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MATH 1A FINAL (PRACTICE 1)
PROFESSOR PAULIN
DO NOT TURN OVER UNTILINSTRUCTED TO DO SO.
CALCULATORS ARE NOT PERMITTED
YOU MAY USE YOUR OWN BLANKPAPER FOR ROUGH WORK
SO AS NOT TO DISTURB OTHERSTUDENTS, EVERYONE MUST STAYUNTIL THE EXAM IS COMPLETE
REMEMBER THIS EXAM IS GRADED BYA HUMAN BEING. WRITE YOUR
SOLUTIONS NEATLY ANDCOHERENTLY, OR THEY RISK NOT
RECEIVING FULL CREDIT
THIS EXAM WILL BE ELECTRONICALLYSCANNED. MAKE SURE YOU WRITE ALLSOLUTIONS IN THE SPACES PROVIDED.YOU MAY WRITE SOLUTIONS ON THEBLANK PAGE AT THE BACK BUT BESURE TO CLEARLY LABEL THEM
Name:
Student ID:
GSI’s name:
Math 1A Final (Practice 1)
This exam consists of 10 questions. Answer the questions in thespaces provided.
1. Calculate the following (you do not need to use the limit definition):
(a) (10 points)d
dxln(3x)e2x
Solution:
(b) (15 points)d
dxtan(e(x
x))
Solution:
PLEASE TURN OVER
Idf fu 3 c e e t la 324 2eZ
7 Cal x Iu Flat x Luca tu thou Tulsa 1
7 x x luca i
adz tance sea e d x luckily
Math 1A Final (Practice 1), Page 2 of 11
2. Calculate the following (you do not need to use the (ϵ, δ)-definition):
(a) (10 points)
limx→0
sin(x2)
tan x
Solution:
(b) (15 points)
limx→−∞
√1 + 4x6
2 + x3
Solution:
PLEASE TURN OVER
Hospital
sin 1 21 I 2x SGT info Zacoscx2ius Limo tangy a O Sed im seek c
k 70
9 O
ExVEE T
Lim Lima a 2 3 2C 3 s z u
TE T
TiLim Lima z x a
I I
tT z
Math 1A Final (Practice 1), Page 3 of 11
3. Calculate the following (you do not need to use Riemann sum defintion):
(a) (10 points) !2x+ 4
x2 + 1dx
Solution:
(b) (15 points) ! e2
e
1
x ln(x)dx
Solution:
PLEASE TURN OVER
I da J one 11 da
u x I dug zu doc IIT J doc JI du
Tulul C In 1 2 1 I C
J doc Tula ti I t 4arctanby c L
u luck DI doc edudoc
J doc Jiu du lulul t C
In Icu x t C
dx tu z luci Lutz
Math 1A Final (Practice 1), Page 4 of 11
4. (25 points) A cup of water is placed in a refrigerator. The refrigerator has temperature5C. After 10 minutes the water is 15C. After 20 minutes the water is 10C. Determinethe temperature of the water when it was placed in the refrigerator.
Solution:
PLEASE TURN OVER
Initial temperature
NCC Tct Ts To Ts ett1012
TC lol is IS S t Cto 51 e
Zo kTCzol to to S Cto S e
O lo k S Zo ke and e
To S To S
1022 s
Zo To s
To 25 C
Math 1A Final (Practice 1), Page 5 of 11
5. (25 points) Sketch the following curve. Be sure to indicate asymptotes, local maximaand minima and concavity. Show your working on this page and draw the graph on thenext page.
y =x2ex
x
Solution:
PLEASE TURN OVER
4 Cx d e Tt x to
DIVE it c o
sketch y ace and remove 0,0
IR
0 en Neither
tympts None
Lim see toa 38
Ix OLim xe Lion Lim ae k a 3 a e
y o horizontal asymptote as x a
1 cts at I Local min
4 x e c ace ltd e
I c T Gal1AT 71cal o x I1
By None
Math 1A Final (Practice 1), Page 6 of 11
Solution (continued) :
PLEASE TURN OVER
4 cts at 27 Cal e t tix e zen ex inflection
Ay 7 coal _o x Z I 54By None
7C l e
7 21 Ze 2
y IIe
C I e l
Math 1A Final (Practice 1), Page 7 of 11
6. (25 points) Determine the equation of the tangent line at x = 1 of the following curve:
y =
! 3x
x
cos(πt)dt+ 2x
Solution:
PLEASE TURN OVER
Y cos Cittldt cos CTH dt 2x
dydy
3 cos 31T c Cns Tx 2
dy 3 cos 31T Cns Cit z
3 C il c 2 0
cos CIT at 2 sin Tt P 2 0 2 2
Tangent line has equation y z
Math 1A Final (Practice 1), Page 8 of 11
7. (25 points) An open box will be made by cutting a square from each corner of a 3 by 8foot piece of cardboard and then folding up the sides. What size squares should be cutfrom each corner to maximize the volume.
Solution:
PLEASE TURN OVER
Objective Maximize Volume
8Objective Volume y x 3 2x
IIe LIs3 Constraint 2x y 8
y S Za
Volume 8 Za e 3 2x t 4 3 22 224 a
FGC
Domain Co IzF x 12 2
44 x t 24 3 2 Hae 63 2 x 3
A 7 x o x n 3
By 7 continuous everywhere2 3
7 Co 0 Volume is maximized when
f 314 0a 8 inches
fC4z 70
Math 1A Final (Practice 1), Page 9 of 11
8. (25 points) A company accrues debt at a rate of
(2t+ 3)√t+ 1
dollars per year, where t is the time in years since the company started. How much willthe company’s debt have grown between t = 3 and t = 8? You do not need to simplifyyour answer.
Solution:
PLEASE TURN OVER
D Ct 21 3 TF
u Et l dat I de du
t u i
Jett3 TEI DE Jett 3 Tu du
J Cu 1 3 Tu du J2u n'k du
z 3 ask 3 ask etty t taste
ttDrEidt 4sCttDskt2_Ltti1k1f
CsI.ss E5 24523
DCS DG
Math 1A Final (Practice 1), Page 10 of 11
9. (25 points) Calculate the following limit:
limn→∞
n"
i=1
6n+ 3i
n2sin((2 +
i
n)2)
Solution:
PLEASE TURN OVER
Gut3in sin 2 1 14 3 2 in sin 2x T f
Let a z b 3 and 7cal 3 a sin se
n 3Lim 2 Gunt s u 2 Ily J3asiulse dau i c
2
u act did Zac da
J3xsiucx2 doe 32 Siu Cui du costal TC
32 scat C
u 3Lim I Gnut sin et 14 casedu g 2
Iz costly Cns CP
Math 1A Final (Practice 1), Page 11 of 11
10. Let f(x) = 1/x and g(x) = 4x h(x) = x.
(a) (10 points) Calculate the area of the finite region bounded by y = f(x), y = g(x)and y = h(x).
Solution:
(b) (15 points) Calculate the volume of the solid of revolution formed by rotating thisregion around the x-axis.
Solution:
END OF EXAM
y 42C
py a Area 4 47
da a da
L Za tubal EarlIs ft tutti IlIn 121
Volume IT 4 32 za da a da
3Stix l t T Iz tic
SIT
gIT G l t f z
