}viAC 2311 Exam #3

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

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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 / 10 2 / 10 3 /15 4 /10 5 /15 6 /15 7 / 10 8 /10 9 /15

Tot al / 100

1. Show t hat the equation 1 + 2x + X 3 + 4x5 - 0 h root. - as exactly one

+(Q) -= 1')0 } .fC'IC) -If> ~I\V(NS ()t\ ("' I,D1 St> -H..t L.iY~~ Vb ~-H_I] :: - .-I f, 0 s-J. .Jl.r .f(,)4) .

ihluw< ~ 6I"t d- "",\s . cJ1 ~ 0. t \' . 1).,." +(.)~o g fib)={)

Sil\(f f is .. 14~-,J) I+'~ CU~u""s l. rJ.;f.(u-~A So ~g"5 ~ Wl,> us tl.vt G( pcn"'~ C. bUwi~ a. Ix '-> ~ .f'eL) 1).IS

/h" ;~ i"1"'ssi iJJ 'f!,..,;e 'S eJx~ ~ coo f-.

2. Find the intervals of increase and decrease. Then identify the function 's local extreme values, if any, saying where t hey are t aken.

+I(~\:: q~~ -12'('2.t~x

f'(X)-::O ~ ~'(3 -Il~ 2-~~ =:0

~x ('l--3X tz}:: D

~)( (x-z) (X-I) ~o

1L +

o

T ', X~-\ ·. (-){- )[-)-=--

1r : ~-= \ '. (+)(...)(-)::+

]I : ~ :; ~ ~ (~)(- ) ( + ) -=

W: ~ -:; 3', (t) (+-)( +) -:: ~

+'(0) : 0

+(\) -:: 1-\{~4 ~ t

~('l~:: 110 - 31~ I('-=-o

-

. . ' (O,t)U(~,oo)tV'ltce.(;!)I J . ruuu.Sij :(-flo, D) V( I,.?) rocJ ~ '. (1 / 1) lacJ M'~ : (0)0) I [~I 0)

3. Sketch the graph of the function. Note: There is more room on the next page, including a blank graph.

y = 4x3 - X4

li ) 1)b~Y"\ '. (- DO I Od) -GIno. r 1c;;. c... f07N-Mi,JJ (i:) X"' l t1 4erCtf ~s '. 0 -= lh? - X'( '::" X~ ( 4-x) :;> X~0I 4

(O}o\ (~,O)

~ -\tl-wc.ert', y-= 0 - 0-=-0 (0 1 0)

(i:i) .} (-X):: ~(-)(y~ - (-x\'i ~ - ~~ l +X'f

n-tik Q.I'fJr\ "'IX ~

(,\}) ViftlcJ. CVI"1-tb~~~' ~ _ . () \ _ \ i f""l Y'> (l\'->-)-:(~)(-Of)_-Co

\ ,it\ 'h - x'i ) - ~ -~Oo f\ X-')()O

tv) f'{x\= 12/"- ~X3 -= ~i{~-x)

t{{x\-::O -=> Ql(3-x ) -:>0 ~ ~=O( ~

"\IIU-fAbi'j ', (-

'0/ t.i I~ z.(,

II lit 1\ 1.'L

in fit( )}Jh / 10 1 0'1\t /i"- r-.. III "-1 V

II.

('1

1 III , IDI

I C.

QV\\1/\ {.1" CkI ...... V/

~ ~ Oi il t ~ /' 1

~ .,.V ..It'!' -It!/ -

n-l, rrw¥......L i'- f- ~. \j f- iv tG J-~, III ~

- t'I -I'

o

-r - '-{ - J - ~ -, 0 , 2- ~ \of ~ Note: Use a scale on the x-axis of ~ and a scale on the y-axis

of 2,

(Vi] I~ ~ : (3 , ;r~)

l~ r\-.j~', ~

(vii) ~ I\bc) = ;).~)(- (lxZ. -= 1;2'({;l - >t)

.f"(x)-:::Q -==> 1;),)( (J--X) :: 0 =i> X-=OIJ.

:n:

t

<

CJr.!I(fJ,1/e. up :(OJ;).)I ~ x ~ -I : l- )(") =

]' : X-=- I : (+)(.}) -= + CMUtV"f d.cv.;~ "", (-~ I D) U~, po)

.m:. ', x -:... ~ ', (-4-)(-)~

4. A 216m2 rectangular pea patch is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for the outer rectangle will require t he smallest total length of fence? How much fence will be needed?

_ 21(0 _ Lo! I I ~ -:. Jib - 2xy =} 7-= 2X - x

f -= 4X./

5. Compute the limit.

(a) lim eh - (1 + h)

h2

e '" - \ o h-+O

ill----- e~ o J. - ~

(b) lim ( x - 1r) sec x

x-+(~r 2

-- -11x- 2.

(c)

lim x1/(x - l)

x->l+

-,X ~-I

_ ti~ I... )C X4''" 'X -I

6. (a) F ind the most general antiderivative of t he funct ion.

f (x) = X -4 + 2x + 3

F (x)-; :~ t-:2(§:)f-3x+C

(b) Compute the indefinite integral.

J(4 sec x t an x - 2 sec2 x) dx yJS~L X~x~ - ;2) sec.:l.~£4

8set X - ~ ~ X +L]

(c) Compute the indefinite integral.

t VIr fi dt

.ltl.

7. Find y if d2ydx 2 = 2 - 6x; y'(O) = 4, y(O) = 1

2- -=- .,{)(- (, (~) +C l' -=-1"1. - 3x~+C

,'(0) -: ,:;(0) - 3(0)'\.+C::Y )c ::. ~

t "2'1 -= J.y. - '3 ~ t--~

1 ~.,1( ~) - ~{%)-I-~ X f C

/ =- /" _ '('J +~x+C

y ( b) = r:, 'L - 0' +'f ({») f-C :: I => c. =1

U=-=. { - x' t-'h ... 1J

1 .!. s ~ -L \4

8. Estimate the area under the graph of the function

f( x ) = X 2

on [0, 1] with n = 4, using

(a) Left end points

A-" ~ t'iOF ~ ,c{~)f ~ fm ~ ~ .j'(~) :: ~(o)-t ~(-h;)~-H~)+ -4(~J

I -L ~ ~~ .... D.} (,'t -t H. -+ G,~ I +'1 +~- - (~ ~ - Co\.( - ~"::~

(b) Right end points

A- " ~+(~)~ ~-I'{~)+ ~+(~)} ~.r{r) - ~O')f ~(~)~ -!d-~)f (tl) - I JL q I

I ..! J ~ + lro + ~ +- lj ~ 1 '1

-

9. Compute the definite integral of the function on t he given interval using R iemann sums.

f(x) = x + X 2 , [0,1] 1- 0 I

11 X -:. h::- Y\

'Af =0+ (~)G =

• . 't.

..... _t._+ .L n ha.