MATH 1A FINAL (PRACTICE 2)
PROFESSOR PAULIN
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Math 1A Final (Practice 2)
This exam consists of 10 questions. Answer the questions in thespaces provided.
1. Calculate the following. You do not need to simplify your answers.
(a) (10 points)d
dx
tan(sin(ex))
x
Solution:
(b) (15 points)d
dxcos(x)sin(
√x)
Solution:
PLEASE TURN OVER
tan since seacsin ex cosce e c tan since
af
Zk
f x Cnsc cs l Luttrell Sintra Lucassen
day Luc tezel cos Ctx et tu sext Sintracos x
daeCas ca
Sintrad
sintrac sext cascara ftp 1uccnscxlli siuirxl sY
Math 1A Final (Practice 2), Page 2 of 11
2. Calculate the following (you do not need to use the (ϵ, δ)-definition):
(a) (10 points)
limx→0
cos(2x2)− 1
4x
Solution:
(b) (15 points)
limx→∞
(1 +4
x)x
Hint: Rationalize.
Solution:
PLEASE TURN OVER
c seem
nita
f If I E'Io o
e_
Recall fig it It e
Eius it 5 find it 1
E I e
Math 1A Final (Practice 2), Page 3 of 11
3. Calculate the following (you do not need to use the Riemann sum definition):
(a) (10 points) !arccos(x)√
1− x2dx
Solution:
(b) (10 points) ! 2/π
1/π
sin(1/x)
3x2dx
Solution:
PLEASE TURN OVER
u arccoscx dat mzdx TE du
J dx J uan tu te
Lz ceos x
u date dx soda
sida Jsiucu du s scuttle
JossC 1 1 C
211T
Jsi
Y da f cos I I Ctl
lit
Math 1A Final (Practice 2), Page 4 of 11
4. (25 points) Calculate the equation of the tangent line at x = 3 of the following curve:
2y3(x− 3)
x√y
+ 1 =1
x√y.
Solution:
PLEASE TURN OVER
3
Zg's x sI 2y3Cx 3 scary I
7 2g x 3 wry da s
6ycd Cx 31 t 2y t Vy t xtzy tda.de o
adj ry 2
6g2Gc 3 t a y I
x 3 3Ny 3 y
dais I 23 I 3 2
Equation A Target y l z x 3at 3,1
Math 1A Final (Practice 2), 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 = ln(x
2x2 − x3)
Solution:
PLEASE TURN OVER
yen la x 1 it x o
Not definedat 0
DIVE it 2 0
should sketch y teal Tu Za ri
Domain 2x sac za z x o o x C 2
0d Neither
Vertical asymptotes Lim Tn Za H so noLuiza sie
2C z
x o and x 2 Vertical asymptotes1 cts at 1 Local min
7 xZac Z
Zx x2 7
Ay 711 1 0 x l s 7 Gal
3 None in 0,2 z
f 2127C M t Za z Za z 2 x 12 1
O22C al Z
Z c set
Ay Neue F XI
By None
Math 1A Final (Practice 2), Page 6 of 11
Solution (continued) :
PLEASE TURN OVER
1 l O
c 6 7C 2
you
it
Math 1A Final (Practice 2), Page 7 of 11
6. (25 points) Show that the following equation has at least one real solution. Be sure tocarefully justify you answer clearly stating any results you use from lectures.
x arctan |x|+ c = 0, where c is a constant.
Hint: Consider behavior at ∞.
Solution:
PLEASE TURN OVER
Let tex x arctanbel 1C
im a Isx I
Cim arctanlal ox sea
Liar x aretankel c tox s
Lim are tan 174 t c Ax so
There exists a b such that 7cal co C 7lbI V T
7 Cx d s ou Ca b There exists c
Such that 7 Cc o
Math 1A Final (Practice 2), Page 8 of 11
7. (25 points) A drink will be packaged in a cylindrical can with volume 40in3. The topand bottom of the can cost 4 cents per square inch. The sides cost 3 cents per squareinch. Determine the dimensions of the can that minimize the cost.
Solution:
PLEASE TURN OVER
stat cost AObjective Minimize As tap and bottom side
d I dr
Objedive 4th 4th 3 2 Trh
µh Constraint Trh 40v
hItv2
SITA 6 uh Sir Gtr 72Str 24 fer
Domain o s
f r 16 r 240M
ay F r o 16 r 24r3 r T
By 7 continuous ou o o
Ifs tier
7411 co 7 60 o
Absolute min cost is when r F hCEE
Math 1A Final (Practice 2), Page 9 of 11
8. Two rockets are fired vertically into the air from the ground. The second rocket islaunched four seconds after the first. The velocity of the first rocket is v1(t) = 6 − tmetres per second and the velocity of the second is v2(t) = 10 − t metres per second,where t is the time in seconds after the first launch.
(a) (15 points) How long after first the launch will both rockets be at the same height?What will this height be?
Solution:
(b) (10 points) Determine the total distance traveled by the first rocket at this time.
Solution:
PLEASE TURN OVER
firstI
S Ct GE Et t C and S co o o
S Ct Gt Iz El
Sz Ct Lot Etc C and 52141 0 40 St C o
32 Sect lot zt2 32
S Lt Sz Ctl Gt Itc lot tzt2 32
4 C 32 E S time after 1st Land they111be same height
S S Secs 16 C height at t 8
G y 6 tTotal Distance travelled
gArea Area
y zz2
Math 1A Final (Practice 2), Page 10 of 11
9. (25 points) Calculate the following limit:
limn→∞
n"
i=1
n
n2 + 3i2
Hint: Remember that AB= 1
B/A.
Solution:
PLEASE TURN OVER
In I L
u2 z ie i t 3 ifu2 3 i 2h 2
Letting a o b 7cal zwe get
fig II n izda
let a Fsu I Fs da duda
J 2da J du arctanculte
avatar Fsu C
fig II n rstanctuers r
Math 1A Final (Practice 2), Page 11 of 11
10. (25 points) Calculate the volume of the solid of revolution formed by rotating the regionenclosed by x = y2 (with y ≥ 0), x = −y and x = y + 2 around the line y = 2.
Solution:
END OF EXAM
y _Fx 4,2 Y
T y Tze zy x 2
I
µ tannate c L010 1 i downby 2y a
y 2C 2
I
Volume f C a 212 Ta 212daLO
it Ca 4 2 CTE 2 Lda
IT C a zkda it ex 4 da it x 4th 14 da
cactus to ca 4131 it Eri Isak 4 11
9 it t O C 9T 8T IT 161T
3 IT