大學(xué)一年級(jí)上學(xué)期-微積分試卷-06-07微積分試卷答案

PAGE

Keytoterminalexamof06-07’firsttermcalculus(I)(A)

(Remark Thealternativebyknowledgeinthefollowingtermisnotallocatedgradetoassertteachingscheduleandoutline)

1.Solution…2’

=……(2’+3’)7’

==0……(1’+2’)10’

2.Solution…1’

=……(1’+3’)5’

=……(2’+2’)9’

=……10’

3.SolutionDifferentiatingtwosidesfortheequation,itfollowsthat

.…4’

Solvingtheresultingequationforandrespectively,weattainthat

.…5’

and.…6’

Since…7’

=

,10’

and,,……11’

wehavethat.……12’

4.SolutionThederivativeofisthat.…5’

Sinceiscontinuous,furthermoreiscontinuous,itfollowsthat

and.……(2’+2’)9’

Solvingtheaboveequationssimultaneously,wecanconcludethat

.……(2’+1’)12’

SolutionTheequationofthetangenttothegraphatthepoint(1,1)

isthat.………3’

Anothertwointersectionpointsforcuresareand.…(1’+1’)5’

Hence,theareaoftheregionfoundis=.…8’

And,thevolumeofsolidgeneratedisgiveninthefollowingbytheshellmethod

=.……12’

SolutionRevenuefunction.……1’

Costfunction.……2’

Profitfunction=.……3’

Setting,……4’

wehavethatand.……6’

Let,thus,and.……8’

Sinceisuniqueextremepoint,isabsolutemaximum.10’

And,and.……11’

Hence,isabsolutemaximum,i.e.theproductionis23eachweektoattainthelargestprofit.……12’

Solution(1).Domain:.……1’

(2).Symmetry:thegraphissymmetricwithrespecttotheline.……2’

(3).Periodicity:isnotperiodic.……3’

(4).Asymptote:isahorizontalasymptoteandtwoverticalasymptotesarelinesand.……6’

(5).Firstandsecondderivatives:,

and,.……10’

(6).Thebehaviorofthegraph:isincreasingonintervaland,11’

andisdecreasingonintervaland.……12’

isconcaveupononand,……13’

andisconcaveuponon.……14’

Thegraphofhasalocalminimumatpointandnotinflectionpoint.……15’

(7).Sketchgraph:……16’

SolutionSettingauxiliaryfunction,……2’

Thus,bytheknownconditionswehavethatiscontinuousonanddifferentiableon,and.……5’

Hence,thereexistsonenumberatleastforwhichbyRolletheorem.

Thatis,.……8’

9.SolutionFalse.……2’

For,isd