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pracfinalexamsol - Math 1 6A Kouba Final Exam Print your...

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Unformatted text preview: Math 1 6A Kouba Final Exam Print your name here. Your exam ID # 1.) PLEASE DO NOT TURN THIS PAGE UNTIL TOLD TO DO SO. 2.) No notes, books, or classmates may be used as resources for this exam. It is a violation of the University honor code to, in any way, assist another person in the completion of this exam. Please keep your own work covered up as much as possible during the exam so that others will not be tempted or distracted by it Thank you for your cooperation. 3.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will not receive full credit. The best way to get maximum partial credit is to write neatly and be organized. 4.) Make sure that you have ten (1 0) pages, including the cover page. 5.) You will be graded on proper use of limit notation. 6.) Include units on answers where units are appropriate. 7.) Use sign charts on all max/min. problems. 8.) You have until exactly 3:30 pm. to complete the exam. Anyone who continues working on the exam after time is called risks being assessed a 20 point deduction on their exam score. page 2 of 10 1.) (6 pts. each) Evaluate the following limits. I I u n o . .. .._.. ~— - .2— X ‘ .— b ‘ .__,..- 2 W Z; a.) lim x 2 r. ,QMM RX c.) lim x3+l x—eZ' x2—4 2.) (5 pts. each) Differentiate. Do not simplify answers. a.) y = V5+2X cos (3x+l) -16 i me—H) 1‘— (EanX) -,2 - {Sax 7M (3M1). 3 \l 1 W M1(3X~Ll) b.) y = tan 3 ( sin (5x)) \(‘~ 3 m1 (M (5x))'ml(4lm(5x))- “(Rig \9 page 3 of 10 3.) A baseball is hit straight upward. Its height above the ground after t seconds is h(t) = 128 t - 16 t 2 feet. a.) (4 pts.) How long is the ball in the air ? MM: OthC+)-* o:i;3£«ie+1:iet(?-t) 2, +33%. b.) (4 pts.) What is the ball’s maximum height ? WM' V(Jc):o -—-> lay—3gfzo...) 154%.): her): .154, {at c.) (4 pts.) What is the ball’s velocity as it strikes the ground ? vet): ixyv 32ch M vés’): may 31(8) 2 ~42? fat/M. d.) (4 pts.) What is the ball’s acceleration as it strikes the ground ? “(-6 ) t: "‘ 31 40" .L 4(3) 3 ‘32 k/A’Q‘C x 4.) (12 pts.) Function g(x) = 3 + 2 + x . Find another function f(x) so that g( f(x) ) = sin x. This problem is not difficult. Just do it. 4 490‘) :an #:AMx—3 «a 9\+_\:(,<) 024»? 4,: C1+¢)(wa 3)") v)" QMX-Q} +‘PMX'34: ”'5 3—L qt—4Mx : .2wa a +(q-,ol~\><)= QMX~6 "” aKMX-vé 46): Liv/.wtx page 4 of 10 5.) (20 pts.) For the following function f state its domain and determine all absolute and relative maximum and minimum values, inflection points, and intercepts. State clearly the x—values for which f is increasing, decreasing, concave up, and concave down. Neatly sketch the graph of f. f(x)=x(x—6)2 simim ; 4M X'NM 41x): x. atx—e)+ (x—e)’~ : Q- e)L2x+x/~(9) : LX~G)(3x—» c) : o H .i . ><=x M +00: CX—<o)L3)+é)(3K'<°) \lz3ét Y: o T’QK‘R‘JCO NQ.M. JUKM —— o 4* —l‘—zi<>’l\ {in ><<2 X>C>) . \L / (2; X:¢l ~i~o x534 5A<><< ) \{zHo ~lFLoUz6’ch><4/ cw» +gnwx<q3 ”WM!“ X:O—} Y:O - : X:(o Y-O-? X o} \l page S of 10 6.) (1 0 pts.) Find two numbers x and y so that x + y = 10 and the sum 5 = x 2 + 4 y 2 is a minimum. Your answer should include x, y, and the minimum S. L \ 8': XQ*4C‘°~><) a S : RX—8(10~K2310X~80: o 4/3.“; 5' X:3 Yil -.. s: so 7.) (10 pts.) Find the point (x, y) on the graph of y = 1 + V? which is nearest the point (9/2, 1). Your answer should include x, y, and the minimum distance. w. L= «Laue—M L:w)(x~§)’~+x —————> -l L‘=~i:lW)/'[email protected]"'3)*l)l 0 _, O 4* Li M —-§ RXVX: O x:q \(33 1'7 MAM. L: z; 8.) (1 0 pts.) There are 30 pear trees in an orchard. Each tree produces 500 pounds of pears. For each additional three (3) trees which are planted in the orchard, the output per tree drops by 25 pounds. How many pear trees should be added to the existing orchard in order to have the largest total output (pounds) of pears ? Your answer should use calculus and include the number of trees, the output per tree, and the maximum total output. MX2#UKW#3M/ M. W T: tidWMW/WW ,7 '1': (30+ SXXSOO —— 45x) J7 Tl: (30+ 3x)(—25)+(3)(500—925><) : “750— 75x+ 1500 —75>< : “750 -— Ifo X: o ——> +0" T! W hiw 3o+562)=qb'% ”Jack Soc-'QSCS'): 3'75an Me. MA Max ‘T:@5)C375): rams/1,64. page 6 of 10 9.) (i 0 pts.) A closed rectangular box with a square base is to be constructed from two different materials. Material for the top and bottom costs $3/ft.‘2 and material for the sides costs $2/ft.2. Find the dimensions of the box of largest volume which can be constructed for exactly $72. Your answer should include the length, width, height, and volume of the box. 3(2x2)+ QCHXY): '72 a (ox2 +?XY : 71—» RXLJ—‘lx‘l: 3G “" \(—_ 3<.>—3><z M& 4M M_ M V: K'ZY ——# ___:_3._:‘/ V ‘ ngqk. #:3zpk. 3 10.) (12 pts.) The circumference of a circle is increasing at the rate of 2 ft./sec. At what rate is the area of the circle changing when the radius of the circle is 6 ft. ? C:21fl(‘ “MS. ,4:'rr’l/"2‘ 3 OlQ __ pk?" __ Fr AV‘ out ' ‘2 our ‘2" ‘2 out 0 page 7 of 10 l l .) (12 pts.) Use limits to determine all horizontal asymptotes for y= X x/x2+25 QM Xi “LN“ ‘251Vio:+1 -— 4r X—=!-l-ofl X1445 X—y-lroo ‘4' 323 l j M ‘“‘ X2 : ~i W x—roo xhgzs / ”’7" Wm M Y: lj\l:”l 12.) (12 pts.) A tank contains 100 gallons of solution in which 4 lbs. of salt is dissolved. A solution containing 3/4 lbs. of salt per gallon begins flowing into the tank at the rate of 8 gal/min. After how many minutes t will the concentration of salt in the tank be 2/3 lbs/gal. ? page 8 of 10 13.)(1Z pts.) Assume that x sec y = y 7— + x y. Find y’ = dy/dx. 2—? X-m‘ffimY'Yl-L i-mY: 2%Y‘4— XY|+ Y />7 XMYWY'Y‘~ QYYL XYI : \(“MCY \ ‘(~/.u»c.\{ ‘* \f ~ x/mjmw—av—x 14.) (12 pts.) Use lim f(x + Ax) - f(x) to differentiate the function Ax-aO Ax f(x)=3x—x2. \ - 3(x+Ax)—(><+Ax)n”-— [3%ij 4X90 _ M fl3-Ax/ax-Ax— @x) nfl/X’Z ~AX-no AK page 9 of 10 15.) (12 pts.) Use differentials to estimate the value of x/1.2 + (1.2)1/3. l/ 31* 4(X):1l><'+><3 ,..g X:1——w1.2 4% AK: 0.; l _ I "l/ "3:? l .___L,_ ~PQ<)';><'Z+ X l *— : V + .2 3 £15? 3x/3’ ’ l m: J;(I.2)~$U): W2+ 002)?— :4 Av ; 41%|). AK ._._ %+%).Co_2): 6;? 2 EL R: O.lé7 l 1,1,; + Cl.&)<?“ 1 :2: O.lCo'7 o4 /3 11.1 + (1.02) 2: 2.167 16.) (12 pts.) Assume that the maximum absolute percentage error in measuring the volume of a sphere is 27%. Use differentials to determine the resulting maximum absolute percentage error in computing the radius of the sphere ? (The volume of a sphere is V = (4/3) 11 r 3 .) [Al/l 5 (17% N If?“ lAvl lAVl lv'-Arl lflfiflAr‘l TQT: V 2%”! :3L—A-‘CL3027070 M r [AY‘l ((770 W - pagelOoin Extra Credit Problems—— Each of the following problems is optional and each is worth 10 points. 1.) Consider the function f whose derivative f’ is given in the graph below. I repeat, the graph below is that of f’, NOT f. However, answer the following questions about f, NOT f’. a.) List the x-value(s) for which f has a relative maximum. x=~ ,1 b.) List the x—value(s) for which f has a relative minimum. x: ~ 3) 7. 5 c.) List the x—value(s) for which f has an inflection point. :" “I Y X 5 J 3 . ' 2.) A square is inscribed in the given right triangle. Find the area of the square. ...
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