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1 Name: ~s'~o--=-i Class; Date: U.;,..;...-h_D_Vl_5 _ MAC 2233 Chapter 4 Review for the test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the derivative of the function. g() = a. g'() = b. g'() = c. g' () == '()=-5-4 _ -7 g'()= Find the derivative of the function re) = '() a. r = I r(j) -::::.- :;;2.. --X b. r'() =7 -.~ +' 7 '( ) _ 2 O c. r d. r'() = ~ - ~ +-:!;b r'()=~-~-22.. e

2 \" Name: _ 3. Find the derivative of the function. r 39 s() = 2'\1 +T a "J.r: s'()=-+- h. s'()= ; ;- '\I -J '( ) 2 c. s =T 39 -;::r; "J "J.i:.r: d. s'() = -2- _ e. s'()=- +-- s '(y,) z:. - ~-, -~-, ~(~))( --t- 3t-i)~ X -!1. -2-)<-~ 'f'2 ~.=."2fX3 :=~r t.ji+c1=- J'fSJ 4. Find the slope of the tangent to the graph of the given function fi) = 2 3 at the point (-3, -54). [a. 1'(-3) = 54 ) J I(p) ::= ;;1(3)X 3 -I -:::.l.o X d.. h. f'(-3) = \ ~ J c. 1'(-3)=-18 JI (-3) -- ~l-3) =-(S"i- = mtat) d. 1'(-3) = 18. e. f'(-3) = 0 5. Find the slope of the tangent to the graph of the given function at the indicated point. 7 g(t) = "3' (0.5, 56) t jl-tj -71:c:.- 3 -'t S'(-t:J ==--~\t 3' Co.s) :::::~dl <t -=t-3;c,. ) {fj.sj 6. Find all the values of (if any) where the tangent line to the graph of the given equation is horizontal. l yv. :::0 ) y = a. =6.5 h. = 1.63 j =-6.5 Iji: = ) e. X=O ~' C()< -t \3 O' 2 ~ "3 =0 ;;:- \3 =-)3 ---' 8"

3 Name: Find the derivative of the function. he) = (lo + 7) a. 17 [.!O+14 J c. 14+ d. 7 e. lo dy 8. Calculate d. You need not epand your answer. ~6?. y = (lo +)( _ 2 ). a. (2O + 1)(1 - ) + ( - 2 )(lo 2 + ) h c. (2O + l)( - ) + (1-2)( ) ) - (1-2)(lO -) e. (2O + 1)(1- ) + ( - 22)(lO 2 + ) h/() -= product- I::)OXO\ r'x r-c\e :3 == 'I. -'"j. J' :=:: eo» +J B' = )-;;;""1- cieri vo:!iv-e d. -= F {) +-5 'f' dy 9. Calculate d' You need not epand your answer. -P -= (-L~+- y=(2-+3.6)(2+4) (0 -, ,.(0 3

4 Name: dy 10. Calculate d' y = 2 (2 + 3)(5 + 5) a (2 +3)(5 +5) t h. 4U~ U \ c. 65 +(2 +75)(5 +5) d e Calculate dy d (j) (i) y= (~+4)[.r;+i 1 a. Jd.l+~)+J;-W.l+4] h. }; J;+4 +( };+~l(~+4) 2 y 2 3 c. };[J;+~ + 2 2J; j(j;+4) d. f (.I + ~){f-~)(.i +4] e. 2:J; [J; +i 1+ [ 2}; - & 1( J; +4) multlp!(t J1rst -= X~(IOXd. +lo +-lsi< + '5) ~X~( IOX cl +0(5)( +15) :=JOX// + 'C15~ ~ r ~ 15(~J)<' ~ (!D:;;Z.3 t-/sx ';;l-tb0!j =- IOLY-) 4-_+_ dso}'><'3-' ~-J ii -~ 1" ~ ~ J< J ~3 -t CJ'P l-l - Yo- )( X Yz. f-l1-y:-2) t- (y:y,,- t-,+)(';';1 -r.. -'6-j) (i'j )(r t- ~ ') +- 4({X + Lt)(~~ -7)

5 Name: dy i~ 12. Calculate d' You need not epand your answer. 5+5 y= 4-l a. b. 5(4-1) + 4(5 + 5) ( (4-1) - 4(5 + 5) (4-1)2 c. 5(4-1) + 4(5 + 5) 4-1 d. 5(4-1) - 4(5 + 5) e dy 13. Calculate d. You need not epand your answer. Quoh ent R-IA \e ~ -=-5'1- t-5 :; z: ~-, r: =:5 3'-=Y 1'5-3 'F 5"2. 5(Th -I) -ifls +-5) (Yh _I) 2 y=, _--:::-=:2:.:...-- =-3 ( - 5)( - 1)( - 4) a. 2( - 5)( - 1)( - 4) + ( O)(2-3) 2 b. -::-2---= O+ 10 c. «- 5)( - 1)( - 4))2 2( - 5)( - 1)( - 4) - ( )(2-3) ( - 5)( - l)( - 4) d. 2( - 5)( - 1)( - 4) - ( )(2-3) «- 5)( - 1)( - 4))2 e. 2( - 5)( - 4) - ( ) «- 5)( - 4))2 ~ ~ d,'a - 5 ~ := l'l---5)( 'i---\)l~-1) --tr-'jr Ie pn::::1u..d- rt.{ 'e J" =.:a ~J -= (\ X - })C)(-Y-) -1- (X -5X \JL'Y~) +- lr--5x)(- J)C \")

6 Name: Compute the derivative. a. 92 b. 36 c. 59 ( d. 72 e. 18 \ ii".,.. lx 3.f- 3r-)l"A A - "') zz: X S _. X Lf- +-~)\ '3 - '3r-~ & ex 5_)< tf +3X 3-3A~) _ c=- -zt u,i 3 ::J- I - cc>x - I A.J- 'll<. -&,X \ ~ 15. Calculate the derivative of the function (y.- 3 a. A -'2-. -g.) -Y{"d) +CfC@.) -(oed) =[7d \ 1a. g'() = -3(4 + 2)( r 4 J b. g'() = (_6. L r 4 c. g'() = -3(4 + 2)( ) d. g'() = _3( r 4 e. ~'() = -12( r c~bulate the derivative of the function. \ 5-5 S()~(:~ir = (Jo'j.+-7; (SX -d.) 5 4 '( ) _ ( ) 47 a. s (5 _ 2) 2 4 b. S'(X)=_s(6.+7) (S _ 2) 2, _ ( ) 4 47 c. s ()--S (S-2) d. s'()=5 S-2 ~Cl) ::::.(Q),.Q+d)( +-3) ~ J L><-) ::= -3 L :f- ::(J-D~t-7) J?' - s-~t'l)tt [ra) = 3OC(O~t?)Lf -3 e. S'()=_5(6X+7) (S _ 2) 2 <' C'f,) 30C(P~ +7)'t (SA -~) (.co'f, +-7) ~ C OA _@.) -Co -5 (Co'f.t7) tf- 0 L\7 ~_~ l.o -d5(5x-d)-~(g~+7)5 [-IOL5><-a) ~d-+fo+3s +S(co)(+7"U

7 <..; Name: _ 17. Find the indicated derivative. The independ~tvariabl~ is a function of t. Y = 0.5(1 +); dy =? 'dt. dy = (1.5O 5) d a. dt dt b. d: = ( ) :._ r."-:!"'- -- ~ d dt c. 1't = ( )~ ;; = ( ) d 18. Find the indicated derivative. Y = & = 5 when t = 1 - dl = 11' - dyl =? ', dt ' dt. (=1 t=1 Please round the answer to the nearest hundredth. a. d: I = t=1 b. d: I = t=1 c. d: I = t=1 d. d: I = t=1..z e. dt = t=1 ~ -:= <tx "3+ -ll- 1 d~ ==- 'dlta ~I< + -II '1--2. dj. dt ot cj;t -- d-3 - dt 7

8 Name: Find the derivative of the following functiorl. fi) = In(5-9) 1 I:~J 9 c d. 5-9 e. none of these Y-C'f--) -= ~ (Sr-. -1) -F' (p ) =- \ Q d-. (5)(-4) 5~-q ~ 5 s Find the derivative of the following function: f() = log a. 4ln(7) 7 h. ~ C<: cn l 4 d. ln(7) e. none of these 21. Find the derivative of the function. f() a. h. = ( 9 + 8)ln 9(9 + 9ln) + 8 8(1 + 9ln) + 8 :F(:yJ ~ J cf' C) = 1- y...~ [FroJlAc j J=-X~+~ :F'=9~f3 cr O(~X) -\- g -== ~X I ) <J - X 7-(Xi~~) c. d. 9 (1 + ln) + 8 e. none of these 9)<.<g.~ X ~ )<.%+-~ X - ' (fj-.x+/) +0 8 X

9 Name: Find the derivative of the function. he) = In[ (-2 +2)(7 +5)] a. (-2 + 2) + (7 + 5) 7 2 b. (-2 + 2) - (7 + 5) h I LX) =- _ \... -d. + I 7 -;)'A -rd. 7I<-t-S ( \ c. (-2 + 2) (7 + 5) d (-2+2) (7+5) 1 1 e. (-2 + 2) - (7 + 5) 23. Find the derivative of the function. (5 + 3)6 I() = In --'-----'--- (4 + 2)9(& + 9) a &+9 [b. ~--th-~ ) 5 4 c. (5 + 3) 6 - (4 + 2) 9 - &+ 9 d & 8 (5+3) (4+2) +9 e. none of these / J C'I-- ) == 5 + I ~Ti 9

10 Name: -,.- _ 24. Find the derivative of the function. a. 28[ln( 6)]3 7 28[ln( 7)]3 h. 7 r' C"') = if-. C7.a..-,J 3. -.:J(7AY,) 'tl7.t...)<.] "3 2- ~. re \.oj c ; -\-e c. 28[ln( 7)]4 d. 7 e. none of these - - 1(X) =e 5 7 ln4 d e5 7ln4 +...:..:4e_ e e 5 6 ln4 +...:..:4e_ :X 10

11 Name: _ 26. Find the derivative of the function h() = e I he) -_. a lo : b. 2 e c : '::':";"'---e dẹ _ : 2 e. none of these 27. Find the derivative of the function. -lo e loe lo \ - )0 ~\A. -e. ~OX -~DA -:::::::- -:;:C=;;;;;.,.. _ JOX-_ 20-1 a. - lo2e20 2O+ 1 b e 2O+ 1 c lo e ld. -1~:4~J e. none of these 28 Find dy using implicit differentiation.. d 3 +4y= 10 4 a. -"3 b (c. :~~ ~ d. 0 e =0 +to- ':: -3 f =e~.:lox I f = -doe -~OOX ~ =-~ I J 11 e -m/< -':)0 X ~ lox I 9- -)0 -.;;0 'A 10 e _ uo) a -JOX r: -~(L..._ L?o-t ~

12 Name: _ 29. Find i using implicit differentiation. t b.~ 7+5y=y. ;=; 7 +-S~ I ~ =~, +Xc; ] 04 J - X.Lr\ I - v2}-7 c. 5-y 0 0 d. -7 ~I CS--) = ~_, 7-0 e -. y-5 I = ~ Find.dy using implicit differen~on. '5)< - d._ yln +y = lo I. -(m~+ 1)] h. - y(lny + 1) Y c. ln 1 d. - (ln + 1) Y e. (ln + 1) J -~ X -~tcr' ~'f. +- ~ X Let' ~ )( -t- 0- ~ICb.f..+-I) J +-0 I - 0

13 Name: Find: using implicit differentiation. y 2 --y =5 8 d ~ y e., (2 + 1) ~ pr~uch \ '. RCA(e - d. +-1

14 Name: _ 33. Find: using implicit differentiation. y e -ye = 10 y-1 a. -I h. e +e Y y ye +e c. y e -e ye -e Y d. ye -e Y y e _ex '@ ~ J:;::)<, 9 = e,, ~ J J= \ 3 =e ~ :J ~ I c +-e oj»: l:j~e I : 3 z: e e. 34. Find: using implicit differentiation. y a. 2+y3 ye X h. 3 Y 2e +y e c. 2e +y ye X ye X e Y

15 Name: ' Find : using implicit differentiation. In(20 + ey) = Y a. +y b. 1 y 20+ e (I-) y ye c. 20+e y Y d ~ I e. y 20+e (I-) Short Answer 36. Use the shortcut rules to calcu~~. vati-i.!)the~:"~3'. _ 4 I c.zo )=_.se""'3 f()= /(' ~~. 0, ~ a. f()=2o ~ 'X e -e,.--~ J.:c- - 3e b. f( =82.5 LA) _ v... e X~ 1 5.:..J. _---',,_J..L- _ f()=2o. -X~ X.,j _ d. f()=20' Y e -e do e. f() = Find the derivative of the function. r: 35 se) = 7", +,J; YhIA\+~ p~ 1~ I 38. Find the derivative of the function. 8 9 k() = 6 - lo 3

16 Name: Given. 40. Given. X ~-d-. ~ X 7-d- 2' Jim c.: Say whether L'Hospital's rule applies. ")..., l< -10'/<. +-CoLt,X"l-_<?, :::::: y;... /<- ")... + J L\- -\- ~If- a -r~/< o o 2 Jim -l ~8-8 2 f2-f;;\ )0 ~ Say whether L'Hospital's rule applies. It is does, use it to evaluate the given limit. If not, use some other method. QX-r-~ ~C~)-I<:O ~} == ~ (-~) T- ( Lfd C:-a.) t-d. It is does, use it to evaluate the given limit. If not, use some other method. - )0 - -~ J S(X)-= 7-.[X:+ 35-7X SX k -)'2- I fx -,k, -* s ('I) =7. ~)X k-+ 35(-~)>< L 2,L~65 Q~ d~)( - d. X%. K()() -= b;1-jox 4 (!-IC~) X 3 - X 3 5 ~ (o -lox 16

17 MAC 2233 Chapter 4 Review for the test Answer Section MULTIPLE CHOICE 1. ANS: E PTS: 1 2. ANS: E PTS: 1 3. ANS: D PTS: 1 4. ANS: A PTS: 1 5. ANS: B PTS: 1 6. ANS: D PTS: 1 7. ANS: B PTS: 1 8. ANS: C PTS: 1 9. ANS: D PTS: ANS' B PTS: ANS: C PTS: ANS: B PTS: ANS: D PTS: ANS: D PTS: ANS: A PTS: ANS: E PTS: ANS: D PTS: ANS: D PTS: ANS: B PTS: ANS: C pis 21. ANS: C PTS: ANS: C PTS: ANS; B PTS; ANS: C PTS: ANS: A PTS: ANS: B PTS: ANS: D PTS: ANS: C PTS: ANS: B PTS: ANS: A PTS: ANS: E PTS: ANS: B PTS: ANS: D PTS: ANS: B PTS: ANS: E PTS: ANS; C PTS: 1 1

18 SHORT ANSWER 37. ANS: ~ PTS~ ANS: 3O PTS: ANS: yes; 0 PTS: ANS: yes; -5 PTS: 1 2

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