# Calc_Review_5_-_3.8,_3.9_answer_key - “AP Calculus ‘...

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Unformatted text preview: “AP Calculus ‘ _ Name: Review 3.8, 3.9 Topics to study: 0 derivatives of inverse functions 0 derivatives of inverse trig functions 0 derivatives'of logarithmic functidns - derivatives of exponential functions The test will be divided into calculator and no calculator sections. Try to do as much of the review as possible without a calculator. Find‘(f'l)'(a) for the function f and the given real number a. 1. f(x)=x2—3x—1,a=.3 3X30 2. f(x)=sin2x,—£Sx\$£a=%-—l- /\ Ble’BX'l I w I f 2 'l L a ‘ m a t. (to—i u ’ corps we 3‘9“ ax W30: 1C“ X 444‘ '3 .(r (l . .1... IT‘- \l~ X0} (xvl‘l’ _, _\- 7" lax"? 'F (Ta): lugs ‘4" \$0023 ( t 5 is e {I- = 1 <0}. The table gives values of the functions and their first derivatives at selecte values of x. 3; ( :3 , ‘ :5 ,~ 1 ,L - (. 5“ (a) V6 3. a) If g‘1 is the inverse function of g,_find (g'l)'(3) 300:3 ' 9‘0?) 7-"? L - 3 (90:3 (TWQ: “a b) Write an equation fjér the line tangent to the-graph _of y = '1 at x=3. a 7 3C» 7—2.: ’Ciflx—g g" (3) 2;; 4. a) If t"1 is the inverse function of f, find (f")’(6~)L \$8199 {307’— '9. giijrjo epoxy): ,9} I b) Write an equation for the line tangent to the graph of y = f'l(x) at‘x=6. 40):,9 Y’l " ‘30‘4‘0 3" (lo) : i 5. y = (csc x 4? cot x) ’1 6. g(t) = arcsin(3t4) 7. y = csc'1 (5x) ‘ 1— lli‘wcx#(Zoﬁ)z(ecscyco+‘>krcgclx) 5%) r. I . la‘l’fg '— w y.’ ~l (-65ch +x— 10 W Y* lg‘lxs’ol" l ‘/ : o W '12 4,3 ‘ Ml ' (Lscv we Les-w)" 3‘40 : Y : B” W '71: max (cc+x-c<,cx) H” CH5? i (so: +u4x 5‘- V’: CSCX cscy +co+>< Pm &\SU&' 7. f(z)=zarccosz—\/1——z2 8. y:arccs‘c\/x+2 ‘ 9. h(x)=arcsec(5)'c4) 3 n -I 1.1; \I: 1......— .———- l , M‘OX 951* ﬁmh “W110 3. W" /, (WW 2075 “@1811 (#42, -’t' i I: ___._:_~—«~' WY. : ' L1” 4((E): V11 + arccosa + W... 7 (“'17 Fr“ 1 (3 pd 25x :7 l p ' . :0. C 52 “C <27 V ‘° ~ , 10. g(x)=arctanx 11. f(x)=ln|secx+tanxl 12. f(x)=1n l _ 1 | suxlrarw +5669} ‘ __ elm” Hx‘ W"): smx' HAS-=2 WW? W") , sumx erlx PM: huts-3' X; -‘— {5 (543 : —————-—--—-*"“" X 321. Y ‘ x J— PM: sax (-mnx—rSeLX) ZSCCx y-F (X): m,"x -; L1 , 13. f(x)=x21nx . 14. y=1o§jff+xi§§3 15.y= 0g4(5x2) . IOX 1:01“ 1 ‘ "J ' 2. \ 1 , \ o \_ . x61“ WUCXVX ° 32 J“ ‘nx ( X) 7": \n3(H‘/~|h§> (614\$) \/' (W4) 5X1 £‘Q:X +2x1nx \_ \n3 : r 'E “3" C ‘/ “ Imnﬁ ‘J‘ﬂnﬁ \/ am) 5%“ 16. y=x33x 17. y=e3"]”1 +5e3" 18. f(x)=46"‘5 3 x x 1 W’5 ' *5 \I" 01K?QBX3-’¥ 15e3x “*3:wa W W - ex—e_x J; 4x—3 6x 19. y: x _ 20 y? 21. 32—5 —6e 1 e +e x x .L 4 —3 - . x —¥ X ‘1‘ ‘: ‘ Y ax 71:.(exwyxietglémfe (6+8 ('03 85‘ 3E leon‘ﬁs . '1’ — (of: ' (o » 7' 2' 7. o — o . ‘ : > '3 \._(e"+e'*> “(MD a”: Re +e1’.‘(e‘fae+e” _ 23x ‘1 (4W5) 5“x -.. 2, (o ef” 1’ W [36“ +2“ :ex “8%): \l 21. Consider the function f(x) = xlnx. 0. Find the instantaneous rate of change of f at x = e. 'F‘CX)Z)(I')XE.L\nXK\> 9‘03): 1+1WL : l b. Find the average rate of change of f over the interval [2.5, 3]. A5 41(3)’£(2,s)'_~ sma' 9.5 mas ; 9 0‘0 [x : 3-2.5 ' ’ 0.5 ‘ 22. If The raTe of change of f(x) = 2" , aT >_< = 5 is four Times TheraTe of change aT X: a, Then ﬁndwmmqua's' PM ‘0” ‘23 9" ~ . Bath 1: Li 029931“ 0 59(5): (l ha) 3:: 319M” ‘ 3281...:0‘1 «C To.) 2. Qn 9g 4 M76 23. If you push open a door ThaT has an auTomaTic closer, iT opens fasT aT firsT, slows down, sTarTs closing, Then slams shuT. As The auTomaTic door-closer slowed The door down, The number of degrees d(T), The door was open afTer Time T, in seconds, is given by d(t)=200t-2" ,OStS7 How fasT is The door opening aT T = land T = 2? d‘L\3~_.3o'.o35 dermis (To): “M .315 “W‘s/5 24. A Table of naTional populaTion esTimaTes in The US. from 1990 To 1999 is shown. a. Use The qua in The Table To approximaTe The raTe of change of The U.S. populaTion in 1999. Show The compuTaTions ThaT lead To your answer. IndicaTe The uniTs of measure. ammo 8l3~R7 u 003 ' [91.0101—m33 3 1W3)“ WWW b. Use your answer from parT (a) To esTimaTe The US. populaTion in The year 2000. Show The compuTaTions ThaT lead To your answer. 37; Mo m + QJ—lL-RETO == .117 5) l 33;, 23 c. A_sTudenT proposes The funcTion P(t) = 24992000000099)’ , as a model for The popuIaTion over Time T, where T = 0 represenTs 1990. Find P'(9). Using appropriaTe uniTs explain The meaning of your answer. (Use your calculaTor) PM): 1%)ng ogqoi ‘ ln [@0101 (are) )Mm pop»)th is, increasing odr 0x9 (0&6 05; gloom, LGLDl 99M“? PIA” ﬂux. 25. A parTicle moves along The y-axis so ThaT iTs velociTy vaT Time t2 0 is given by v(t) = tan“1 ((x—3)2)—1. AT Time 7‘: O, The parTicle is aT y: -3. (NoTe: tan‘1 x = a'rctanx) Find The acceleraTion of The parTicle aT Time 7‘: 3. _ I “soc—3)“ 013.2311, QUE)” Hum)? 11 How)“ 26. A parTicle moves along The x—axis wiTh posiTion aT Time I Z 0 given by 120‘) = 1+ e2!+1 (a) Find The velociTy of The parTicle aT Time T = 2. it 'll : \1 (t > z e - 9x 3 v m : ;e5 (b) Find The acceleraTion of The parTicle aT Time T = 2 2.“ i (LL-b7: 383\$} a; 7" L'le Ru): Ll e5 l €28 (c) Find all values of faT which The parTicle changes direcTion. JusTify your answer. wt): 0 z 9 ezHl H O 2t+l [no r The. 4\ u n alt-Fm (A The. pal/+che nave/V changes ClW‘LC‘l’lWS \feLOLil-l‘kd \5 k) mamas Pogpm; .3 took a+ ave/P ...
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