®P!'5 , I AP Calculus AB 1~ E \8) Q)t> '3 G 0v f\ ~~~ I p @}( (i)A @)P ~ 1\,(±)t> \\ 15 @c.. @c ~~ 2~ c @B $ g t Unit 12 Assignment 15 E @ t @ tJ..ame: L , So I IA+w1A', c 2_,; AB #6 Limits, Continuity, L'Hopital, Asymptotes, Symmetry I. Limits limX➔a f(x) = C ~ a +-IN.,.. w ,-,;,.-4 1 l (discontinuous) does the limit still exist® Differentiability derivative exists 1 y V point x = a, then f(a) = ~1_:,,__f()() +MA-~~+ 1 Y x, 2x, at x = j, x~1 X< _) Z=ZV (A) continuous {t'Jf'fV'IS nA.vt ,·h+o -et.l_cko-t!A.ov--1 y-vo.-llN!..$ M<::1-. -h:1-1? B · erentiable -f'Lx') =~2x+\1x>I M~·h:n o..-t x;::-1? No c . . . ._~,- \ 2. ><@I 3=/:-2. ~~er +o be di'#. 11t.~s +o \ l.!fi~-eq-,,,.._~s fo '1 o+-f s 1n (-€ classic problem f(x) ={x + j;1 hole/ J -~Iv., a~y~~nd continuous)=P :., ~ _,,...,. Ji\__ Non-Differentiable 1/er+.osy"fl, 1 ~ if-f-(x_)_i_ s_con-t-in_u_o_ us_a_t_s_o_m_e_~ Cv1.s p c"r-Vll'l' ver+. ~ -v I 2 ..b/..A,. c 1 is f(x) ~ b-e { 0 1'1- t • I S" 0 For which of the following does limx➔4 f(x) exist? ,/Ml\- s w--(''( ilo + ($ 1/ -f-1,t_p,_ 'c_&,.,, fV\'€ ,q_v,.e T <j -t--,'o vi The graph of a function f whose domain is the closed interval [1, 7] is shown below. Which .of the following statements about f(x) i true? Y -0-1--.--· -0-1--4_, I 4 Graph o[f ) y-6'? Graph off 5 - Graph of/ 4 y't!7 (A) I only 3 ~mx~; (B) II only ~4f(~ J /\ l 2 3 4 ./ :z: 5 6 7 (C x) is con'!lPueuscit x = 3. (D) f(x) is continuous at x = 5. (E) limX➔6 f(x) = f(6) (C) III only \ (D) I and II only (E) I and III only 1 The graph of a function f is shown below. 4. The table below gives the values of three Which of the following statements about functions, f, g, and h near x = 0. Based on f(x) i false . Y the values given, for which of the • functions does it appear that the limit as x a roaches zero is ? -0.1 0 X -0.3 -0.2 f(x) 2.018 2.008 2 h(x) 1.971 1.987 undef 2 f is continuous at x = a - (B) f has a relative maximum at x = a / (C) x = a is in the domain off. / (D) limx➔a+ f(x) is equal to limX➔0_ f(x). / (E) limX➔a f(x) exists.✓ I (A) (B) (C) (D) f g h f only only only and h only (E) f, g, and h I 0.1 0.2 2.002~) 2.008 2 0.3 2.018 2 1.971 yeS C&:_J_ v(_-f,~ 5. The graph of a function f is shown below. If ~ <\nd f i@ontinuous at b, the b = D : • l ~ l(B) (C) (D) (E) j 0 { ltWl / Cvvt-1,X 1 . 1 ,'vvi ✓ Cc, vt-+. ✓ 2 I r'vvi >( Co111.:-r .. x__ 3 /1'V'vt. ✓ COI/\_T, ✓ (make your algebra 2 teacher proud!) II. L'Hopital's Rule (BC topic, but can use) 2 . x +6x+9 _ o 1-/;1.def--e,rl/lA;-V'--,,,_-1-e Ii x3 -8 - o -= h'Vv\ _(~(x:z. + I 1mx➔-3 -----mX➔2 x-2 - o x._.2.. ~ 2.. X+3 0 zx.+1:J-Q-z..i in pre-calculus, we factored = limx➔-3 (x:tJ~+~) -==@] ex what if don't remember how to factor a3 L'Hopital If limX➔a f(x) = limX➔a g(x) = 0 then lim why is it okay? x-sc b-ertt ., _ s e ,~ i's I.Im t,'v...f vv1+11'\..1 c< h v 1-e , 1,\ , a_ '+ f(x) g(x) = 3 X - 8 o --:..__-:= X ➔2 X - 2 C> f' 6<..) 1 <x) \1'VV\ x-. o Q.. ~ \,IW\ 3x. 2 x-z.-,- I. @rt f(x) = (A) 3 2 9 x - is ~ t x = - 3, then f(- 3) 3 x+ ==- E__ = I ,'vi., · Lx.-:s .~ 0 = B - )(-t>- ~ (B) - 3 (C) . (D) O 6 ] [ (E) - 6 _ hol-e °'-+ x:::- f I"'-;) c; -s )---G--rL·• -ti« ~ - ho { _,; , r-F(_--~-,'l1c--__ _._ _ if:> - b 3? p(Jse;,-bt-e III. Asymptotes Vertical ~y Horizontal y h o w· d. o wy7 cu.-J c /A,r QI..-+ e - - - -- - - - /i'wll+-CA.-f-00 (A) - 2 of:._ lrm 2-X -)(-Q>oO (C) 1 \-~ = X-lil<-oo-8' /' /V\,I\ 2 =~ M--t-t-------,~~ fl~., f· DO -Qj -- ..q 2 (D) 1 (E) The limit does not exist. 13. The graph of which function ha~ 14. If(.)! = 7 s ~ orizontal as m t of a as an symptote h" rt ~tm+~ I rational function f, then which of the /t'VVL y following must b true? /1'vvi =- T ·;JA'JY= e-x -:- e~ =O~ -~---... ±-oo (A) lim~ a: x~ too = -/ vfBfY = -_: ·_ l =f- I✓ j/2?y = In (x +· 1) == l.n: CX9 =00 =j:--1 ~y=&i= I =t=:-1 ( (E) y= !&[-\ ✓ t (B) limx➔oo f(x) = ~/ (C) Ii~= 7 (D) li~O (E) lim~j2}f 15. Fo@he orizontal line an 16. Which of the following graphs best ~-===II+ lxl+l resembles y = ~-? lxl-1 x==±-1 Vlt- asymptote for the graph of the function f. Which of the following statements ust be tru /1'111,f v (A)~ (A) = z... COWL _Ji I I I I (C) y !I l\_ -----rI ,----i o·'o c; i; H-A- (Bl y , x_-.+c1<0 (B) ~or all x ~ 0 (C) ~ndefined. (DJli~x) = oo (E) limx➔oo f(x) = 2 :,; I I I I I :,; :,; (D) (El y -J.:::..xL-/ y<o :,; 17. Which of the following are the equations of II horizonta 18. ~~e graph ot y ~t n y-o ~ ) \ (C) = /1'W1 =0 y,..;---t"' _(-E')- .,blooe 't = + W-o -ft/VD (A) J).0--horfz. and one vert. asymptotes (B) one horiz. and one vert. asymptotes (C) two horiz. and one vert. asymptotes (D) one horiz. and two vert. asymptotes CTE) two horiz. and two vert. asymptotesl 0 I% Y=-/ x~-oo Horizontal) y y X = 1_I x x~ +co asymptotes for th ff (B) 12 19. The function f is given by f(x) = ax: + 20. A function f(x) has X +b The figure below shows a portion of the following could be the values of the constan s a and b? /-flt: 1= 3 y ' .~a.=-3] ! \ v~~ x-;: z_ L~i1:".:~1 3 -) 2 , I I ,,· J 4 5 X 3(x2+4) ex -t- 2--)c x-2...) (A) a = - 3, b (B) a= 2, b = ( C) a = 2, b = -~ ~ =2 -3 -2 t{D) a= 3, b = - 4 (E) a= 3, b = 4 (A) I only (B) II on I verti~--, 1 0o-f >1 21. If f is a continuo lim, _ f(x) = - 3, which of the following If limx➔2 f(x) = fG°e2) = 0, which of the x- 2 i · JI following, us etrue. I =r+: S statement~ - - o )- --+--....... /r. 'limX ➔+oo I. f(2) f(x) = 3 / ✓~ , (c~ill!-14&- .... s . There are no vertical asymptotes OVV\._cl :i=- (8) II only (C III onl D I and III only l{E) I, II, and III ill: 0 ~ =0 -= }i'v,,, -f''(x) x~2- I ✓ ofl-uwt'~ I ,-I . '1 ~+ -=-f'(v-o -;;- /ri-v.._ -=eo II. f(x) is continuous at x = 2 /(c1.1 vt'v ~><,.)~ (x) has a horizontal tangent line/ at X = 2 III. The lines y = 3 and y = - 3 are horizontal asymptotes ✓ (A) I only 0~- 0 ' ' (A) I only , ,. , . 7 1 (8) II only .\ (C) I and II only\ (D) II and III onl I, II, and III 1 23. If f is a function which is everywhere ) 25. A population grows according to the equation~= 6000 - 5500 e -o.1~ for@ This population will a roach limitin value as ti . During which year will the population reac half f this limiting value? (Ge.) 6 (A) (8) \ ( C) ' (D) (E) ) second third fourth ( eighth twenty-ninth /i'Vk.. -6 ~ (X) Pl-t-) = 1,·Vvt 10000 - -t, -i> OQ'(__ S5oa )~ ,/sq_ -b