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AP Calculus AB
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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