Math1210,ComprehensiveReview1.Definewhatismeantbyeachofthefollowing:a) lim

x→ cf (x) = L

b) lim

x→ cf (x) = ∞

c) lim

x→∞f (x) = L

d) lim

x→∞f (x) = ∞

2.Findeachofthefollowinglimits.Showallwork.

a) limx→ 3

x2 + x −12x − 3

b) limx→ 4

2 − x4 − x

c) limθ→ 0

sin 3θ( )θ

d) limx→1+

x − 1x −1

e) limx→∞

3x2 − 7x2 + 4

f) limx→ π

2

+

sin xcos x

g) limx→∞

5x + 9x2 +1

h) limx→1−

x + 3x2 − 1

3.Definewhatitmeansforafunctionftobecontinuousat x = a .4.Whatcanyousayaboutthefunctionfwhosegraphisgivenbelow?

5.a)Definecarefully,intermsofalimit,thederivativeofafunctionfatthepoint x = x0 .Itisdenotedby f '(x0 ) .b)Whatarethetwoimportantinterpretationsof f '(x0 ) ?c)Let y = f (x) = x2 +1 .Usethedefinitionofthederivativetofind f '(1) .

6.Statealloftherulesfordifferentiation.7.Findeachofthefollowingderivatives.Showallwork.

a) If f (x) = (3x2 + x −1)4 , find f '(x) .

b) If y =1− x( )3

x2 + 4( )5, find y ' .

c) If w = ln (1+ t 2 ) , find w" .

d) If u = x e−3x , find d 2u

dx2.

e) If r = sin3 2θ( ) , find drdθ

.

f ) If r = sin−1 3θ( ) , find dr

dθ.

g) If w = x tan−1(7x) , find w" .

8.Considerthecurvedefinedby x2 + y2( )2 = 16xy .Findtheequationofthelinetangenttothecurveatthepoint ( 2 ,2 ) .

9.Let f (x) = x3 + 7 .Findthederivativeof f −1 at x = 8 . 10.Let y = ( sin x )x .Find y ' .11.Twocommercialplanesareflyingatanaltitudeof40,000feetalongstraight-linecoursesthatintersectatrightangles.PlaneAisapproachingtheintersectionpointataspeedof442knots.PlaneBisapproachingtheintersectionat481knots.AtwhatrateisthedistancebetweentheplaneschangingwhenplaneAis5nauticalmilesfromtheintersectionpointandplaneBis12nauticalmilesfromtheintersectionpoint?

12.Findtheabsolutemaximumandminimumofthefunctiondefinedby f (x) = 4 − x3 on [−2 ,1 ] .13.Let f (x) = 2x3 − 6x +1.Identifytheintervalsonwhichfisincreasingordecreasing.Findthelocalandabsoluteextremevalues.Determinewherefisconcaveupandwhereitisconcavedown.Findallpointsofinflection.

14.Evaluatethefollowing.

a) limt→ 0

1− cos tt sin t

b) limx→∞

x3x

c) lim

x→ 0+x2 ln x

d) limx→ 0+

x ln x

15.Aposteristocontain150squareinchesofprintedmatter,surroundedbymarginsthatare3incheswideontopandbottom,and2inchesoneachside.Findthedimensionsfortheposterthatminimizesitstotalarea.

16.Supposefisacontinuousfunctionon[ a , b ] .Definecarefully(intermsofa

limit)thedefiniteintegraloffover[ a , b ] .Itisdenotedby f (x) dxa

b

∫ .

17.Express limn→∞

1− k2

n2⎛⎝⎜

⎞⎠⎟1nk = 1

n

∑ asadefiniteintegral.

18.Evaluatethefollowingintegrals.a) (1+ ex ) ex

0

1

∫ dx

b) sin x cos3 x dx∫

c) 2x x − 5 dx∫

d) ln tt∫ dt

e) xx2 + 40

1

∫ dx

f) t (t2 +1)1/3dt0

7∫

g)0

1/3∫ 6

1+ 9x2 dx

h)0

1∫ 6

1− x2dx

i) tan20

π /4∫ θ dθ

19.Findtheareaoftheregionenclosedbythecurves y = 7− 2x2 andy = x2 + 4 . 20.Thebaseofasolidistheregioninsidethecircle x2 + y2 = 4 .Everycrosssectionbyaplaneperpendiculartothex-axisisasquare.Findthevolumeofthesolid.

21.Theregionboundedby y = 1x,thex-axis,andtheline x = 1 ,andtheline

x = 4 isrevolvedaboutthey-axis.Findthevolumeoftheresultingsolid.

22.Determinethelengthofthecurvedefinedby y = x2

8− ln x , 1≤ x ≤ e .