Calculus Answers

 

(1) y = 3x 2 - 2x + *. Evaluate Evaluate y' for x = 2. y' = 6x - 2 for x =2 f'(2) = 10 (2) y = -x3 + 2x2 - 3x + 2. 2 . Evaluate Evaluate y' for x = 3. y' = -3x2 + 4x - 3 For x = 3, y' = -18 (3) y = (x - 2)(x - 3). Evaluate Evaluate y' for x = 2. %a F, y = x 2 - 3x -2x + 6 y = x2 - 5x + 6 y' = 2x - 5 For x = 2, y' = -1 (4) y = 5 + 2 . Evaluate y' for x = 4. /%e y = 5 + 2 %# a or%otal l%e, te value of %t# &er%vav &er%vave e at ay $o%t x %# alay# 0.

(5) /u$$o#e tat  orer# orer# $ro&ue a $ro&ut a& tat te uer of u%t# $ro&ue& %# %ve y f() =100.5. 7e $r%%$le# of eoo%# ol& tat %rea#% u%t# of a re#oure y%el&# &%%%#% retur#. %%%#% %%%#% retur# ea# tat e ea a a&&%oal u%t of te re#oure %# le## $ro&uve. Ex$la% o t%# "$ro&uo fuo" %# o#%#tett %t &%%%#% retur#. o#%#te 9# te ra$ of f() #o#, te #lo$e of t%# $ro&uo fuo %# %er for loer value# of x: % oter or&#, te loer te x value, te reater %$at a ae % x (uer of orer#) %ll ave o te orre#$o&% y value ($ro&uo). 10000 000 8000 *000 6000      n      o        t      c      u        d      o      r        P

5000 4000 3000 2000 1000 0

f'(L) = 50L^-.5

f"()=-25!-1.50 #o 

f() 100 1100 1100 2100 2100 3100 3100 4100 4100 5100 5100 6100 *100 8100 100 100

1000 3316 3316.6 .625 25 4582 4582.5 .5*6 *6 556* 556*.* .*64 64 6403 6403.1 .124 24 *141 *141.4 .428 28 *810.25 8426.15 000 53 53.3 .32 2

 

0

1 00 0 20 0 0 300 0 4 00 0 5 00 0 60 00 *0 00 8 00 0  00 0 10 00 0 Workers

 

 

#lo$e %# &erea#%

 

(1) # f(x) =1;x 2 a ovex or a oave fuo of x, for all value# of x < 0 f(x) = x-2 f'(x) = -2x-3 f''(x) =6 x-4

7e #eo& &er%vave of t%# fuo, f''(x) =6 x -4, %# $o#%ve for all o-ero value# of x. 7u#, f(x) = x -2 %# ovex for all o-ero value# of x.

 

(2) /o tat f(x) = x 3 %# e%ter a ovex or a oave fuo. f"(x) = 6x for x>0 t%# %# >0 a& for x0 t%# %# 0, #o f(x) %# e%ter ovex or oave.

 

(3) /o tat f(x) = -2x 2 - 3x %# a oave fuo of x. f'(x) = -4x - 3, f"(x)=-40 #o f(x) %# a oave fuo.

 

(4) /u$$o#e tat %t o#t# te autoo%le aufaturer @000 to $ro&ue a yr%& a& tat te &ea& for yr%&# (% tou#a&#) for a $r%e of $ tou#a& &ollar# %# ex$re##e& y f($) = 40 - 3$. Aat $r%e for yr%&# %ll ax%%e ax%%e te o$ay'# $ro?t Bo oul& te a#er ae %f f($) = 500$ -2 Part 1

Cart 1D f $r%e %# $ tou#a& &ollar# a& &ea& %# 40 - 3$, te reveue %# $(40 - 3$) a& o#t %# (40 - 3$) (e %ll a##ue tat e a aufature to &ea&). Cro?t %# te $(40 - 3$) - (40 - 3$).

$r%e $ (% tou#a&#)

Cro?t u&er te#e a##u$o# %# tu# 40$ - 3$ 2 - 360 + 2*$, or -3$ 2 + 6*$ - 360. 7e ?r#t &er%vave of t%# $ro?t fuo %# -6$ + 6*.

 .1

7e #eo& &er%vave %# -6. /%e te #eo& &er%vave  0 for all value# of $, te $ro?t fuo %# oave, a& %t# ax%u value a e fou& y #e %t# ?r#t &er%vave to 0.

.2

7a% -6$ + 6* = 0, e ?& tat $ %# rouly 11.16*, or tat a $r%e of @11,16* &ollar# %ll ax%%e $ro?t#.

Cart 2D f &ea& f($) = 500$ -2, te $ro?t %# $(500$ -2) - (500$-2), or 500$-1 - 4500$-2. 7e ?r#t &er%vave of t%# $ro?t fuo %# -500$ -2 + 000$-3. 7%# fuo eual# 0 ere x = 18. 7%# %# te oly $o%t ere te ?r#t &er%vav &er%vave e eual# 0, a& -3 -4 #%e te #eo& &er%vave (1000$  - 2*000$ ) %# eave ere, e o tat $ = @18,000 y%el&# y%el&# ax%u $ro?t..

.3 .4 .5

.6 .* .8 . 10 10.1 10.2 10.3 10.4 10.5 10.6 10.* 10.8 10. 11 11.1 11.167

11.2 11.3 11.4 11.5

 

11.6 11.* 11.8 11. 12 12.1 12.2 12.3 12.4 12.5 12.6 12.* 12.8 12. 13 13.1 13.2 13.3 13.4 13.5 13.6 13.* 13.8 13. 14 14.1 14.2 14.3 14.4 14.5 14.6 14.* 14.8 14. 15 15.1 15.2 15.3 15.4 15.5 15.6 15.* 15.8 15. 16

 

16.1

 

Part 2

$ro?t, ere &ea& f($) = 40 - 3$

$ro?t, ere &ea& f($) = 500$ -2

$r%e $ (% tou#a&#) 0 1.2*

 10

0 5

2.48

11

8.26446280

3.63 4.*2

12 13

10.416666666* 11.834315266

5.*5

14

12.*551020408

6.*2 *.63 8.48 .2* 10 10.6* 11.28

15 16 1*

13.3333333333 13.6*18*5 13.840830448

18

1.888888888!

1 20 21

13.8504155125 13.*5 13.6054421*6

22 23 24 25 26 2* 28 2 30 31 32 33 34 35

13.42*520661 13.2325141*** 13.0208333333 12.8 12.5*3644* 12.3456*0123 12.11*346388 11.80606420 11.666666666* 11.4464086 11.230468*5 11.01283*466 10.813148*88 10.61224488

11.83 12.32 12.*5 13.12 13.43 13.68 13.8* 14 14.0* 1".08

14.08000 14.03 13.2 13.*5

 

13.52 13.23 12.88 12.4* 12 11.4* 10.88 10.23 .52 8.*5 *.2 *.03 6.08 5.0* 4 2.8* 1.68 0.43 -0.88

36 3* 38 3 40 41 42 43 44 45 46 4* 48 4 50 51 52 53 54

10.416666666* 10.226442658 10.0415512465 .8613238 .68*5 .51814361 .353*41466 .1415004 .03256183 8.888888888 8.*42111531 8.6011**0032 8.463541666* 8.3286255*3 8.2 8.0*381**624 *.51183432 *.831686*21 *.*1604382*

-2.25 -3.68 -5.1* -6.*2 -8.33 -10 -11.*3 -13.52 -15.3* -1*.28 -1.25 -21.28

55 56 5* 58 5 60 61 62 63 64 65 66

*.603305*851 *.4362244 *.3868882*33 *.28264328 *.1818442*6 *.0833333333 6.8*368868 6.83860561 6.802*210884 6.*1386*18*5 6.62*21834 6.5426*245

-23.3* -25.52 -2*.*3 -30 -32.33 -34.*2 -3*.1* -3.68 -42.25 -44.88 -4*.5* -50.32 -53.13 -56

6* 68 6 *0 *1 *2 *3 *4 *5 *6 ** *8 * 80

6.4602361328 6.3**5**855 6.3011*22*5 6.22448*5 6.145*34*3 6.0*6388888 6.0048*8642 5.3480431 5.866666666* 5.*861458 5.*34525215 5.6*0611438 5.6080*5628 5.5468*5

 

-58.3

81

5.4866844

 

7e ?r#t &er%vave %# -500$ -2 + 000$ -3

7e #eo& &er%vave %# 1000$-3 - 2*000$-4.

6.1*2835062 4

-2.*43484225 -1.*

2.626018032

-1.028215286

1.*361111111 1.13*15331

-0.*233*626 -0.401*8153

0.*28862*38

-0.3384006664

0.4444444444 0.244140625 0.101**08121

-0.23*03*03* -0.16*8466** -0.11*3036*2

0

-0.0857882

-0.0*28623* -0.125 -0.161654*

-0.0613868832 -0.043*5 -0.0308513428

-0.18*828*002 -0.2054*38226 -0.21*013888 -0.224 -0.22*58306*8 -0.2286236854 -0.22**66*3 -0.2255115011 -0.2222222222 -0.218186*00* -0.213623046 -0.208685558 -0.2035416243 -0.18250*28

-0.0213441*05 -0.014238311 -0.000422454 -0.00512 -0.00218828* 0 0.001626263 0.00282**304 0.003*03*03* 0.004331246 0.004*683*16 0.00505358 0.0052382036 0.005331112

 

-0.12012346 -0.18*550583 -0.182242304 -0.1**00052* -0.1*18*5 -0.16685**066 -0.161654* -0.15*2188612 -0.152610818 -0.1481481481 -0.1438316*58 -0.13660*688 -0.1356336806 -0.131*4*8262 -0.128 -0.1243865482 -0.120035048 -0.11*54666 -0.114311842*

0.00535836*6 0.005335*20 0.0052*54353 0.00518*0*85 0.0050*8125 0.0045441*6 0.0048205223 0.0046800033 0.0045356362 0.004385*48 0.0042434811 0.004086286 0.00355823 0.0038162636 0.00368 0.00354*5664 0.003412168 0.00325106 0.0031*532

-0.11114505 -0.108105** -0.105255565 -0.10250522*8 -0.08154631 -0.0*2222222 -0.04*215846 -0.0230*58 -0.08830832 -0.08**3803*1 -0.0855*12335 -0.0834*4223

0.00305003 0.00248803 0.002841853 0.002*33638 0.00264083 0.002546263 0.002455611 0.0023686521 0.002285284* 0.0022053*1 0.002128*** 0.0020553646

-0.08145488 -0.0*50844* -0.0**6234441 -0.0*5801*43 -0.0*4040*531 -0.0*233*63 -0.0*06081 -0.060*5855 -0.06*5555556 -0.06606283*1 -0.06461*453* -0.06321*5188 -0.0618612321 -0.0605468*5

0.001850012 0.0011*556* 0.00185204 0.001*0204 0.001*3148*2 0.0016*448 0.001618186 0.00156*368 0.00151*03* 0.001468*21 0.0014223521 0.0013**81** 0.0013350422 0.00123453

 

-0.052*280*3

0.001254450

 

(5) et G = te o#t of $la% a or&er for &%%tal aera#,  = te aual &ea& for &%%tal aera#,  = te o#t of ol&% a &%%tal aera % %vetory for a year, year, a&  = te o#t $er aera or&ere&. /allto'# eletro%# #tore #tore o# tat te aual o#t for #to% aera# %ll e (G;) +  +.5. /o tat a or&er uaty (alle& te eoo% or&er uaty or EI) of H(2G;) %%%e# te total aual o#t of #to% aera#. f() = G-1 +  +.5 f'() = -G-2 + .5 /e f'() eual to 0, e ?& -G-2 + .5 = 0 .5 = G -2 .52 = G 2 = 2G;  = H(2G;) f''() = 2G-3 /%e f''() %# $o#%ve for all $o#%ve or&er uae#, e o tat te EI ota%e& y #e f'() to 0 %# a %%u.

 

(6) /u$$o#e tat you ave 100 yar&# of fe% a& at to u%l& a retaular fee tat elo#e# te ax%u $o##%le area. Aat #oul& te &%e#%o# of te fee e B%tD f  = te let of te retale a& A = %t# %&t, 2 + 2A = 100. F%r#t #olve for  % ter# of A: te oo#e  to ax%%e te area of te fee. Jo#tra%tt o &%e#%o# Jo#tra% 2 + 2A = 100 2 = 100 - 2A  = 50 - A Ae at to ax%%e A #uKet to 2+2A = 100 or +A=50. /o e o A = 50 - a& e at to ax%%e f()= (50-) = 50 -!2 f'()= 50-2 %# 0 for  = 25 f"() =-2, #o  =25 a& A=25 %# a ax%u.

 

(*) /u$$o#e tat you at u%l& a retaular fee tat elo#e# 2500 #uare yar&#. Aat %# te %%u aout of fe% ee&e& Jo#tra%tt o area Jo#tra% A = 2500  = 2500;A Fuo to %%%eD 9out of fe% reu%re&=2+2A /u#tu 2500;A for D 2(2500;A) + 2A f(A) = 5000A-1 + 2A KeveD L%%%e 5000A -1 + 2A F%r#t &er%vaveD -5000A-2 + 2

/%e oly $o#%ve value# of A are relevat to te $role, a& #%e te #eo& &er%vaveD 10000A-3 %# > 0 for all $o#%ve value# of A, A, te $oro of te fuo tat e are oere& %t %# ovex, a& te %%u value tat e are oere& %t a e fou& ere f'(A) %# ero, $rov%&e& tat A at t%# $o%t %# $o#%ve.

f'(A)tat =0A at A = -50 A = 50. /%e e o tattof(A) %# ovex $o#%ve $o#%v e value# of A, A, e ?& = 50 %lla& %%%e te fe% reu%re& elo#e 2500 for #uare yar&#. Fro te o#tra%t o area elo#e&, e o tat  = 2500;A, or 50. A 5 10 15 20 25 30 35 40 45 50

 



area elo#e& fe% reu%re& 500 2500 1010.00 250 2500 520.00 166.666* 2500 363.33 125 2500 20.00 100 2500 250.00 83.33333 2500 226.6* *1.4285* 2500 212.86 62.5 2500 205.00 55.55556 2500 201.11 50

2500

200.00

 

(1) /u$$o#e tat te o#t of $ro&u% x u&re&# of ael# %# (x) = 150 + 5x 3 - 12x2 + 50x. 9t at level of $ro&uo &oe# te ael o#t urve ave a %Meo $o%t oe# te o#t fuo ae fro ovex to oave or fro ovex to oave (x) = 150 + 5x3 - 12x2 + 50x '(x) = 15x2 - 24x + 50 ''(x) = 30x - 24 /e ''(x) to 0, e ?& tat a %Meo $o%t our# at x = .8 Fro te tale of x a& ''(x) value# to te r%t, e a #ee tat for value#  .8, (x) %# oave, a& tat for value# > .8, (x) %# ovex.  oter or&#, a# x %rea#e#, te t e o#t fuo ae# fro oave to ovex.

 

x

''(x)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

-18 -12 -6 0 6 12 18 24 30 36 42