Exam with Answer Key - Analytical Geometry / Calculus | MATH 0031, Exams of Analytical Geometry and Calculus

Download Exam with Answer Key - Analytical Geometry / Calculus | MATH 0031 and more Exams Analytical Geometry and Calculus in PDF only on Docsity! NAME: _ 4 score: /100 (20 points each) 1. Determine whether cach integral is convergent or divergent. Evaluate those that are convergent. ‘ = / G Byte © 4 Py AcCAGn) - at ck GE Ld ‘ 4 Layo 110 points} “aJx-+sin? x 2. Use the Comparison Theorem to determine whether f ds converges or diverges. 2: Xx 2 oe) , Tinus singe we Yaow \ oo ar 3 i x wey Aig Aca el nor wa Gap \ \ co ace sWeiwie ws ) (wid OW PRO EC I ee 6 x (10 points) . : ca 3. Find the length of the curve y =In(cosx) between the points where x=0 and x=—. 3 y = In(cosx) J? : rer ——— =F i | x | Po yy ax al L. su = . —Sjiax) =o Loe « ~t \ jG 1 | \ ay ' | ‘ } | \ Hs a ' “or \ a dy - lov [ ) a axl if : ie , Ms WH, vhs ‘ 7 | (ce 4x © do to 410 points} 7, The cubical metal tank shown below has a parabolic gate, held in place by bolts and designed to withstand a fluid force of 160 /b without rupturing. The liquid you plan to store has a weight density of 50 Ib/fi’. What is the fluid force on the gate when the tank is filled 2, fi deep? For extra credit, what is the max height the container can be filled without exceeding the design limitations (use separate paper)? Ty eli id} Leye f ats ae ! — ee os 7 “ ‘ -L Be 0 1 Parabolic gate . QRLGIN 7% Enlarged view of parabohe gare fo DS . _ qinee Sa 4S Neo nce fax Vax oF 3! Ae ee 3 C a ¢ oy _ tae Wy lasy \ \b Caves 2 EMV adames f bee \ i io “Ye \ F _ Ade = tee yy eX TN 8 Fes ds 3 (ja 32 3 roo) TE rs \ po tov] Fs) ao fin\ — | eee eo rc joo ( 5) TF foaw (10 points) 8. Beer that is 6% alcohol is pumped into a vat that holds 500 galfons of beer at a rate of 5 gal/min. At the same time and at the same rate the mixture is pumped out of the vat (assume it remains thoroughly mixed). If the vat originally contains beer that is 4% alcohol (initial condition), what is the percentage of alcohol after 100 minutes? What happens to the alcohol percentage as time increases without bound? { > \ Nr { yo ve aaWons oh ony \ VN the bere W 4 yaa @ 29 VV boy = Lowy f Soo) > 2 0gallons.,, We \Arsase eA 6 4 ON c ae \ ~( Beep os oR mearee ain Aq _ oe oe y\ : ° (W a ke ° indecinal) A / oe! 5a , i 1 \i sey \ KL —— BL Lo e\iS YN mf aga\L OM /ate = At 4 oe Boo bo (10 pois) 9. Recall that Newton’s Law of Cooling states that the rate of change of an object’s temperature with respect to time is proportional to the difference between the object’s temperature and room temperature. So if Vis the temperature of the object al time f and 7, is the surrounding temperature, then dT T ar-1) elf I took my cup of coflee out of the microwave at 150 degrees. Ten minutes later the coffee was at 110 degrees. Room temperature is 70 degrees, Find the temperature of the coffee 20 minutes after I took it out of the microwave, What is the rate of change of the colfee’s temperature with respect to time 20 minutes after I took it out of the microwave? Since OV eo we have ST = k (T-7e5 . ° at s \ Th we Say yo Tat. fark of course 1 = FOTN J yen PALS brome Sick \= Tey -70 = 1907707 BO e\t dy wy where YI at Taiared Condixion 4h NBO. | cov. cur ior in class we Janow bis Means. ke ' — w&E yiy= yeve 0 ~ = Yith= Bo I wow oT (ie) HHO whack be ls Woe alse bok yok AL\. [o koa ke Se aad ove eg vation Loe us lata) o Nor 60e —» a> * becomes sOS= Be ¢ q Yon aos \ wis ve Yee colleo 36 T-7or20 2/40 de ~ts ’ pv fre Have Bola) fma\ER) = <8 6) 0) , Fe ccell lo - Yio \ \ \ = \ (205 = -8 \wGh (2) C aye Vey Ae cole of ‘ \ \n 2. Dine | Atl vos : Ges Tales a \ —\a (4) , | ( \ Aegeas J : Change ob ike colle | bene p Vs \— eT [IO \ oa  ow Ag redid HY rl co a7} > ln et SA ON ) x fd 4x? d 9 : 2 > _ Loos A , u WANs ye \ ‘ x ~ Se Neo we yl & dura dx dye du See ° a i a ' noon ( ie n | ) oa 3 yp aut Lo bar ; : ne kre é band =e earls cx . i — a SA\e > \ toe 0 [Tt bento es,? Tv / : am ae G@n°O Ser dg — 4 ‘ f i do TO fan” a 2 Fd é oo} fo ~ f > 8 S SA { Py _ 7 fe 5 i van 2 _ a of ~ — SA. 7 . . fo. 3 uh) ec OdB ~ j Sec Oda) wa Rave Seen ght integeal inc a | ~ ) ° rr a and is she book many times S49 , 2 eegeal ary { Sulestiatiar ’ ve ‘  “Oo TO ee deen tf ~ ey dg back to SA deere \ divie by 4 } iy fe. . o 1e dG - —— we a tond Gav It - el mn ul Sec | ft ! | Ah -—— See rH{f ~ Ol f +c, oS ly 8 ta | rl 2 Seeg@doas ¥ 4 alse 1 booy} “oO . Tt fl Mo _ a % i i. 3a ~ gg (0 (se o ‘ ( a i a | > § | |} =e Secar f  4 Bo 2 Oy | L 2 oR ° Dh 2 ~ ~ ~ > i - a a i eo” a ie 6 1 ° Boo” PeSeT Vn exceeding Aecian Viemidedion