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Math 119 2008-2009 MidTerm-1

# Math 119 2008-2009 MidTerm-1 - M E T U DEPARTMENT OF...

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Unformatted text preview: . M E T U DEPARTMENT OF MATHEMATICS Math 119- Calculus With Analytic Geometry I Exam 1 (120 minutes) Inn-“nun“ 1956 Name, Last name: Id number >: Justify your answer. No partial credit will be given. for unsupported answers. 1. (6pts). The graph of y 2 ﬁx) is given below: ' 3 Sketch the graphs of the functions: ~ (a) y = f(—z) (b) y = -—f(—a:) (b) y = —f(2-—z) “ ' “ax ‘ w W ff?“ 2 clack about. 52:1: .3. as» (.q a“ W imiﬁml" 2 w é 2. (15 pts) Determine the limit if it exists: (a) 1im(\/-’1?2‘--'10——:z:) 2 MM Q xéio «~x) (\j x240 +y<> —— UN; woo ””0 38.40 +x , -XWX (b) umwlm“2‘“\$ aim 3,...” =‘21 9-4 IB~1 “ x-M x—’\ ( \X—2\ = z—x wij x 1.5 sugfﬁaeﬁa—‘dé dose 4:0 4 ) . 2 1 (c) £1332: cos(\$+E3-) “X 2 g x2C05(X+-g4-g‘) g X; , X*O SUM X2 :: Um C’xz) :0 => UM x—‘bo xao x-90 1cm :1: 3. (15 pts) (a) ‘Find f’(z) if f(z)=%a (do not simplify your answer) . 5/00 : [3x25mC25-3) +x3 2, Cos sz—sﬂg’iﬂﬁz) __ S31; XBSirﬁZx-s) (\f? +2)2 . 03) Find 31’ at (1,2) if Sin(2\$-y)+zy3-4m=4. C05®><~g3 ~ (2—:3’) + g3 +3x523’u “i 2: O x=i,3=2: 4'C2‘é'3-r8—i—42 35—4 :0 'i 4 3 I = _‘g 5' :‘ 76: (c) Find 9’ (1:) , if f and g are differentiable functions such that f(g(\$))=m and f’(\$)=1+f2(\$)- 8% Jaime. cilxcx,kn mie, SICQCKD alCK) = 4 \- W «Jr 52(500) A 4+>< a ’cx) : 2. _ 4. (6 pts) Use 5 — 6 defintion of limit to prove that lim (32: + 10) = 16. z—rZ 5. (10 pts*) Let 11:2 :1; g 2 ﬁzz): {mm—l—b, :z:>2 Find the values of m and b that make f differentiable at a: = 2 . MM §O<3 = 59(2) 2 4 ’ x-—>2 - j is enabled: 2 <3 UM Six) -— UM K2 = 4 , by“ gm x-BQ.” xe—u THUS, 3? 2xm+n=<l; 3 is dig. a}: 2, 6:) MM x-2. X“>2 2 3 - 2 x —-4 RAM QCX) g( 3 : LiM M : LCM <X+2) 24: xez‘ x-z x—zz" x~2 X92. (“A EOQ— 5C2) 19AM mx+£4—2m3 -—4 “62+ x—z. x—eQAr x-Z. : UM m<M .1: m X62“? y? \hus L? m==4, J: 3" beww auxS 04* X=2.. 6. (8 pts) Show that 277 + 411:3 + Ta: ~ 10 = 0 has exactly one zero. Lela £09 2 X\$+¢<X3+¥x 40‘ Mam-g, gqo) =~4o 3‘ £03 ='2,. Since 5 1A covet, M 10,33) b3 the :W 1:er Mis‘cs a. po‘m’c x1, 6-. (on) each that ( L musk. bod/.2. on OLD. VOLLuee balm/0w ~40 3< 2-» SU990\$~42 Jchoc‘c §C><O =0 ~3’9“” SOMQ' X4 =7-"><O. C\e_a1‘26, 5- Ls cant. Q LXIX‘JXO’X Kg X4<7k¢>. A d.\£ On (X91X45. 3% '\ Oﬂ “ixOJXdl the MVT ch/ll— arr/<42) = mg) (xv-x03. gut 5’00 : 7L><6+42><2+13L 2;:0 ‘v’x 34H“ Some, ’52. lad-ween Xe anal >44 The. \méhgﬁ tvod‘. g’cg3=o “Hus ConU‘adJCJnoA shows taut ’xﬂ does 00’: axist\_ 7. (6 pts*) A particle movesvalong the curve y :2 V1 + :53 . As it reaches the point (2.3), the y —coordinate is increasing at the rate of 4 cm/s. How fast is the a; —coordinate of the point changing at that instant? 2 x ”it ' ea... “-ld"; ‘2 H—x3 8. (7+5=12 pts*) Let f(:1:) = (In/4 —- 2:2 , a: E [~1,2] (3) Find the local extremum values of thejugiion. .- 4. ,_ lez ____ 2 (2'65 §’(><) ; A.\l4—><2 +>< 242:2 _ W \lér‘xa LCP: x = {i1 <_\j’i is not in the cloiuaim> is a Local. max 0:!) Explain why the absolute extremum values of the function must exist, and ﬁnd them. - insol-le: ~ Since Jew {rm g M can‘t. on 14,21) 19% the. \:.\IT ’cm at extreme! exist- \l? <——- Absoku-l—e Wm mum ll ll Endw’i r1335 ‘. S L—A 3 . O _ A bS-O WAC: Mm? MUM 5&2} "ch4 chB =12 ‘5'“ 9. (7 pts) Use tangent line approximation to estimate the value of V3 7.8 “”1 2. __ I .- Lat 5cm =3\/>< . Them 52x) :ELX~§‘ lose “-8. W) = gm) + s’cm (x—xa ‘ J... J. x~8 : Z, ‘1‘ 3 4 < > 4 ) 2 4 _ M9 1:) : 2+ ..__.(—-O.2, :: —- *QO .. —..6_.6_. 5-Way a Mate) 12 a 10. (3+4+3+5=15 pts) Let f(a:) be a function having the following properties: f(0)=0, f(2)=3y f(3)=7. f’(0)=0, lim f(\$)=0. IEH;°(f(\$)-2\$)=0. 11m f(\$)=00. ‘1 f(\$)=~oo, 34“” z~>~l+ z—>~ — f’(a:) <0 on (—00,-1) and (-1,0), f’(a:)>0 on (0,00), f”(:z:) <0 on (~oo,—1) and (2,3), f”(:z:) >0 on (—1,2) and (3,00). (a) Determine the asymptotes. VA: ><=~l HA1 \$39 (b) Determine the intervals of increase and decrease. (-ao’w") QAA (‘4’0) g is éecreasiné Ora S {‘5 ”cncrtfaikné an '{Ol 90) (c) Determine the intervals of concavity and inflection point(s). -1 - X ’7" 3 Since 3 M cont. oj‘c ><=z cmd xza, and ﬁll \ __ : ..... l . - ‘ o (X) “V l “t concqwlgg‘ changes the/UL) zap-4.5g Pts t i \J ’ ELK) co l co l CD I C‘u‘ (SK/(‘6 \ngiecﬁen ?O\F\t5. (Note. that E M not coated; x.—.-4) (d) Sketch the graph of the function. ...
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Math 119 2008-2009 MidTerm-1 - M E T U DEPARTMENT OF...

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