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Sample Math 131 Final (2004) - (Solutions)

Sample Math 131 Final (2004) - (Solutions) - Math 131 NAME...

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Unformatted text preview: Math 131 NAME Final Exam — Form A Social Security Number Autumn Quarter, 2004 Tue. Thurs. Instr.: time: This is a closed book, 110 minute test with 12 problems on 10 pages; the 11th page is for scratch paper. You may not use other scratch paper or notes but may use a calculator. However, answers must have appropriate work to receive full credit. Points available are in the margin. (30) 1; Compute the following limits. (Ifit does not exist, but is 00 or —°° then say so.) (65 1( lim “7 — a) x"3'x2——9_ WM I‘m X+7 '= 3H:IO >0 Xr-95‘ w £10: = w lx+5)(x-9) ‘-' W mm W x-% J V 'Y‘fij‘ = 6 UV“ (>93) >92,— : Q— : O" ( Clvib 1’0 O I. Sm all?” Tlwa 0 CV‘ 2M m law) (6) 1(b) lim--————~—~—— = r:- I”4x2-—5x+4 3 “94 m ) 9M) : LEV»; 2 3("4 X‘ - 2:. ' 3 5 1 g‘3x3 6 1 lim - = _ —L ()(c)x_,m7311 7 —.x -— 3 5 ...L 3 _ \Vm ' l ,3 1‘5“ ,3 ‘A - V LET. —— v:'\ _1 yaw ,3 Page 2 Math 131 Final Exam—Form A 2 2 , . (x+h)m(_x; ‘ fl“ V” <6> 1w = x l W - A ‘Vm - .3; LXL>0 V < >044 x > _. L; s (Aim; ' ” xao I h ‘\ WWW" > 3 MW ‘ f 21‘ ¥~7v T: (X‘l'h) \< w . _ W _ I} ” (>1 "T li‘ U T 313% x (X Mfi { D/VE 1““ in, 1 lim——————(3x—3)e(3X- ) — 1 (6) (e) xal x_1 _ '- ‘38 _ 5‘ , BX-i " M :3 6 x—el : ’2 {3’1 (——3--) if x50 \ x+3 (18) 2. f(x)=v 3x+1 if O<x<2 x+4 if x22 1' . = ‘1,“ ‘3 ~ j.— w~ (6) 2(a)x;1{)1_f(x) :10" _ 0 f3 _ ( (6) 2(b) lim f(x) = Liw 3x+| x‘*0+ "uh—70+ (6) 2(0) Give all values of x for which the function f(x) is Mcontinuous- mm; xxzqu Page 3 Math 131 NAME Final Exam - Form A (12) 3. Use implicit derivatives to express fi—Z— in terms of x and y , where x2 + 2107 + y2 = 5 . ( Please circle answer) d“) :- Lx’r (1‘3 flxgfi) f 33 fix 0 a . M O - \ om 2173131 . _ _\_ 1X+2L) ZFQ “x ; I h” _ Cikj ‘- 1+1x+2320, "W ‘5"X’ Fi*'l {W 7 ‘\ at y In (WW , ‘ r “ I (25+sz (10) 4. Solve the inequality: 2 0 (Please circle answer) —x (M4) X1:D 9 K 2?! x=o 7—xro >7 x:7 x 45) -y ,, 1 J— .J— 1 60 a ,L 5 c” - :77 6 Page 4 Math 13 1 NAME Final Exam — Form A (36) 5. Find the derivatives of each of the following functions. ( just circle your answer.) (6) 5(a) y = (5x+ 3)7(x3 + 6x) I «~ ‘3' ‘1 ((S‘thafl‘lxgwx) + (WHY (X749) C 7 '7 -3 2. r (S'XHf-S‘ (XB-Jréx) + (5M3) (5X +6), (6) 5(b) — x3‘8 '7x—3 ; j 7x~5 H l ' 3 7: , I T ((5845)?) ('7X‘3) ‘" (X’B‘) ' (7x~ 3) \ 9 K " ‘ ' fl” '2 ” ' ' "N W‘ (W?) . > 3 1% ’% Him“? (W3 5 I” 7!. X "3) (7x --‘3 )2 (6) 5(c) f(x)=ln[(x—1)2]+[ln(x—l)]2 7 'lffx) I :1 L44 (“0 1L. [LV‘ “H” \ l r— : u—-—-‘—r"“ 2 L,“ (X‘ly' X-I Page 5 Math 131 Final Exam - Form A (6) \\ I I (6) (6) 58(x7—2x4+1) 98—21%“) > (x'7u— 2 x4+i \ 11 NAME \ . l (x7~2x4+\) 5(e) y== (5x) 7 + 2111(3):2 + 1) fl; ji2171(§x)7 (9X). + 2 5(a) [ 5 6 g Q 9 (Z .l_‘_ .7 .i _ (Sx)7~5 ’f‘ ‘2 . .4 , “(1-2x .+\) (7X6 '_ 3x3) 3X24" 3X14" ) ‘ . 6x 3x1+‘ 5(0. Use logarithmic differentiation to find the derivative of y = (2x + 1) (3"7) bag 'r (3x~‘i ) Lu(2x+l ) . 13‘ L. 3 [M (mm ) + ( SW7) % j D ( . 2x+l L14. (2H1) 7L 8%?) ‘) 2X +/ J Page 6 Math 131 Final Exam — Form A (12) 6. Use the Second Derivative test, to find the x—values of the points of Relative Extrema of the function f(x) = x4 — 2x2 -— 1. (4) 6 (a) Fmd the critical values. Ans x: it“) ‘: 4x3»4\< ‘5 4x (XL—i) ': 4XM‘W1XT‘) v (2.) 6(b) Find the second derivative of the function. f "(x) = i 2 X r A" (6) 6(c) Use the second derivative test to find the values of x for Which f(x) has relative max and relative min. (if there are none please say so). Relative Max at x = D ‘1! ~ V - a I” 4 7 i (L) <b RelativeMinatx: "l 5 I (12) 7. Let y = x3 — 5x + e04”) (6) 7(a) Find the slope of a line tangent to the graph of the given equation at the point (2,—1) . l I - 7 .1 ,_ ‘_ 4'.er ‘. Ans 3‘: 5x»; + 6 ~ 4r . we j[(2):l2'§+€a '4" r: ‘ (6) 7(b) Find the equation of the line tangent to the graph of at the int (2,—1). ._ . e , p0 Ans flaw-” (x ‘> Page 7 Math 131 NAME Final Exam — Form A (12) 8. If x=0 and x=1 are the critical values of f (x) = 2x4 - 4x2 on the interval [0,2] , then determine the absolute max and absolute min of the function on the interval [0,2]. EVCVMCxTe Ci " i l’ 1 Abs. Max ‘A «‘5 (12) 9. Determine the concavity and the x— values where points of inflection occur for y=x4 —3x3 +7x~50 (6) 9(a) Use derivatives and/or sign chart to find interval(s) of concavity. . concave up on (e M ,- o ‘2 U (’2— 'H?“ I) ‘3“: 4x3~~01><l+7 (Q; .12) concave down on V)" 2 {2. XZ’iBY : 6x (I V 3 i) _ - lest“ Ll'wLmLe/vs - W ;E_ X' L/ , i X +il + r + “i A ~ s- q i --b .3 . o ,i. .2 :2 L (6) 9(b) Find x—values of inflection points. At x = O ’ 1 Page 8 Math 131 Final Exam - Form A (24) 10, Let f(x)=x3 —3x+14 (3) 10(a) Find its y—intercept(s) (4) 10(b) Use derivatives and/or sign chart to find the interval(s) where the function y=f(x) is increasing and where it is decreasing. Increasing Swim: t Ni.) 0 (l, T20) " i”): 3x14 Decreasing (“"7 ‘) x "P ‘m 2 C! 0 -~ 5 .7 of , (4) 10(0) Us; information obtained in part 10 (b) to find the x-values of its point(s) of relative max and relative min. , rel max at x: A l , rel min at x: Page 9 Math 131 NAME Final Exam — Form A (4) 10(d) Use derivatives and/or sign chart to determine the intervaI(s) where the function y = f(x) is concave up and where it is concave down. concave upzw concave down: (‘ "(12¢ / O ) t' ( X) f 6 X X I O .yfil Test Mum/Jo‘er # + ~ . .M_/,._—_.—-—.»--—-—--—r_.i:w-—~....__<..-......._..,_c,,,_w___~_‘> V H( x ) U . ._ ( ._. C I a: (4) 10(e) What are the x-values of its points of inflection. , Inflection point(s) at x: O (5) I10(f) Sketch the graph of the function y = f(x) using information in 10 (a)—(e) *l O I rage 1U Mam 131 NAME Final Exam - Form A (12) (10) (5) (5) 11. A company has determined that their production costs (in dollars) for units produced is given by C '= q3 —18q2 + 60g + 350 where q > 0. 11(a) At what level q is the cost minimized? " “ , ‘ ~~ Ans. q: C": 36(1- 3M té‘b’ 5/ Q rm +2” 1“ r + t.— 7 C ) . ( ~»_._..(,.,_, * ‘4; 7 7i L" )(’('(U):U ‘7 ‘0 (Z : 2 1 I L) 11(b) What is the Minimum Cost? Ans Min Cost [ 5 Q x2_+ 3x—7 12. Let f(x)= 3x_1 12a. Use limits to find its horizontal asymptote(s). (If there are none please say so ) Show work below. W Horizontal Asymptotes flare \ A Z : . LU“ 'MX) 3 MW % Z Lima “'3‘:(/) X—W: Y ~> w ' - X9vo Lil/L1 +‘m ‘: 1w L : LLLM 5:: ~90 1y: Xéz — ea 3 X \("7» W 12(b) find its vertical asymptote(s). (If there are none please say so) \ >< I <— L, ‘ Vertical Asymptotes 3 3x/ I I o X : 3_I 2,. __, ,L _ ,_ S i? 7( +3 X’ 7 x: r q i 7 ’ of ...
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