1.
2.
Math1AMidterm1Review
Let
⎧⎪
if x < 0
⎪⎪ −x
⎪
f (x) = ⎨ 3 − x if 0 ≤ x < 3
⎪⎪
⎪⎪( x − 3 )2 if x > 3
⎪⎩
a)
Evaluateeachlimitifitexists.
i)
iv)
b)
Whereisfdiscontinuous?
c)
Sketchthegraphoff.
lim f (x) ii)
lim f (x) v)
x →0+
x →3+
ii)
iii)
iv)
v)
vi)
vii)
viii)
lim g(t) lim f (x) vi)
x →3−
lim f (x) x →0
lim f (x) x →3
lim g(t) t→0+
lim g(t) t→0
lim g(t) t→2−
lim g(t) t→2+
lim g(t) t→2
g(2) lim g(t) t→4
Findthelimit.
a.
c.
(h − 1)3 + 1
h→0
h
e.
lim
lim e x
g.
3
−x
x →1
lim
x →3
iii)
t→0−
3.
lim f (x) x →0−
Thegraphofgisgiven,findeachlimit,orexplainwhyitdoesnotexist.
i)
lim
v→4+
x + 6 −x
3
x − 3x
2
4 −v
4 −v
b.
d.
f.
h.
x2 − 9
lim
x →3 x 2
lim
+ 2x − 3
t2 − 4
t→2 t 3
−8
lim tan−1(1 / x) x →0+
⎛ 1
⎞⎟
1
lim ⎜⎜
+
⎟
x →1 ⎜
⎝ x − 1 x 2 − 3x + 2 ⎟⎟⎠
1
4.
Usethegivengraphof f (x) = x 2 tofindanumber δ suchthat
If x − 1 < δ then x 2 − 1 <
5.
6.
1
2
Provethestatementusingtheprecisedefinitionofalimit
x
3
lim = a.
b.
lim(14 − 5x) = 4 x →3 5
x →2
5
a.
b.
StatetheSqueezeTheorem
If 2x − 1 ≤ f (x) ≤ x 2 for 0 ≤ x ≤ 3 ,findthe lim f (x) x →1
c.
Provethat lim x 2 cos(1 / x 2 ) = 0 7.
8.
a.
StatetheIntermediateValueTheorem
b.
Usethetheoremtoshowthatthereisarootoftheequationinthegiven
2
interval
e- x = x ,[0,1]
9.
10.
x →0
Showthateachfunctioniscontinuousonitsdomain.Statethedomain
x2 - 9
sin x
a.
b.
g(x ) = 2
h(x ) = xe
x - 2
Findthevaluesofaandbsothatgbecomescontinuouseverywhere.
⎧⎪ 3x + 1 x < 2
⎪⎪
g(x) = ⎪⎨ax + b 2 ≤ x < 5 ⎪⎪
⎪⎪⎩ x 2
x ≥5
⎧⎪ 2 ⎛ 1 ⎞⎟
⎪ x sin ⎜⎜ ⎟ if x ≠ 0
⎜⎝ x ⎟⎟⎠
Showthat f (x) = ⎪⎨
iscontinuouson (−∞,∞) ⎪⎪
⎪⎪⎩ 0
if x = 0
2
11.
i)
12.
a.
b.
Whatdoesitmeantosaythatthelinex=aisaverticalasymptoteofthecurve
y=f(x)?Drawcurvestoillustratethevariouspossibilities.
Whatdoesitmeantosaythattheliney=Lisahorizontalasymptoteofthe
curvey=f(x)?Drawcurvestoillustratethevariouspossibilities.
c.
Findthehorizontalandverticalasymptotesofeachcurve.
x +2
x2 + 1
ii)
iii) lim x 2 + 4x + 1 − x lim
y= 2
x →∞
x →−∞ 9x 2 + 1
2x - 3x - 2
(
Findequationsofthetangentlinestothecurve y =
coordinates0and-1.
13.
14.
15.
2
1 - 3x
)
atthepointswithx-
Findtheverticalandhorizontalasymptotesfor f (x ) = (a - 1 + x - 1)- 1 ,whereaisa
positivenumber.
Insection2.7,wefoundthedefinitionofderivative,namely
f (a + h) − f (a)
f '(a) = lim
.
h→0
h
Usethedefinitiontofindanequationofthetangentlinetothecurveatthegiven
point,
y = 2x 3 - 5x @(-1,3).
Abicyclestartsfromrestanditsdistancetravelledisrecordedinthefollowingtable
atone-secondintervals.
a)
b)
c)
d)
16.
If lim
17.
Sketchagraphofafunctionfthatsatisfiesallofthegivenconditions:
lim f (x) = ∞, lim f (x) = ∞, lim f (x) = −∞, Estimatethespeedafter2seconds
Estimatethespeedafter5seconds
Estimatedthespeedafter6seconds
Canwedetermineifthecyclist’sspeedisconstantlyincreasing?Explain.
f (x) − 8
= 10 ,find lim f (x) x →1 x − 1
x→1
x →2
x →−2+
x →−2−
lim f (x) = 0 , lim f (x) = 0 , f (0) = 0 x →−∞
x →∞
3
18.
Several graphs of f(x) are listed below, sketch the graph of f '(x ) .
Note:Thisisjustareview;anyconceptsthatwehavecoveredinclassmayormaynotbe
ontheactualmidterm.So,Isuggestyoustudyreallywellandfullyunderstandthe
concepts.
4
AnswerKey:
1.
a.
i)3 ii)0 iii)DNE
b.
atx=0
c
Yougraphit.
2.
i)-1 ii)-2 iii)DNE
1
b)
0
iv)0 v)0
vi)0
iv)2 v)0
vi)DNE
c)3
d)1/3e)-5/54
vii)1 viii)3
p
f) 2
3.
a)
4.
5.
6.
d = min{0.29,0.22} a.
lim f (x ) = 1 bytheSandwichTheorem
b.
Yourargumentshouldstartwiththis −1 ≤ cos(1 / x 2 ) ≤ 1 g)-1 h)-1
xÆ1
−x 2 ≤ x 2 os(1 / x 2 ) ≤ x 2 ,
since lim x 2 = 0 and lim− x 2 = 0 itimpliesthat
x→0
x→0
2
lim cos(1 / x ) = 0 bySandwichTheorem.
x→0
7.
a.
b.
Let f (x ) = e- x - x andsince f (0) > 1 and f (1) < 0 ,thereforethereexistscsuchthat
f (c) = 0 a)
x–iscontinuousbecauseitisapolynomial
e sinx -iscontinuousbecausesinxiscontinuous(trig)ande x iscontinuous
(exponential)sothecompositionoftwocontinuousfunctionsisalso
continous.
Theproductoftwocontinuousfunctionsisagaincontinuousinitsdomain.
D = (−∞,∞) b)
Radicalsarecontinuous,polynomialsarecontinuous.Thequotientoftwo
continuousfunctionsisalsocontinuousinitsdomain. D = (−∞,−3) ∪ (3,∞) a=6,b=-5
a.
Youdoit.Lookatyournotesorbook.
b.
Youdoit.Lookatyournotesorbook
c.
i)½ ii)-1/3
8.
9.
10.
11.
12.
13.
Statethetheorem.
Theintervalshouldbe[0,1]
2
Verticalasymptote: x=-a
HorizontalAsymptote;y=a
5
14.
15.
16.
y=x+4
a)Answerwillvary.Onegoodanswerwouldbetocomputetheaveragespeed
between
1and2(14ft/s)andtheaveragespeedbetween2and3(18ft/s)andaveragethem
toget16ft/s.Thisisalsotheanswerobtainedbycomputingtheaveragespeed
between1and3.
b)Answerswillvary.Usingreasoningsimilartothepreviouspart,wegetestimate
of22ft/s,butitcouldbearguedthatanumbercloserto22.5wouldbemore
accurate.
c)Answerswillvary.Theaveragespeedbetweent=5andt=6is22.5ft/s
d)Iwillletyouthinkaboutthisquestion.Thisistherealessenceofcontinuity.
f ( x) − 8
Tofind lim f ( x) youhavetorelyon lim
= 10 andhereisthetrick
x →1
x →1 ( x − 1)
f ( x) − 8
lim[ f ( x) − 8] = lim
⋅ ( x − 1) whichmeans
x →1
x →1 ( x − 1)
⎛ f ( x) − 8 ⎞
= lim ⎜
⎟ ⋅ lim( x − 1) x →1
⎝ ( x − 1) ⎠ x→1
= 10 ⋅ lim( x − 1) thefirstonegoesto10bywhatisgiventousandtheothergoesto
x →1
zero.
= 10 ⋅ 0 Thismeansthat lim[ f ( x) − 8] =0
x →1
Butyouwanttofind lim f ( x) ,sowearegoingtousetheabovefact,namely,
x →1
lim[ f ( x) − 8] =0.
x →1
Hereisthesecondtrick(subtractandadd), lim f ( x) =
lim ( f ( x) − 8 + 8) = lim( f ( x) − 8) + lim8 x →1
x →1
x →1
= 0 + 8 So lim f ( x) =8.
x →1
17.Youdoit.
6
x →1
18.
7