YOU MAY USE YOUR OWN BLANK SO AS NOT TO DISTURB OTHER COHERENTLY, OR THEY RISK NOT YOU MAY WRITE SOLUTIONS ON THE

Transcription

1 MATH 1A FINAL (PRACTICE 1) PROFESSOR PAULIN DO NOT TURN OVER UNTIL INSTRUCTED TO DO SO. CALCULATORS ARE NOT PERMITTED YOU MAY USE YOUR OWN BLANK PAPER FOR ROUGH WORK SO AS NOT TO DISTURB OTHER STUDENTS, EVERYONE MUST STAY UNTIL THE EXAM IS COMPLETE REMEMBER THIS EXAM IS GRADED BY A HUMAN BEING. WRITE YOUR SOLUTIONS NEATLY AND COHERENTLY, OR THEY RISK NOT RECEIVING FULL CREDIT THIS EXAM WILL BE ELECTRONICALLY SCANNED. MAKE SURE YOU WRITE ALL SOLUTIONS IN THE SPACES PROVIDED. YOU MAY WRITE SOLUTIONS ON THE BLANK PAGE AT THE BACK BUT BE SURE TO CLEARLY LABEL THEM Name: Student ID: GSI s name:

2 Math 1A Final (Practice 1) This exam consists of 10 questions. Answer the questions in the spaces provided. 1. Calculate the following (you do not need to use the limit definition): (a) (10 points) d dx ln(3x)e2x Idf fu 3 c e e t la 324 2eZ (b) (15 points) d dx tan(e(xx) ) 7 Cal x Iu Flat x Luca tu thou Tulsa 1 7 x x luca i adz tance sea e d x luckily

3 ius Math 1A Final (Practice 1), Page 2 of Calculate the following (you do not need to use the (ϵ, δ)-definition): (a) (10 points) Hospital sin(x 2 ) lim x 0 tan x sin 1 21 I 2x SGT info Zacoscx2 Lim o tangy a O Sed im seek c k 70 9 O Lim a a 2 Lim (b) (15 points) a VEE Lim 3 2C 3 s TE lim x 1 + 4x x 3 z Lim Ex z x a I T T u Ti I t T z

4 Math 1A Final (Practice 1), Page 3 of Calculate the following (you do not need to use Riemann sum defintion): (a) (10 points)! 2x + 4 x dx I da J one 11 da u x I dug zu doc IIT J doc JI du Tulul C In I C J doc Tula ti I t 4arctanby c L (b) (15 points)! e 2 e 1 x ln(x) dx u luck DI doc edu doc J doc Jiu du lulul t C In Icu x t C dx tu z luci Lutz

5 Math 1A Final (Practice 1), Page 4 of (25 points) A cup of water is placed in a refrigerator. The refrigerator has temperature 5C. After 10 minutes the water is 15C. After 20 minutes the water is 10C. Determine the temperature of the water when it was placed in the refrigerator. Initial NCC Tct Ts To Ts ett temperature TC lol is IS S t Cto 51 e 1012 TCzol to to S Cto S e Zo k O lo k e and e To S To S S Zo k s Zo To s To 25 C

6 Math 1A Final (Practice 1), Page 5 of (25 points) Sketch the following curve. Be sure to indicate asymptotes, local maxima and minima and concavity. Show your working on this page and draw the graph on the next page. y = x2 e x x 4 Cx d e Tt x to DIVE it c o sketch y IR ace and remove 0,0 0 en Neither tympts None Lim see to a 38 Lim xe Lion Lim a e x k a 3 a e I O y o horizontal asymptote 4 x e c ace ltd e AT 71cal o x I By None as x a 1 cts at I Local min I c T Gal 1 1

7 Math 1A Final (Practice 1), Page 6 of 11 Solution (continued) : 4 cts at 2 7 Cal e t tix e zen ex inflection Ay 7 coal _o x Z I 54 By None 7C l e 7 21 Ze 2 y IIe C I e l

8 Math 1A Final (Practice 1), Page 7 of (25 points) Determine the equation of the tangent line at x = 1 of the following curve: y =! 3x x cos(πt)dt + 2x Y cos Cittldt cos CTH dt 2x dy dy dy 3 cos 31T c Cns Tx 2 3 cos 31T Cns Cit z 3 C il c 2 0 cos CIT at 2 sin Tt P Tangent line has equation y z

9 Math 1A Final (Practice 1), Page 8 of (25 points) An open box will be made by cutting a square from each corner of a 3 by 8 foot piece of cardboard and then folding up the sides. What size squares should be cut from each corner to maximize the volume. Objective 8 IIe Is Maximize Volume L Objective Volume 3 Constraint 2x y S Za y x 3 2x y 8 Volume 8 Za e 3 2x t Domain Co Iz FGC 2 24 a F x x t Hae x 3 A 7 x o x n 3 By 7 continuous everywhere Co 0 Volume is maximized when f a 8 inches fc4z 70

10 Math 1A Final (Practice 1), Page 9 of (25 points) A company accrues debt at a rate of (2t + 3) t + 1 dollars per year, where t is the time in years since the company started. How much will the company s debt have grown between t = 3 and t = 8? You do not need to simplify your answer. D Ct 21 3 TF u Et l dat I de du t u i Jett 3 TEI DE Jett 3 Tu du J Cu 1 3 Tu du J2u n'k du z 3 ask 3 ask etty t taste ttdreidt 4sCttDskt2_Ltti1k1f CsI.ss E DCS DG

11 Math 1A Final (Practice 1), Page 10 of (25 points) Calculate the following limit: lim n n" i=1 6n + 3i n 2 sin((2 + i n )2 ) Gut 3in sin in sin 2x T f Let a z b 3 and 7cal 3 a sin se n 3 Lim 2 Gunt s u 2 u i c Ily J3asiulse da 2 u act did Zac da J3xsiucx2 doe 3 2 Siu Cui du costal TC 3 2 scat C u 3 Lim I Gnut sin et 14 cased u g 2 Iz costly Cns CP

12 Math 1A Final (Practice 1), Page 11 of Let f(x) = 1/x and g(x) = 4x h(x) = x. y p (a) (10 points) Calculate the area of the finite region bounded by y = f(x), y = g(x) and y = h(x). 42C y a Area 4 47 da a da L Za tubal Earl Is ft tutti Il In 121 (b) (15 points) Calculate the volume of the solid of revolution formed by rotating this region around the x-axis. Volume IT 4 32 za da a da Stix l t T Iz tic 3 SIT g IT G l t f z END OF EXAM