MAC 2312 Exam #4b

Name: --,tth~.~_wif.s.=-_K~,---,--+-________

~"'lr ~Ol\ID# _-'---__---"-'-______

HONOR CODE: On my honor, I have neither given nor received any aid on this examination.

Signature: ________ __________

Instructions: Do all scratch work on the test itself. Make sure your final answers are clearly labelled. Be sure to simplify all answers whenever possible. SHOW ALL WORK ON THIS EXAM IN ORDER TO RECEIVE FULL CREDIT!'!

No. Score

1 /8 2 /10 3 /8 4 /8 5 /10 6 /8 7 /8 8 /8 9 /8

Bonus / 10 I Total I / 100 I

(1) Solve the differential equation. (8 points - 4 points for the answer and 4 points for the steps)

dy - = 2xy2dx

(a) y = x2 + C (b) y = x2~C

(d) y = (x + C)2 (e) None of the above

<

-- -=­

..,

\

(2) Solve the initial value proble L(10 points) m. eave you answer in the explicit form y = f(x).

~""'j -I.,.. \1" st=I ~ 'j1-_\-::;~"l.e

\ <;tee J ~,-=~(,~9de

j /I~ =Jt..~ . ~\}~de

~ ~ ~d\l :0 r'Ott-9 ' CbWd9

-:: Ji:~ ~~"JLsd}~Q

~-\W\\t~et~\

-::. __ \" \ ~ ""t .-L\\If"'\ ~~2"-l

dy y2 - 1 dx = x2 _ l' y(2) = 2

:: -\~ \s~~) ~ I~ J~J

Jfi~=J-Ft-M ~ 5S-\ 0 \tC= lp\ 1)+'1 X~II

~(,):: 2 ~ /" I-~ ):I" I£tT. ~c ~ \" l ):: I" [ i)rC

_I~ ( ) :1" ( ~)

(3) Determine the integrating factor for the differential equation. (8 points - 4 points for the answer and 4 points for the steps)

x 2 dy +.,;x2 + ly = eX dx

(a) f..L(x) = - Vx~+l + ";x2 + 1 + x (b) f..L(x) = (";x2 + 1 + x) e - v'x2+l

@ f..L(x) = (";x2 + 1 + x) e- v'x:+l (d) f..L(x) = e- v'x:+l+v'X2+l+x

(e) None of the above

-;) (:.~: t<ift e)~e i

u.=Cfll\9 cW.~toSeJf) t \ J - _ 1. - - ~ :: - C.Sc. 9J~ CA."'- - U. - sma

'" _ esc tl .\-l~ \~lIj ·H·- e \ - _ rM_ -I- \., \ ~~ll t~ \ - x

- Eil +l~\~ ~Z.tl +)c \)A tx):: e x

-Hi ~lJti nc\ -=- e ~ (t"

~ : (J X~H +k)e- ,.

(4) Solve the differenti al equation. for the steps)

(a) y = 1(In x) 3 + Cx

(b) y = x(lnx )3 + Cx

(c) y = 1x3 + G.x

@Y = 1x (ln x)3 + Cx

(e) None of the above

~ - ~ ~ (l~)c)~ ~ X ~

?(~

_ ~ ~~ ~ -\.Jy.\ -0: \" \'1.-' \

1-< \ X-'\ \ ~ (¥.) =~ ~ X

J.. h- - -+~ ~ ~(l~~YX rJ/K. X J

\A.~ \... ~

~U~ ~~

J\),1- d\A ~ -.k \).:~ tC

~ ~ (\J\ ") ~C

(8 points - 4 points for the answer and 4 points

dy Y 2 - - - = (ln x)dx x

-k J ~ ·HI" St C

'j ~ -tX(t" >.Y tLl<

(5) Solve the differential equation. (10 points)

dy 6 X x - - 4y = x edx

1--t'J ~~'51 \....--or-'"

ff,,)

_4 ~ ~ ~~ -'\ \" \-"\ :=\" l~-'lJ

r") -= 'it-~,e-.\.. ("...) "

- 'i ~ \ \ -s XX -a;.. - "\ ~ j -:::- xe.

(, -~ \' )(

\~ 'j J -=- ~e

J('it-~ ~fcAk " ~ 'f.l~ JJt­\A':.- '( ~ au.-=- clrK

KJ.... - )(.cll\T" ~ t IN" -=t ~ - t

)'/.(:~-:. xl-1z'~ J x f'

-:. Xt - t, o\'v

---- - -

(6) Find the length of the curve on the given interval. (8 points - 4 points for the answer and 4 points for the steps)

Y = ~x3 + ~ 1 < x < 36 2x' - ­

(a) ~ (b) 13 (c) ;8 (d) ~ Q None of the4 ~above

_(13:. _-L)_ (.1 - j.\- <- Gr b i)

r~ -....

, 3\2'")- -. - - +- .. - - fo ~(., (,

(7) (i) Find an equation for the tangent line to the curve at the point defined by the given value of t. (8 points - 4 points for the answer and 4 points for the steps)

X = 9t2 - 6, y = t 5 , t = 1

(b) y = 5 X - ! (c) y=~x+l. 18 6

(e) None of the above

CK

A+t ~I : ~~

At t ~ \ X-= 1(\)'t - t,~ 1-~ -::::,0 .

At ~ ~ I ·. J :: \S- -=. \

(ii) Find the slope of the polar curve at the indicated point. (8 points - 4 points for the answer and 4 points for the steps)

8i

r = 2 + 8 sin (), e= 0

(a) 4 (c)-4 (d) -I (e) None of the abovei <;:;\11\1 Hu>s1) _ "u>~GSlAe + (H'l..",e) r...9~­rJi.­ &r- <i ~-z. e - (~+-t~"" e)Sln(j

~u>~\j _("&\-v\ B \ DD t

~ ~~ + (2+f~) .c.ert) .... _ I-A+ e~O'. ~ ~ ~7. &-_ (tf-g~)~

00 1

--- -

(8) Let the curve C be defined by the following parametric equations.

t 3 t2x = - 12t, Y = - 1, -5 < t < 5

(i) Find all points where the curve has a vertical or horizontal tangent line. (8 points - 4 points for the answer and 4 points for the steps)

@)vertical : (-16,3) & (16,3); horizontal: (0 , -1)

(b) vertical: (3,-16) & (3,16); horizontal: (-1 , 0)

(c) vertical: (0, -1 ); horizontal: (3, -16) & (3,16)

(d) vertical: (-1,0); horizontal: (-16 , 3) & (16,3)

(e) None of the above

~ ~ ­-ax - ,}.x

d~

Y\-4- t--:; - ;}'. 'X-= (_~)1. _ \~(_~) -=- -~~2\.{ :: llo

~ = (-;i)l. - \=- \.( - \-= 3

ft+ t- ~ 2', X =- ;/' -- (J(:J) =::- ct, -J,-{ =: - Ito

1 -=- ;;':J. - \ ~ l( - t ::- 3

~ V~cJ " Cllo :~) I (- l(o,3)

\ - ,e' ~t~V\fW\"tMW ·

+:-::..0 itt t -=-O '. x~0'- IJ..(O) ~ () - 0 :="0

f =. D'Z. - l ~ 0 - l ~ - 1

~~oi\ ~{-J: CO, -I)

(ii) Determine the t ~ntervals on which the curve is concave upward or concave downward. (8 P01l1ts - 4 points for the answer and 4 points for the steps)

~ up: (-5, -2) U (5 , 2); down: (-2, 2)

(0Y up: (-2,2); down: (-5, -2) U (2,5)

(c) up: none; down: (-5,5)

(d) up: (-5 , 5) ; down: none

(e) None of the above

~_ rNr.~ -

~ ~ (~\ _ ~) -

_~(rt)rNc

_=

~

_ 1.

- fb (t~ -r4) ~&'}~ ?,~ .~(\.~~'f , - '1((,;~lY("·l~

tl::n)L\o-:i\ ~o ~ {;; = -~,~( lWs i~ ~ 9" \l>JLt...(j oI...J....r< " ~~) 1: 11:- "Ttr

C I \ f'?~ ,~, ~ -s-r-I1 -). ~ S­

1'-~ riUc l<=- ~ ' (j\J;\=tj . ( ) ~} ­

'It ', Vi<l- X-:: o· ~ (f'-1Jll0 -res

N : '11\t.l Y:"'3', t" ) ('~( fOS) " "'j

CMtlv-t. v{- (-d. , l \

~~~ ~'. (-'5")-d-lV [:lIS)

(9) Find the length of the curve. (8 points each - 4 points for the an::;wer and 4 points for the steps)

(i) x = ~t2, Y = /2 (8t + 16)~, 0 ::; t :S 1

(a) 5 (b) ~ (c) ~ (d) 9 (e) None of the above

L~ ~ I ((t-l"t)J cIt '"JI (U'I\utt o ~

-;.. {t'-t~tt :: ~ (I)~t~ll~- lHo)' n(o)1

(ii) r = 5B 2 , 0 :S B :S 2V3

(a) 807[5 ®2~O (c) ~o ( (7[2 + 1 )~ - 1)

(d) 720V3 (e) None of the above

~r ~{ de :o J(~ ~J~) r ,/1. '1

:: ~ (1 (.,) - ~ ('1)1(1­

:: ~ (~y - ~(~y

-= ~ ('\) _ ~ (2)1

-=- ~(~~) .- ~ (~)

_ W .- ~ - ~ 3

2.1'.Q. -;::.. '0

--------------------------

--

B . F" I d3

7J f .onus. m e dx3 or the curve gIven by the following parametric equations. Leave your answer m terms of the parameter t. (10 points)

x = 3t2 - 3, Y = cos t

----­

it( -:;) (g-t; (-Co~i;) - (-~lht)(fo) C{Dt) 2..

..

~ - tc.osi:; +C;'ih~

\ 3(,t

.~~(-to~t tt.Si'w\'l +to!)~) - (- t;COlt ~~~) (rote)

~~ ~,rl\ t 10ft] Q)~t -IDtt'1 SjY\~ _ ~ ({\lnt t ~b-lc&t - ~<;'\~t.) ~'l. . (o-t1 1Vf -L, ttf~

e· ~in(: -t~t~t. - 3S'i'1t

JllctS

1