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1 Math 0, Final Exam, Term 3 IANSWER KEY I I Q II MM I VI I V I V3 I V 4 I : a e b c e a b a c d! 3 a d e d c i 4 a a a e a 5 a e d e c! 6 a a d e c 7 a e c d c 8 a d d b b! 9 a c d b e 0 a c a c a : a a a e b i a b e d a 3 a c d d d 4 a d b c C i 5 a b c d a I 6 a e d a e 7 a d c a d 8 a d c d e 9 a d a d e 0 a d b b c. a c c c e i a b a d e IE' a.e d d e 4 a a d d a L.::.:.. 5 a c e. a c ffi a a c e a I 7 a c 0.. c 8 a c c d a I --"\>

2 King Fahd University of Petroleum and Minerals Department of Mathematics and Statistics Math 0 Final Exam Term 3 Sunday 9//03 Net Time Allowed: 80 minutes MASTER VERSION

3 Math 0, Final Exam, Term 3 Page of 4 IMASTER I. An equation for the tangent line to the curve y atx = 0 is x-i x+l (a) y = x-i (b) y=3x-i (c) y = -x-i (d) y = 4x+ (e) y = 3x - x. lim x-+-l+ "';x + t--( ~m (a) (b) 00 (c) 0 (d) (e) _\ o-t )en o ~ (-7 ~ ) -00 -::r :=.\$0 \\VV\ ---===- "j...'f -\-t F+I _o(j

4 Math 0, Final Exam, Term 3 Page of 4 imasteri 3. Where is f(x) = In( - y'x) continuous? \n Hs do m~ ; (a) [0,) \., {X ) X?'-'..) _"" \- 'J X / n (- y)<.----"-/.r::: > ~ <. \ ~') x < ~ (b) (0,) (c) (0,00) Ccm.bll'IYlj, \jje ~~ o!sxl..\ (d) (,00) (e) (0,00) l'\ --- n... n 4. f: (~+ k) = Z* ~ak.. k=l n K. -= f\ (= n...::::- \( -+ al _ i. L ~ k. ::::) - n I<~I,.. Y'\ '\ (Y\-+-I) (a) n +n+ -+ ~...~ (b) n+n 3 = '" (c) -+n ~+Y\~~ n (d) _+n n n (e) n n+ -+ ntv\~l)

5 Math 0, Final Exam, Term 3 Page 3 of 4 IMASTER I 5. The slope of the normal line to the curve y3 + tan(x) = cos(3xy) when x = 0 is.. X=o (a) 3 (b) (c) (d) 5 (e) If lim f(x) ~ 4 = 5, then lim x f(x) = x-t3 X - x-t3 +b9-lt == 5 AI \'ly\"~ \'W\ "-3 Q~ '.. (.. )C~'?, (a) 'VI ().~ \ \l'i\ (.\' od -4) == (J (b) 5 ~~3,\~ ~\><) :4 "f-l3. (c) 0 =-, -. 3.l.\ (d) 5 (e) -4 -rhw» \' "m 'i.~s -;r+6<)

6 Math 0, Final Exam, Term 3 Page 4 of 4 JMASTERJ 7. If f(x) = 0x + tanx, then f(3) (x) = ' ;f 't<) ~ 0 X -I-~ S~c.. X. f('v)::::?-o+':l. (a) sec x(sec x+tan x) I, J.. ;{ seo( ~cx~nx." _ 0 + s.ec"-x.fc:tnx - '. L a s..eck. # (b) 0 sec 4 x + sec x tan x tt, S- ( '.. X S-R<: x + 'Jl'lt\)( J., I L--tdr '><.f he} ~ O ".s.ec~ "" r' (e) sec x(sec x - tan x) 8. Let H(x) = f(xg(x) -). If f(4) = 6,'(4) = -5, g() = 3, andg'() = -, then H'() = '(:;r lx) -). L'-::i- <J' (-xl - ~('X)J HI ft.) :;. f ~ (a) 5 I [:cj()+:)cz)) H'(-) -:. f (J.t)c z.j--). (b) -6 := f I ( 4) ~ [a(-.) + 3. J (c) 0 ';:, _ 5 (-4+?:» (d) -5 (e) - 5 (-I) -)

7 Math 0, Final Exam, Term 3 Page 5 of 4 jmasterj 9. Using a suitable linear approximation, (.000)500 ~ (a).0 (b).0 (c).0 (d) 0.00 (e) \ e6 0 -= It- \00.:'>-;::' = I + To =- \~.o x 3 + x - X - : I 0. Let f(x) = 3 If R is the number of removx -x able discontinuities of f and I is the number of infinite discontinuities of f, then fix) "",y... 'L (x+..j _!X+'Z.) ~ -X (')(L_\)-- (a) R = and = (b) R = and = (c) R = 0 and =3 (d) R = 3 and =0 Q<-f- L ) (x- -I) --~ "X ()( -I) (X+L) (X-I) (xt-i) -=~) ~+,'l.. - I x 9'(.~ :!,.I (e) R = 3 and =3

8 Math 0, Final Exam, Term 3 Page 6 of 4 IMASTER I. If Newton's Method is used to estimate the x-coordinate of the point of intersection of the curves y = sin (x + ;) and y = In(x + ) with Xo = 0, then Xl = "T\ f (x):;: \Yl (-x..-r.f) - SIf\('X+ ~ } pi. (a) 0.5 (?<) - x..+~ _ CoscX,... "'i: ) t (Xol (b) 0.3 ::>; -= 'Xc, - f I(~).\ (e} :::: 0 (c) 0. -::: o - -f/(o) (d) z. (e) -0. -l. lim (../x + 3x - 3Vx-x) = x-+oo (a) -00 II tr\ )(~ 0iJ (b) 00 :;::;. c0 Ixl ( ~ \+ ~ (\-.3).- ) - 3~ \ - ~ (c) 0 (d) -3 -co (e) 4

9 Math 0, Final Exam, Term 3 Page 7 of 4 IMASTER I dy 3. If dx = cosx - 3 secxtanx and y(rr) = ' theny(o) = (a) 7 (b) ~ 4 (c) 3 5 (d) (e) 4. A hot air balloon, rising straight up from a level field, is tracked by a boy 00 m on the ground from the lifting point. If the balloon is rising at a constant rate of 50m/min, then the rate of change of the boy's elevation angle (} when II 7r. v = 4 IS cj ~ _ 56 mlm,,, dt: (a) 0.5 rad/min (b) 0.0 rad/min (c) 0.05 rad/min (d) 0.05 rad/min (e) 0.5 rad/min

10 Math 0, Final Exam, Term 3 imasteri 5. lim ( + x)~ \'" (\+:,:9 X-l>OO \ \ '\ (\ + l,:x) :. - Z \ \i\?c.. In ~ ~- -.\\"\)(' \.,(\+ x...,) (a) J \tw- \" J := ~~...." (b) e (c) J (d) e (e) \ -.. Vz e 6. The absolute maximum value of f(x) - sin(x) + cos(4x) on the interval [0, ~] is equal to (a) 3 - (b) I f("i<):::o (c) 3 (d) v'3 (e) + v'3

11 Math 0, Final Exam, Term 3 Page 9 of 4 IMASTER I 7. (b) yin 0 (c) yx In(x + 0) (d) yx(x + 0) xy (e) In(x + 0) ex 8. The graph of f(x) = X e + (a) is concave up on (-00,0) (b) is concave up on (0, 00) (c) has two inflection points (d) has no inflection points (e) is concave up on (-00, 00) X e --c "'.)( (e+i)~ e (ft-- it~ ) (ex+) ;. J J II (x) =D ::}e" ('/ up o d<3>v,,\ t ty't\.e. \(v \..ectlvn. t

12 Math 0, Final Exam, Term 3 Page 0 of 4 IMASTER I 9. Let f(x) = a:x + /3x +" wherea: 0, /3" are constants. The value of c that satisfies the conclusion of the Mean Value Theorem for f on the interval [3,7] is (a) 5 /('><) =- ;{on< -+ ~ r CC) :! (b) 6 Q,c(L+~ - - (c) 3.5 (d) 4.5 (e) 4 ~d c +Lfp.:zrolc..\'() -~C:~) i-3,., ) t- _ ('-<;'" t- f3 +Y) - (0.x+-'~ Lf -= 400( 400( c.:: ~D( ,.'-(f == 5 I 0 to, 0. The number of critical points of f(x) (x - X 3 )-/3 is --<.f/'5 t) f I i ( x"}),( t - :,:x J (-,.) = - 3 'X (a), -- X -t I_~X - - S3 5'('.) =.0 - (b) 3 ~/("y') ONE-b (c) 4 (d) 5 (e)

13 Math 0, Final Exam, Term 3 Page of 4 IMASTER I. The function f(x) = x4-4 x3+ 4X + 4 has (a) a local max at X = and a local min at X = 0 and X (b) a local max at X = 0 and a local min at X = (c) a local max at X = and a local min at X = - (d) a local max at X = and a local min at X = 0 and X = (e) a local max at X = and a local min at X = and X = 0 f ),,..."S _' X'l.. + '6'X... (~) -::. '...-A S, :-4-':(( rx 'Z.-?:'x+z-) \- 0 :::: 4): (X-I) (')(-'.) /, /-> ~ "./ WI,y\ YI'lMC Jf('i) -:: C :=::::) x::: (};) \) l hi'>". The function g(x) = e-; is (a) increasing on (-00,0) and (0, (0) (b) increasing on (-00, 0) and decreasing on (0, 00) (c) decreasing on (0, (0) (d) increasing on (0,) and decreasing on (, (0) (e) decreasing on (-00,0) and (0, (0). -:z. ( 3 6<).: e X -IlL I >D ~cjj X;fQ ~' - -t 0 /' t-?

14 Math 0, Final Exam, Term 3 - cot X) 3. Ify tan- ( ' then y' + cot x (a) ~ esc)<. (b) 0 " Q C'5.c. )< '\ (c) - ~ -+. Ctj!;)<. + (~::..., =- G:-Cot x)"z.-+_(\-(o~y.. "'l ~ -+.\_~e t. :l al f'( ::: esc x (d) csc X -:: :::. ~ C>c.)< ~ :) L \+ (at )( (e) +csc x <. 't. ~ (tx + Cot ')( 4. A rectangle has its base on the x-axis and its upper two vertices on the parabola y = - x What is the largest area that the rectangle can have? (a) 3 (b) 30 (c) 0v' (d) 8V6 (e) 8 A=?::-~ A (x) =?( (P--~ 3 _ ::.4 x - 'l. x.. I ~4 A (x) :::?L = ']. -,) AI ('<) = ti AIr (x) = _. x..., f\ () G 'x.. ~ (?-} (\~-4) -:: X ex ~ 0) AII( ) -:: _4 (Ii.<.. CJ { >(Y\ oj'/..

15 Math 0, Final Exam, Term 3 Page 3 of 4 IMASTER I 5. ( X) Ix ~+ dx= x (b) --+-+C x 8 X (d) - x - ln Ixl + 4" + C (e) + (In Ixl)-l + 4 x + C x 6. Using 00 rectangles of equal width and right endpoints, the area below the graph of f (x) = 4x 3 and above interval [0,] is approximately equal to I:::... '.-0' 'x =: 0 + Ln k :: -;:; bx ':: ~r;; t:. '/ ( a).00 A - L ~ ('X~) [) " (l - I(:..=" l"\, \<-)~,.., k.. L =..:::::::; "" \''' \ Y\ - - c-) ~L (b).0 "'" ~ _, "~Z\ J'.. - " k 3 =:4" l'y\c;+') - ~ ~ V\. () C 03 - '4 '- =\ '\ ".. (d).0 :::: (n +~) '= (I +~ ) ) L ~) == \,0\ (e).00 Alto - ( + '-'ll I _.0. 0

16 Math 0, Final Exam, Term 3 Page 4 of 4 IMASTER I 7. If the curves y = x + bx + a and y = ex - x have a common tangent line at the point (,), then a-tlb+e = (a) ~ 0 (b) 4; (c) 4 (d) - (e) -3.So. o 8. If a is a positive real number and then a = (a) 6 (b) 8 (c) 4 (d) 7 -t-hey) \c -(V'\ ~) '" eo. :l..j 0{. ~ 64, s;.; "So 0\.::: G (e) 9

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