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

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1 MATH 1A FNAL (PRACTCE 3) 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:

2 Math 1A Final (Practice 3) This exam cnsists f 10 questins. Answer the questins in the spaces prvided. 1. Calculate the fllwing. Yu d nt need t simplify yur answers. (a) (10 pints) d ln(x 2 + 2)) dx tan(x) d th x'c 2 Zac tan x1 tu site See Cx DX fan se f fan call (b) (15 pints) d dx cs(x(ex) ) f Gc ate Lu 7441 e Luca luctec e envy t e T F see e eucxl t e T c.sc.ie sinc.ie see e eucxl e

3 Math 1A Final (Practice 3), Page 2 f 11 x Cim 2. Calculate the fllwing (yu d nt need t use the (ϵ, δ)-definitin): (a) (10 pints) Hspital tan(2x) lim x 0 x fan 2x him 2 Secc 221 K 2C 70 z (b) (15 pints) Hint: Ratinalize.! lim (9x2 x) + 3x x im V 3k c k 4 2 3k Lim t 3k a 7 N Lim x s J 3se C Cim f T l z Cim l l x a M

4 Math 1A Final (Practice 3), Page 3 f Calculate the fllwing (yu d nt need t use the Riemann sum definitin): (a) (10 pints) " x 3 x 2 + 1dx du dc U x2 22C D 2x f c 3M dc sie Tu du Jae il Tu du fuk a edu E Zuk Zuk C x2 ij 5 Cat The C (b) (10 pints) " 2 1 e 1/x2 x dx 3 u did dc du J die Jeu du eat C e c J a.c e Ee

5 Math 1A Final (Practice 3), Page 4 f (25 pints) The ratin f carbn-14 t carbn-12 is an ancient wden artifact is 60 percent that f living rganic matter. Hw ld is the artifact? Hint: The half-life f carbn-14 is 5730 years. M t Mass f carbn 14 t f tree years after death Nlt ett and M 5730 MCO Ce C t e 5730 Mitt e Luth t M t O G M LO C e luh t O G C t tuc G c Hw Lng ag tree f died fantifact was made

6 Math 1A Final (Practice 3), Page 5 f 11 1 Csc Dmain 5. (25 pints) Sketch the fllwing curve. Be sure t indicate asympttes, lcal maxima and minima and cncavity. Shw yur wrking n this page and draw the graph n the next page. 3 CZ t 1 R y = x3 1 + x 2 0d Odd Fl x 7 Cx cdasympt.tn UL Nne B a text a in Lim a x 7C x Lin 4 x se Lim is 23 2 a Lim s 2C s 2cL sett l y x a slant asymptte flea x 3 2 f Mtl Catti 2 l 1 Gal x t 7 by 3 Nne f s

7 Math 1A Final (Practice 3), Page 6 f 11 Slutin (cntinued) : f a nil Zx 2 Gilts se x Mtl C G 22cL xl Aj x O F By Nne v nflectins r w t 7 Ge t B tr y x tri _T t Bl 31 4

8 Math 1A Final (Practice 3), Page 7 f (25 pints) Shw that the tangent line t the curve " x 2 y = te t2 dt 3x at x = 3, des nt cntain (0, 0). 32C y te t J C et dt O dy Tze 22C acte C s 3 1 e y dug z 33 e 33 e's 33 zzes s Odd 4 J t 9 et dt O Tangent line ges thrugh 13,0 81 and has scpe 27 e O Cc n m tangent line

9 Math 1A Final (Practice 3), Page 8 f (25 pints) Find the area f the largest rectangle that can be inscribed in an equilateral triangle with sides f length Z3 if ne side f the rectangle lies n the base f the triangle. J z bjective cnstraint Zay pg T y Fs l a Zay Zfs se zi 764 Dmain l c clsed interval 7 Cal 253 Fs x Ay 7 x x B 410 O 1 l 7 E zer E Bz Area at largest rectangle

10 Math 1A Final (Practice 3), Page 9 f Tw cars are travelling directly twards each ther n a straight rad. The first car is travelling at 3 metres per secnd. The secnd car is travelling at 6 metres per secnd. When they are 6 metres apart they simultaneusly apply the brakes. The first car decelerates at a cnstant rate f 2 metres per secnd per secnd. The secnd car decelerates at a cnstant rate f 4 metres per secnd per secnd. (a) (15 pints) Hw lng after applying the brakes will the cars cllide? justify yur answer. Carefully a t Z V Lt Zt V 3 mis 3 6mg _VO Vz t 4 t G ar t 4 CAR Gm CARZ c Silt t 3T 5,103 0 x 6 Sz Lt ft G CS 07 6 S c Szc S Ct SzCt t2 13T 2E2 6t f C L r et Ct c Ct Cars cllide when C (b) (5 pints) Hw far will be secnd car travel between applying the brakes and clliding with the first car? sell 2 G The secnd car travels 4ns between hitting the break and crashing

11 Math 1A Final (Practice 3), Page 10 f (25 pints) Calculate the fllwing limit: ZCatibal lim n n# i=1 n + 2i n 2 + in h 12 i l t f act in 1 atiba let a b z f x i n h 22C bu Z Da j neg n z Ed zu ennui z tuk

12 Math 1A Final (Practice 3), Page 11 f (25 pints) Calculate the vlume f a pyramid with a square 2 by 2 base and height 1. A x Vlume 4 ed 43 3 t END OF EXAM