MATH 1A FINAL (PRACTICE 2)

PROFESSOR PAULIN

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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 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