MATH 1A FNAL (PRACTCE 2) PROFESSOR PAULN DO NOT TURN OVER UNTL NSTRUCTED TO DO SO. CALCULATORS ARE NOT PERMTTED YOU MAY USE YOUR OWN BLANK PAPER FOR ROUGH WORK SO AS NOT TO DSTURB OTHER STUDENTS, EVERYONE MUST STAY UNTL THE EXAM S COMPLETE REMEMBER THS EXAM S GRADED BY A HUMAN BENG. WRTE YOUR SOLUTONS NEATLY AND COHERENTLY, OR THEY RSK NOT RECEVNG FULL CREDT THS EXAM WLL BE ELECTRONCALLY SCANNED. MAKE SURE YOU WRTE ALL SOLUTONS N THE SPACES PROVDED. YOU MAY WRTE SOLUTONS ON THE BLANK PAGE AT THE BACK BUT BE SURE TO CLEARLY LABEL THEM Name: Student D: GS s name:
Math 1A Final (Practice 2) This eam consists of 10 questions. Answer the questions in the spaces provided. 1. Calculate the following. You do not need to simplif our answers. (a) (10 points) d tan(sin(e )) d tan since seacsin e cosce e c tan since a f Z k (b) (15 points) d ) d cos()sin( f Cnsc c s l Luttrell Sintra Lucassen da Luc tezel cos Ct et tu set Sintra cos d dae Cas ca Sintra sintra c set cascara ftp 1uccnsclli siuirl sy
Math 1A Final (Practice 2), Page 2 of 11 2. Calculate the following (ou do not need to use the (ϵ, δ)-definition): c (a) (10 points) seem cos(2 2 ) 1 lim 0 4 nita f f E'o o (b) (15 points) Hint: Rationalize. e_ lim (1 + 4 ) Recall fig it t e Eius it 5 find it 1 E e
Math 1A Final (Practice 2), Page 3 of 11 3. Calculate the following (ou do not need to use the Riemann sum definition): (a) (10 points)! arccos() 1 2 d u arccosc dat mz d TE du J d J uan tu te Lz ceos (b) (10 points)! 2/π 1/π sin(1/) d 3 2 u date d soda si da Jsiucu du s scuttle Joss C 1 1 C 211T J si lit Y da f cos Ctl
Math 1A Final (Practice 2), Page 4 of 11 4. (25 points) Calculate the equation of the tangent line at = 3 of the following curve: 2 3 ( 3) + 1 = 3 1. Zg's s 23C 3 scar 7 2g 3 wr da s 6cd C 31 t 2 t V t tz tda.de o adj r 2 6g2Gc 3 t a 3 3 N 3 dais 3 3 2 2 Equation A Target l z 3 at 3,1
Math 1A Final (Practice 2), Page 5 of 11 5. (25 points) Sketch the following curve. Be sure to indicate asmptotes, local maima and minima and concavit. Show our working on this page and draw the graph on the net page. = ln( 2 2 ) 3 en la 1 it o DVE it 2 0 Not defined at 0 should sketch teal Tu Za ri Domain 2 sac za z o o C 2 0d Vertical asmptotes Neither Lim Tn Za H so z 2C no Luiza sie o and 2 Vertical asmptotes 7 Zac Z Z 2 7 1 cts at 1 Local min A 711 1 0 l s 7 Gal z 3 None in 0,2 f 2127C M t Za z Za z Z c set 2 1 2 1 22C al Z O A Neue F X B None
Math 1A Final (Practice 2), Page 6 of 11 Solution (continued) : 1 l O c 6 7C 2 ou it
Math 1A Final (Practice 2), Page 7 of 11 6. (25 points) Show that the following equation has at least one real solution. Be sure to carefull justif ou answer clearl stating an results ou use from lectures. Hint: Consider behavior at. arctan + c = 0, where c is a constant. Let te arctanbel 1C im a s Cim arctanlal o sea Liar aretankel c to s Lim so are tan 174 t c A There eists a b such that 7cal co C 7lb V T 7 C d s ou Ca b There eists c Such that 7 Cc o
Math 1A Final (Practice 2), Page 8 of 11 7. (25 points) A drink will be packaged in a clindrical can with volume 40in 3. The top and bottom of the can cost 4 cents per square inch. The sides cost 3 cents per square inch. Determine the dimensions of the can that minimize the cost. stat cost A Objective Minimize As tap and bottom side r d d Objedive 4th 4th 3 2 Trh µh Constraint Trh 40 v h tv2 STA 6 uh Sir Gtr 72 Str 24 fer Domain o s f r 16 r 240 M a F r o 16 r B 7 continuous ou o o fs 7411 co 7 60 o 24 tier r3 r T Absolute min cost is when r F h CEE
Math 1A Final (Practice 2), Page 9 of 11 8. Two rockets are fired verticall into the air from the ground. The second rocket is launched four seconds after the first. The velocit of the first rocket is v 1 (t) = 6 t metres per second and the velocit of the second is v 2 (t) = 10 t metres per second, where t is the time in seconds after the first launch. first (a) (15 points) How long after first the launch will both rockets be at the same height? What will this height be? S Ct GE Et t C and S co o o S Ct Gt z El Sz Ct Lot Etc C and 52141 0 40 St C o 32 Sect lot zt2 32 S Lt Sz Ctl Gt tc lot tzt2 32 4 C 32 E S S S Secs 16 C height at t 8 time after 1st Land the 111 be same height (b) (10 points) Determine the total distance traveled b the first rocket at this time. G 6 t Total Distance travelled 2 g Area Area zz
Math 1A Final (Practice 2), Page 10 of 11 9. (25 points) Calculate the following limit: Hint: Remember that A B = 1 B/A. lim n n" i=1 n n 2 + 3i 2 n u2 3 i 2 h 2 u2 z ie i t 3 if L Letting a o b 7cal z we get fig n iz da let a Fsu Fs da du da J 2 da J du arctanculte avatar Fsu C fig n rstanctuers r
Math 1A Final (Practice 2), Page 11 of 11 10. (25 points) Calculate the volume of the solid of revolution formed b rotating the region enclosed b = 2 (with 0), = and = + 2 around the line = 2. _F 4,2 Y T 2 µ tannate 010 1 i down b 2 Tze z c L a 2C 2 Volume f C a 212 Ta 212daL O it Ca 4 2 CTE 2 Lda T C a zkda it e 4 da it 4th 14 da cactus to ca 4131 it Eri sak 4 11 9 it t O C 9T 8T T 161T 3 T END OF EXAM