Calc 113 - Fall 2010 exam solutions

Calc 113 - Fall 2010 exam solutions - MA 113 ——...

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Unformatted text preview: MA 113 —— Calculus I Fall 2010 Exam 1 September 21, 2010 Answer all of the questions 1 — 7 and two of the questions 8 — 10. Please indicate which problem is not to be graded by crossing through its number in the table below. Additional sheets are available if necessary. No books or notes may be used. Please, turn off your cell phones and do not wear ear—plugs during the exam. You may use a calculator, but not one which has symbolic manipulation capabilities. Please: 1. clearly indicate your answer and the reasoning used to arrive at that answer (unsupported answers may not receive credit ), 2. give exact answers, rather than decimal approximations to the answer (unless otherwise stated). Each question is followed by space to write your answer. Please write your solutions neatly in the space below the question. You are not expected to write your solution next to the statement of the question. Name: —/’%L—*__— Section: Last four digits of student identification number: Total I |-‘ r—1 r—- H H I r—Ai—a COGS) -- (1) Consider the functions f = -;—6 and g($) = x2 + 2. (a) Compute f (9(5)) and g( f (5)). Give exact answers. 1 s @ [ {/5331}: ((27): 11131 0 [ 3({(€))= 3&5}; =3(-‘} 7:3; (b) Let h be the composite function h(a:) = (f o Find the domain of h. As usual, justify your answer by showing your work. ‘ 1%!) = ¥(5{x)) = {{xzw‘zu/ 1 L3) {k = KHZ-6 l xz—‘r’ 1‘ z H-omo: MK/ :1 MMJ W {19:0, ft», x =9,” X2204“ @ L Xanl. ku Mu “Ma/M0; ,4», ix IXqét23=(—w,-2/u(—Z,2/u(2,oo/ 1 (a) f(g(5)) = g*, g(f(5)) = _3_.__ V (b) Domain of h is Maj—— (2) (a) Solve the equation 5z+2 = 7. Show all steps of the computation and give the exact answer. u. 'Tqfiz‘ma Away mdoeé x-r-Z 2 tags '7, @ Mm x:107€7-2_- (b) Express the quantity 1 10g4($6 + 1) — 10g4(955) ‘l' 510g4($) as a single logarithm. _‘ 71; 14m 6; layoff/CM Mm‘oQ 103‘! (X614) " 103‘, (7x24'é1o3? (X) @ =/959(7;:‘(]+/o§§/§K) _ (x64!) FX/ ._ fix (a) Solution is flig— (b) A;— 3 (3) Consider the function _ 2:3 —— 4 fwd—53:“. (a) Find the domain of f. “ " x .L @[ l0 $K+(20( “MC/to X:“S~' @Z "W A AM or If» 1x / wéé = (wnéM—aw (b) Find the inverse function f “1 of f. ~ 7: =2X‘LI 0w 0W ; M v ((x/ 5W x IM—x y(5x+/) = 2x41, Mm X {S‘y—Z} = ~y—9. z N ammo/m4 3" (WM "W‘ 5Y“2 1‘90 “(We affirm->3: ‘7: "5“ «Md/Q M—flmx O t! moldimcd. mum; ML. (woken 4‘, m Wad m nfll~flmw1 mm ;, _%-c( fO/q Coufnoouct‘fm. /V‘0¢Q ( m (M out/face ‘6 €y—2, W4 0616;,“ A + L! ,A X a -— Y U/ Sr-L‘ N ~r x+‘r f“ Now-c 60> *- m __,L (a) Domain of f is K I K 5 __ )0“! (b) f‘1($) = $2. (4) Use the limit rules and continuity to determine each of the following limits if it exists. If a limit does not exist, but is 00 or —00, then clearly indicate that. I ._. (a) 2fi‘5 5 =2—( kcw/Q M4 AMOL'WQ z—>16 @ 0.x muff/Mum an/aen— dmac‘ma (MM on M Mr CM/wofkau 2 (“402—16 (D A?” “‘46 EDA. A2414 . Mme (b) hm ,2 AM _ M >LM h——>0 h £40 A ’2—10 [4 Ad“) /¢( 4“ 7‘ (:j‘N a 0+8 = f 616m I64 A—vo AMOCW 24ft? (J Can/\qu 1. x2+1 (C) 11—13 117—4. J89.“ xz+( a...“ X“! on Cwé‘flum AMOL‘WI/ “a”. low (xi-H) =17 a”! low (x44: 0* M X.(, )0 xaptt x—alt" 0M4 JIM (x41) 3* 0- 4‘” X"" {‘0‘ @ X—JQ‘ 7' I X1+( 3‘ “amt A”M X 4/ _: 09 W4 /&\M_ x-“ 7. "M \ ' x—ay“ ‘K- ‘1 x—ac, . ,‘ A‘ thfiau MA M4¢-Jf¢v0.o( Lani/g an cor/WA / @ xe‘ztua‘z 4m “'01 W191, ® (5) Let f and g be two functions such that the following limits exist limg(a:) = 7, lim [3”‘f(x) — = 13. z—>2 :c—>2 Use the limit laws to compute the following limits. ‘ L‘M (K) (a) mg?) 2 z<_:z~_§___.__.. = 37 1-; ‘na- (X14) 2' “I / WL I % AM‘Y. A” Mal-«MM! I‘M—"4"“! W m cM'DuQM (b) hrn‘f(x). x—>2 Id L;-: ,aM ,£(x/_ 7514,. M Am’l— (can Wu‘oa {—3 2. a (low 3x ~ 121‘»— {(x/)— Afiy.[,(.;w 96¢) >(3/ x—I; 7‘41 x—aL x—n’. W afhfi aL ‘1 7 x ufim Kc Aug/7 at: 061°sz é? oa‘recL yuhk‘kkm ({ch 3 %X x art (egg/Mam. V °(L~2-7=/3 Jordonch 9L; 27 L23 . ._—._—. (01017.3:- flu c7M~°fiw (6) A particle is moving on a straight line so that after t seconds it is s(t) = 5t + 1 meters to the right of a reference point. (a) Find the average velocity of the particle over the time interval 1 g t S 2. A _. ( .— ® All )0 1‘9 “S (1/ S U = “‘“—"H 6 = 5 r}: 2 —~( I ==s (b) Find the average velocity over the time interval [1, t], where t > 1. Simplify your answer. (c) Use your answer in (b) to find the instantaneous velocity of the particle after 1 second. A ( F. @ [ TA('g (‘3 A‘M M g g :2 5 '3‘“; __ é*( _, 6 7/ (D f l G (D (a) Average velocity over [1, 2] is 5 % (b) Average velocity over [1, t] is 5 % (c) Instantaneous velocity at time t = 1 is 5 § (7) Using the definition, find the equation of the tangent line to the graph of the function f($) = :32 —— 555 at x = 2. Write your answer in the form y = ma: + b. q 010/“ 01 M4 9K X'al 1'; ® {’(U=AWW=OM ’4—70 A A—MD A WMML—w—sm +6 (9 eh» ———————————-—— A—oo ’4 ' =A'm (A4) A40 Hwa kn ¢%GA~“‘I a] A Lang“; AM 15 ® [ V‘ {(Z/ 1 ’(I/Z/(k‘l/ I .50 ~ y- {—6) = —/~(\<—z/( Mao ._ Y -:: ~X~L( 0 #— Equation 0f the tangent line; MM Work two of the following three problems. Indicate the problem that is not to be graded by crossing through its number on the front of the exam. (8) (a) Define what it means for a function f to be continuous at (1. Use complete sentences. <9 i A Ame/Vow 15 A“) (WAMQM aL a (‘f A‘M “alga , Y—J (1 Let 03”” — 16, if cc < 2, f(:v) = 5, if a: = 2, x2 — c, if x > 2. As always, justify your answer to the following problems! (b) Find all values for c such that lim f exists. (is—>2 X \ Jag“ c3 -/6 4040‘ xZ—~< art COHAMQO’M. Mama; ,0“ x I MM " Au“ /(x/—. u (c—Z"~/6) = C-SZ—Jé : 7c -/6 «N4 Kai" X“?— Z @ I: (on ((322 A». (kl-c/ 2 2 —c 2 (1“C- V ¥_.)l-l x_)21‘ O l , TA our-mun! A'Mn‘l: aw 2/; 7c—I6 :Q—c, (fig (Dc: +2.0 [/1 cal C3) [ [flu Aim X4035 1% (:1. C29 Y ~72. (c) For which of the values for 0 found in (b) is the function f continuous at 2? C29 / 2! {Ila (wéhud‘b 0‘ 27 My. AM Mw‘ Mctbfl. fl¢fl(6],/QU‘, {99% C22. X-JL s3; 7m r! caa, m ,4»- !W 71*" =1 3‘ $— fifllfl We 1" '3 MW \<~)Z (WK/mum 0/- 2 . ((1) Find all values for c such that the function f is continuous at 1. _. X .flfl‘q YX/Q ch'M-ccm I Mu And-“w Cg -—/6 1'9 chs‘Mum ,{N was? c_ @[ No“: { r3 (oh/swam cl /4me flumévc (b) _L__ (c) MW (d) (R (9) (a) State the Intermediate Value Theorem. Use complete sentences. 0) CD a F 3% 0- ,fiMO‘VN «K 4‘: COM/{Mum om. Ck (Xm4 Juk/VGX [6/67 mud N I" Q) My MtM-u‘u rfo‘dé 41(0/ Md 15/5) ( W M "3 Q/um6—d c A, a WM 2qu (4,5) mac. Idol (Kc/aw, 0 0 (b) Explain in detail Why and how you can use this theorem to show that the equation 2m—3\/§5=1 has a solution in the interval (1, 4). @l- (aw-‘0‘“ K4 Mo‘rw «((X/= ZX—gr’: fir/“q law/ml: (:0! M4 {19¢ M0414: on. Cm/tflam an Mair dM/Aa( {N r I . . l (O‘KMQM ‘ Q,H flQr5-a4w( ,{ 19 (WK/{ALLOW am (09/. W4 ((g/=2"—3/?:/o >I_ "‘ MA“ N=( 4‘; afi'l'ctfi; MM “Md Ma" h c l “Iv/«Mu. 490m C 4;“ (“9/ out}. #o/ (={(G/ a 2 “Br: ‘ (D t: c fa ML 00297114 oWfi‘w. 10 (10) (a) State the definition of the derivative of a function at a point a. Use complete sentences. ‘ _. \ —- {Q} t . i(¥)~f(q/ A ‘ a ' 44 to A’AM [(—9qu! J ( ‘9 ’ U‘dt“ AN“ C; A (/u alt/n0 60c v; ,{ a A—qo A or y. H}, ‘0‘)“ x—q / ” mm'oac! M—c 4‘4“)! xxx/06 . r q 6 (— 3“ 1‘2 lavage: I" Wfiw Mat/W. fwo Aha/é. (b) Using the definition, determine the derivative of the function 'f <0 W): 96+ 1””— 7 if$>0. atx=—1 andax=0ifit exists. " {r(_(/= (4.,“ {(‘lffll/‘(H/ A-vO A 2 AM 3(-I1‘I1+Z- rel/+1 (A «#9».ch 06m lo 2w) A—wo A _ a I” -( —— (Kg-1 glam/w 43A _‘ AW 3 @ L 5—“: 5(344) (-0 1—20 flaming 4_,0 314 (1%“ mfl‘wod awe! (M14445 MJM on cwfibum , 6K. 96L ( _L a ( a ‘ 7 7- 7 - = -M .——————- \K ' crud [cm ¥ K) I‘M 3(ng— lflx} Kf20— le L kqof X—w-l Ll flow {0:} at»; MY will. [Av-u M4 au¢—.n‘a£w( Iv‘pacé ave 01c X—ao <2) L— ”cucc fio «fl “VII-kaum 0" 01/40» 15’? “’7 WW“"°‘"‘ “'4 0‘ f’(-1)= M4 f’(0) = W5 11 ...
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