TOTAL. Math 21A. Fall Dec FINAL EXAM. NAME(print in CAPITAL letters, first name fiit): NAME (sign)

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Read each question caefully aid answe it iii the space povided. YOU MUST SHOW ALL YOUR WORK TO REEIVE FULL REDIT.

1 Math 21A. Fall 218. Dec FINAL EXAM NAME(pint in APITAL lettes, fist name fiit): NAME (sign) ID #: stuctions: Each of the 8 poblems has equal woth. Read each question caefully aid answe it iii the space povided. YOU MUST SHOW ALL YOUR WORK TO REEIVE FULL REDIT. laity of you sohiltiolls may lie a facto when detetinning cedit. alculatos, books o notes ac not allowed. The pocto has been diected not to answe any intepetation questions..\lalce sue that you have a total of 11 pages (iiicntliiig this one) with 8 pohletits TOTAL 1

= 1. Please be cueful: cii each of the pats (a) (c) below, you will eceive little

(a) oiiipute the delvative of the function Do not simplify!

2 = 1. Please be cueful: cii each of the pats (a) (c) below, you will eceive little o ito cedit if you make a dideeitiation mistake, even a small one. (a) oiiipute the delvative of the function Do not simplify!, xx (h) oiiipute the deivative of the function y = actan(x ± 1/a:). Do not simplify! I (1) (c) Find the ecluatioti of the tangent line to the cuve p3 It2 cos y 2x = 1 at the point (, 1). Give lie answe iii the slope intecept foiti. t (< ;. ) 2tiR 1 1 x=o, %=I 4 2= * 1 =-(x o) 41 9

3 2. ompute the following limits, in any coicct way you can. Give each answe as a finite numbe, -1 - /T 1 2 ( x) - (y 1f& 1 = L 4 tcfj al %..) (h) n (a-v&e-v >OJ V cos :1; (c) bin T{) 1 cos(5.c) 4&lvx x) () H 2 2x t 2x tlx Z 3

F-I 9 i -o I-tm.4-3 It It I -, -p 1 -S 1.3 II id V 1-.

4 F-I 9 i -o I-tm.4-3 It It I -, -p 1 -S 1.3 II id V U). ;:o 3 I S cj 1 c-liz n UI li Ii -nil - - to H 4. I _,.5s--. -t c< : 7 _J Ij > > ) n,< I ) - II K 3< 2- E& 3 U cij1 4 d 11 J A? c J 1 A, c 1 V Al II - :3 4-3 II

4. Tn all pails of this ;nvblen, the function f is given by

(a) Deteiiiitie the domain of y = f(x) ompute him f(x) and

which y = f(x) is inceasiii and t lie intevals on which it

1-1- 4 SI 7 zoi (c) Detemine the intevals on which y = f(a)

5 4. Tn all pails of this ;nvblen, the function f is given by f(x) = l z ± X /. (a) Deteiiiitie the domain of y = f(x) ompute him f(x) and tim f(x) (, L(Stk)t) -3 (h) Deteiiiiiie the intevals oii which y = f(x) is inceasiii and t lie intevals on which it is deceasing. nn4.34t1 4k -t Wt&.c, a /2- U Oil SI 7 zoi (c) Detemine the intevals on which y = f(a) is concave up and the iittevas oti which it is concave down. Identify all inflection points. Vi (x J Do ) 1- - n ->1 )

of iiiipot ancc on the gaph. (You may use.1(3)

6 (d) (Still: t lie [tinctioti f is given by f( ) = (/2 :.) Sketch the gaph of y = ompute all iiecessav hunts and label all points of iiiipot ancc on the gaph. (You may use.1(3) = 2.3.) H (e) Deteiitiite the ange of f. LI (f) Is q = f() otw-to-one on (. 1)? tl, \ sl-l - &1 c1 ) AtcA-(4&j fa (ô,1) 6

5 Ac yliude is inscibed in a cone. The cylinde has height.

Detemine I lie (hliiiensio&is of such cone with the siijaflest

flecall that the volume of a colic with adius of t lie base and h&ghit

7 5 Ac yliude is inscibed in a cone. The cylinde has height. 3 inches and adius of the base itichi. Detemine I lie (hliiiensio&is of such cone with the siijaflest toluiiie, lustily all you con elisions. (Hint. flecall that the volume of a colic with adius of t lie base and h&ghit /i is 41c2h. Use siiniia tiangles.) side view h h h S t 1 a 3 %Jt L&a4 4 w4l1t&a_1t 4 = V ikl ()2 - z(h- h 3 = = jln. ( 41 US& j o I. - h1 ) 2-

6. A balloon is ising vetically above a staight oad,

A ca is slowly diving on the oad awe fom A.

3 miles fom A, diving at the speed of 2 mph.

and the balloon is changing at that instance. (Note:.

$ca \Mktvs x=.2 ock <=O,? tz ) OI = c=q,: = o,t-) =?$

8 6. A balloon is ising vetically above a staight oad, stating at the point A in the figue. A ca is slowly diving on the oad awe fom A. At sonw point iii time, the balloon is at.1 miles above A, ising a.t the speed of 5 ntpli, while the ca is.3 miles fom A, diving at the speed of 2 mph. Detenane the speed at winch the distance between the ca and the balloon is changing at that instance. (Note:.32 =.9, 942 =.16, 2 = (J.25.) 2 )(2_f;j) I balloon 2 ta_x at d -x at at -k-/f ±L A o ca \Mktvs x=.2 ock <=O,? tz ) OI = c=q,: = o,t-) =? (b) At what ate is the at t lie same moment? aea of tile tiangle detemined by the balloon, tim ea, and point A changing A= b<y at- ) 1 ( o

9 7. all pats of i/i/s pi&lcn. the fuiction f is given by I) = = I ±v2. (a) Identify tue (1111W!), iiionotonicitv and concavity Popeties of tins function and sketch its gaph. I,, (b),4 > (b) Is this fat ict ion odd, even, o iieit he? it

4 -axis LI1)cI two of its vetices on the gaph of y = f(x)?

10 i2) (c) (SUN: the functioit f isgveti by f(i:) = 1 = 1 - What is the smallest aea of a ectangle with two of its veces on the.4 -axis LI1)cI two of its vetices on the gaph of y = f(x)?. N 1t&3 a - 4= 2&( a(&tj k ( O ca4 (t.jll.k ØL=i I ij a4a( tkJ.J &4-&L Z 2 1

assuiie that j = 1(i) is a cointitnious futictioti tlehuied fo al j:.

(Note: assumptions in (a) apply only to (a); the same is tue fo (b) and

How manly v-intecepts does f have? oua o.

all a. Is it possible that f is one to one?

11 8. Povide staightfowad. awl full: justified, answes to the following questions, each of them, assuiie that j = 1(i) is a cointitnious futictioti tlehuied fo al j:. and fc) aid f (:z) exist and ae continuous fo all a. (Note: assumptions in (a) apply only to (a); the same is tue fo (b) and (c).) (a) f(- 3) = 5. f(3) = 2, and f (x) < fo all x. How manly v-intecepts does f have? oua o.i,fcaaf a US3* j x-ia4pt, J &-2O ojus)< 4 VT ct &at -- (h) f (x) > fo all a. Is it possible that f is one to one? alt example of such a function.) (Eithe pove that it is not possible o give -i &-&cj 4 (/L QX%>Q (c) f(1) = 4, J(3) = 8, f (.v) 1 fo all :. How umiany solutions does the equation f (x) = 2 have? tlvij-llt-t 4J c = 2 j It

4.3 Area of a Sector. Area of a Sector Section

4.3 Area of a Sector. Area of a Sector Section ea of a Secto Section 4. 9 4. ea of a Secto In geomety you leaned that the aea of a cicle of adius is π 2. We will now lean how to find the aea of a secto of a cicle. secto is the egion bounded by a cental