*This preview shows
pages
1–16. Sign up to
view the full content.*

This ** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This ** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This ** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This ** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This ** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This ** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This ** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This ** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **Mathematics 115 — F2011 Midterm Examination 1 —‘6 October 2011 o Read the instructions to each question carefully — be sure to fully answer the question that is
asked. 0 Show all work and calculations. Provide all necessary reasoning. Use units where they are given. 0 You may use a calculator for arithmetic calculations. You may not use a calculator for any other
purpose. If improper calculator use is evident in an answer, there will be no credit for that answer.
Use 2 digits to the right of the decimal place when using decimal representation. Total points: / 100 ,m'“ .r i
i
L {ii 1 (1) Let me) = 1
m—l' ﬁfla) Use the deﬁnition of the derivative to ﬁnd 1” “it”; trewédwﬂ 2 «If? Let f be the function with this graph:
M £7 (a) For What values of ac is f not continuous? [Give a reason for each value] / Lg m3; Eavmgtiiénkﬁ ‘2 "‘\\
1’3
w/
53
3
W
‘x. g
i o
F
vb
E
n,
diff“; . ; . w- ‘ ' W” R ~ 5:? “
:2. K g {:2 igﬁﬁg .(ikgkAw b «m {KM a” W $3 Qiﬁéﬂ 74%” *9 I 3,- / x. a; ' was if I (
{my}; :3: “a as; :6 {HM 4: a“ .ﬁj/ y m _ H
\M X”? {1, ,55 ’ n {I ,, 5 A m w ,,7 a. ' :
$653 as m} Mais- ig n; , a W \ m
VQLWQ § Qwﬁi‘ mi: *1 ﬁwg m4» gaﬁwié “iiéia’j‘g 6: 63.25;? K E} @1‘
I “k ' z ' é ‘ ( :‘ggéﬁgf/ Luz}, "3% g—éjﬁwﬁﬁ, A; '3 E? ‘ Q}: V; : 7325 U éﬁETa-Hb a, 3 u: V dd: (3) Find the derivatives of each of the following functions. Name the main differentiation rules used. cos a: (a) ﬂy?) = 231: + sing: W}: \z
X i ll) S Lama a, (a) ' Find all critical points of Show your calculations then place your ﬁnal answer in the box: ‘1 Z” 1M” if; 315%.”) 7;: Q; 54“; :2" “i m" We “i . (4) (b) Determine the intervals where 9(23) is increasing and where g(:1:) is decreasing. Give your
a??? reasoning and calculations then put your answers as indicated below: ‘g’iwll “v o W 92,) A G 1 l3 .4" {€12} i?» e 1: g(m) is increasing on: &”00 ‘“ l“) y ) ,5 \ a»; ‘ M g a \ \ gm My Jr Find all relative extrema of 5.; “”’ “a .E" g r ea 9 ﬂ» .1 a
,.~ a, K” (d) Find the absolute maximum value and the absolute minimum value of g($) on [0, 2]. fr; 5 yﬁ) Suppose that f has derivative 1“ = 2x3 — 24:13. @\ l f (a) (a) Determine the intervals Where f is concave up and Where f is concave down. Give your
/ reasoning and calculations then put your answers as indicated below: i 1?“ (K7) 3; (5 Cigévrmcl $45 i“ W3 1; £2} \L a :2 4% W“ gr Fig 5% Kg A M a!) 1‘; AK ixvggiiikﬂil 2Q {xi {awfva im at” i" M {"‘iié'm‘ $6»; 3:? L} $65“) <5 (:3
r r ' “ l; W Z“ {it ‘33: Z” 431% x: f is concave up on: f is concave down on: (b) Find all inﬂection points of f Give your reasoning then place your ﬁnal answer in the box: w
gr Mi diff/$33 C hi1 C ire/roars 3i “we; m g IM ,1 ‘33: @ gm i“? (gm. :3: m 3 m ,:
WM Wmﬁx 3’: "a m E} 2 ya I; (6) Suppose that an amoeba proteus is moving in a straight line and its position at time t hours is 3(t) = t + cost ﬁf (a) Find the velocity function 11(t) and the acceleration function a(t).
f x “a; r 3‘ NA
é V r ,9er (15%} e": ’23“ ft my its”? 1.1
i i e {AM (b) Suppose the amoeba travels for 27r hours. What is the maximum distance of the amoeba
from its starting point during this time? [Justify your answer.] it/ US} a: aw Etta i at to ; Z‘TT/i
5;) L (at _
‘ g a C3; % 91 g gig! N «T N
hiwgé i r Q; My Mn} ﬁes; 3:3 3L} ﬁx; fig om Lo! V ‘ 9 rm “6 air“ ; g» ‘ “i _ y Cgaatewig’r: iii—L ; a"; '2 i 3;“: f: j 3% r5; a {i t ' 9/
“YEW th‘zﬁgLfﬂkLMwWL digs! (136:5: LE; (C) What is the starting point of the amoeba? Does the amoeba ever move toward its starting
point? [justify your answer.] HQ ‘ ,-~, at 1“)
gig} :‘L‘fi g fir
\w/ gift”) 2*: fags/dc 2:5,) £7 ., g, M
Q i ,7,“ (VWVQAQLg/ZE a g"; L ‘xi
i i; 3‘ e
\M’” «Commie; a hum/Tr temer 3%??ng sf; gl 1 $2—31:+2 (7) Let Mm) = 2x2 _ 1 A6 (3) Determine if h(w) has a horizontal asymptote and justify your answer with a limit as 3: ——> oo. 4 em lit, ,3 ,, ,4, 5; 5i kw??? $23 w ’l Em (31‘, L“ if 89;}?
ﬂ ‘i M“ e?“ if R A - (b) List all vertical asymptotes of [Be careful!] 1 E
Civulzv» mam-ea ludﬁi mmﬁiéfgigtéwﬁlk ‘ Wt g; Q 53% 34% r >51» mMmm m ﬁftgtg m1 ﬂ } M a: SS g; 1W 2 {24W 8
‘3’ (8) With an unllmlted supply of the rlght nutrients, the mass of a type of tmchoderma (a mold) grows
- ” exponentially With growth rate of 0.07, assuming the time units are hours. Let m(t) be the mass
[mg] of trichoderma at time 15 [hours]. ' : ' ' I? l‘ f _- ,x (a) If 771(0) 25.35 mg then Whig: the mass at time 4 hours. 1
/ a C} ' .N r mm a»; 2% 3i; L %l “’“ if: (c) What mass of trichoderma would you need initially to guarantee a mass of 30.00 mg after 24 ﬂ hours?
a“ KIAK 4:9“ .s f“ A
oval
t Mathematics 115 — F2011 Midterm Examination 1 — 6 October 2011 o Read the instructions to each question carefully — be sure to fully answer the question that is
asked.
0 Show all work and calculations. Provide all necessary reasoning. Use units where they are given. 0 You may use a calculator for arithmetic calculations. You may not use a calculator for any other
purpose. If improper calculator use is evident in an answer, there will be no credit for that answer. Use 2 digits to the right of the decimal place when using decimal representation. Total points: / 100 i 2
XL)» (1) Let W) = g. (a) Use the deﬁnition of the derivative to ﬁnd f’ .9 x :7 I - (54%;? A} ta ta 2:. 3i W W" “’7‘” W m M
: Egg“ w K W m an s
Awe:sz my
W. “as {up
if»,
If it \‘5 .4
l ’E g ‘
\m/ Ki? (b) What is the equation of the tangent line to the graph of f at (2, f (2))? , x” ,
T \ W X} ’33 as ’
m is We Ea v (/19 . .i 5 (2) Let f be the function with this graph: £9 (a) For what values of a: is f not continuous? [Give a reason for each value] if: ‘ .2 I (25!? g M {52 6mg; {pf/gm,humﬁ m I, 3 Find the derivatives of each of the following functions. Name the main differentiation rules used.
A a, 5 -
3 ~ r t :f 3 / ‘ f} ,
g (a) y = 1Ink/36”“) 3 if «4h é Em, itﬁsh} *4 @er
w WM e;
I a c' 7 ‘
I 5'” g A s: / A (a ‘ I V’”
2/ “I! ﬁx “Ear Q E J 1—,— Qfﬂi.%~ x: a; :- l1 sin :3 a: + 2 cos m «~55! I
X Z; A 6 \
'vx ‘ I . gr w :2; '( Kalli/mi? [E Mistral; t i] ' ' >4+ ’2‘: of; {3% :2. mg {Mgr/er 3!; ~14" :2 Swiglillm‘lW {Karma 36;}: H V ixél: mfgwmcl 5;) a r «
5
F? (4) Let gm) = 53:3 — 3:135 with domain R. I (a) Find all critical points of Show your calculations then place your ﬁnal answer in the box: m‘ a“ (b) Determine the intervals Where 9(37) is increasing and Where g(:v) is decreasing. Give your reasoning and calculations then put your answers as indicated below: gltegf} 34:; “l g, 3% if e {maxi} 1;: mi l (13: >43?” *1 (i) 6” all: x U; '2} anewu \l ﬁg: NE W ﬁgsan w Z £3 ﬁ,
sew—«mg: w W (5) Suppose that has derivative = 1:3 — 2793. g (3) Determine the intervals where f is concave up and Where f is concave down. Give your
:7. reasoning and calculations then put your answers as indicated below: Q
i ’2 w \ W . W Qﬂﬂék) ﬁg; rﬁa EQ'Mgie—e {mi} K “i Mi £5 ‘33 ﬂ fgi‘xi)
' I M Mr K
ﬂair)“; a illcﬁawm €19 ‘ (4WD 7;
._ .i‘ mMﬁMWigwmﬁmnw.“ km“ 5. a“,
«((71 WE: ‘” law z 4%” 42%) é: (swam
E ii: A r M I 2 V
{Kg if”; ME?) {Safaring ‘ Assn ("7 f is concave up on:
xii“? i 1% Sigh
f is concave down on: l «‘13. (b) Find all inﬂection points of f GiVe your reasoning then place your ﬁnal answer in the box: if;
i» 6 sf” (6) Suppose that an amoeba proteus is moving in a straight line and its position at time t hours is
/,J 3(t) = t + sint + 1 [mm]. (a) Find the velocity function v(t) and the acceleration function a(t). i a «as = g M 2: i a: Limits: ‘5 “a Mti «:3 a“ if} 3 "ti/Era f- 5:1,?
a; ” (b) Suppose the amoeba travels for 27r hours. What is the maximum distance of the amoeba
a; a frOm its starting point during this time? [Justify your answer.] Lil/ti 7 it» *5? {a it; {at
, :6 ft to A I ( A J aging) J L W \ ﬂitéf ,x “#157 {Wraith diatm‘zrﬁ W
W A K / r ‘ W " é, i f v “ «mm “" “3 ~
r- W M 2w its
EtZﬁTlg‘ZT/T’mi? “8’” ‘ Q (C) What is the starting point of the amoeba? Does the amoeba ever move toward its starting
E” point? [Justify your answer.] ‘
W i M ii
I IV I * I g} 1 (é , E K ("Vng 2%
{r 3;ng :— i, is t a, Watt «p51 Q—mmmgmmégmm > EN “‘3 i C} “i z r;
x ii k2, {f‘ a ﬂaw} a. {+5 Eff; Q (g; j} f3 mam a.
,l t» 2 — 4 (7) Let h(w) = (a) Determine if h(:c) has a horizontal asymptote and justify your answer with a limit as a: —> 00. WW W x 15’ 4., »
via—ng iwﬁ/xt tiff“ “73 k“ 0’ ,_
M372, ‘2
é [a Q i :3” i“ 7:3 \ gm at») .m {y} (b) List all vertical asymptotes of [Be carefull} g 2 tr 7‘“ Mn”? {W 43% a
$523» x1972 1y wﬁﬂwﬂéfi ,3, V "i i gt, 3% ﬁtﬁgmé (Egafmgémiﬁ. f3; 8 <8> K Lg / {/
is»
f With an unlimited supply of the right nutrients, the mass of a type of trichoderma (a mold) grows
exponentially with growth rate of 0.06, assuming the time units are hours. Let m(t) be the mass
[mg] of trichoderma at time 15 [hours]. (a) If m(0) = 15.35 mg then What is the mass at time 6 hours? / \ in {g 7;: g: EM. Q? C 5 uéw£§// {i sit“; «a z? £2 2:1:be t (b) If 771(0) 2 20.00 mg then how long does it take for the mass to reach 70.00 mg? /" (c) What mass of trichoderma would you need initially to guarantee a mass of 40.00 mg after 12 (’
if hours? k 0 Lo!
.1 . i 1,, _ my {am to i bng "WK it 3:: do} 0%“ i i I ’ 47': “is .3 I { .4" '9.
W gm M ” EJW: 1%; l vs “ii” i ﬂag a a in,» lav if?) 0 is)" ...

View Full
Document