJ Instructions : Readquestions carefully. Name: 5ol..J..'1)o,-J'J Please show your work/reasoning asto help me awardyou (partial) credit. so You will necd to compute a dmvative of a function "using the definition" only in question5; in all otherproblemsusethe variousrules/shortcuts we've discussed that Note point values- 105points are listed.the maximumyou may earnis 100. Good luck! " 1. (20 pts.) Suppose that the distance,in miles, that you've traveled along 1-90 east of Rapid City after 1minutesis given by 601 f(t)=-, 1+8 O~t~40. a. Computethe average velocity over the time interval from 1 = 12to 1 = 22 minutes. } - ('l "2..) -k ( ~ { - .f:f-=-_~~.:::. 3' ~ 10- '2 ~ - 11- """i~.Jk b. Computethe instantaneous velocity at time t = 16 minUtes. k I ( t) =- ~ ~ ~ -+"""~8' ) -:~.!_:~:: ..!~~- ( ~..,.. ) 18' C K 1... --E:-..I) f} D 1/./, ) (1'0/ - -"'2 f- """~ Ler.s:J.'-\I~ . --' > """'-,,~ --1"'i ~. 2 c 2. (36 pts.) Computethe derivativesof the given functions.Don't wastetime simplifying. a. f( x= 3-2x-x2 ) 4x2 -I ,f'()(').= (-2 - - '2.l<)(9~1 1) - (:i-2,,-?C'2)(r?C) '\.. (4?< "L.- ,) b. g(x) =( -X+I ) (2x-5) x+2 -l- ~ , t ) - (d C~ J) (Lx-5) I X ~ '/(. 1< .f' '1. (~ -> ) )(... 1- ) '2... - (I) (~+1...)- (7<.fI)CI) ( ~ 1..",( ~ '2.. L?( '1.) (- ) ?(./-( 'IC,. 1/ 'L c. h(x) =x2 +5-3x-2 A I(X) -:: '2." + ')(-3 d. w(x) =(sin .J4~2-:;5-;-:;Z)3 Lc) '::. J ()" ,~ ~~ ItiC.) )"2.C W~ C;IJ- ) . - V".. + S) 3 -+ ( 1 x.1 Ii" '2...) # (r" 1(."" . 3. (12 pts.) For eachof the following indicate whetherthe statement true or false (no is work necessary): a. If fis differentiableat x, thenfis continuousat xer False (circle one) b. Foranyfunction!: ~f(x) =f(c). Trill' oreCilClC one) rJZ) - D pI...i c.PVf~,Jrf- J t' olJ1) J c. ~(fg) 1J1Jt:A.J.5 ~ ~ = ~~. True orScircle df dx one) f'JD - f26t~t,.i- r~'tJt>""r- nl)~E. d. -(f dx d + g) = -+-. dx dg e True r False (circle one) e.~(~)=~/~. TrueoeCin:le one) - VI? fl.&t"t...t.. crvDl1 ~ rw~ f. -f(f(g(x=f'(g(xg'(X). eFalse(CircleOne) CHA,f' JZut..c / 4 .. 4. (15 pts.) Find ~ using implicit differentiationif dx X3-3x2y+2xy2 =12 d J')( ["K 3 -..>~ 1 ,..1 l1C~'I. J '::. ~ c19 .J,'t .,.,.. '2. ./ -.3 .1x."'" C'l ~" of- "'1( '1... ~J dx 1- C ~ -I- 7.1'7 "2.. ~] J?C- -::: 0 ':J "X')..- b'1(" - > ?C '2.. ~ d)<. '2'?'2. + -x, ~ J~ = 0 (47<1-3 x:'2..) b ),x = - 3')('1.-#- "7<""2,/1"1C'-'-=~ ~ ,j~ = -~..f 1?<'7-3'"'r<'1... ., 5 5. (16 pts.) Let f(x) = 3x2-4x+5 a. Compute f(x+h)- f(x). h for both partsof this problem. f(7""~)":: ~{'X'+,,)"'- t()CJ.I.)""".; ':. '3(?C "1..f-2 ~ ~ f- L. '1.) 1-- ( X'" "') - .Eo- .of"' S ~ 3")('"2- - + 6)C~"""lt -: b?< ~ -#- -t~-4-~"'.s -- Jt1';4) - r{?t) 341..- 1L, -:~i=~ (~~~~~=~~ / b. Using,earta, compute lim f(x+h)- f(x). h~O h L A~l> tf~~ -J-I"~ ), :. (Z hoo-tC> 6'"X -+34- 4- = (~-=~-) 6 6. (6 pts.) Koch's Snowflake,namedafter the Swedishmathematician Helge von Koch (1870-1924) is obtained as follows. Given an equilateral triangle, place equilateral triangles on the middle third of each side. Repeatthis process.The first four Koch Snowflakes shownbelow: are I . . , , . , , . , , , ,. , , , , , , , , , , , , , , , , , , ' ,: , , , , ' 2. 3. 4. If the first snowflake(the equilateraltriangle on the left) has sidesof length one, then it canbe shownthat p(n) =perimeterof nib snowflake 3( ~)n-1 = a(n) = area of nib snowflake =4+~ .J3 3.J3( 1-( '9 n-1) ) 4 Here,finally, are your questions: a. What can you say about th process? That is, tell me about ake peri limp(n)? n, you continue to repeat this ~ h~(:7D :3 ( -f) ~ 11\"'( C~:"') -= ~~ .,,>S r 'J~v ~,~\I"' a \-1 \"~ b. Whatcan you sayaboutthe'::~::'::j That is, tell me about ':=~~:::'::.J~ lima(n)? n, as you continue repeat process? to this ~ ~"'PC l1{J...:. ~ 3Ji ( I - (~)k_l) l ~ ~ '2.0 ~ I) ~6 r; 7 -: .pi -+ ~::. ':ti=
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