MATH 1A MIDTERM 1 (002)

PROFESSOR PAULIN

DO NOT TURN OVER UNTILINSTRUCTED TO DO SO.

CALCULATORS ARE NOT PERMITTED

YOU MAY USE YOUR OWN BLANKPAPER FOR ROUGH WORK

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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 Midterm 1 (002) 12.10pm-1pm

This exam consists of 5 questions. Answer the questions in thespaces provided.

1. Determine the domains of the following functions:

(a) (15 points)x+ 1!

x−42−x

Solution:

(b) (10 points)ln(− arcsin(x))

Solution:

PLEASE TURN OVER

x 4 o x 4 No solutionZ 2C So 2 a

2 4K L l l

oZ K K L C O A

s a 4Z K co 2C in CZ cZ C x

2C 1 IDomain A a 4

Z 7C

casedarcsin x o arcsin d C 0 open

s c

Iz F arisingin C I of

Hence d main rt Tul arcsin xi

is C l o

Ic

Math 1A Midterm 1 (002), Page 2 of 5

2. (a) (15 points) Describe in words, how, starting with the graph y = − tan(2x+1)− 1,one can draw the graph

y = tan(2− x)− 2.

Solution:

(b) (10 points) Simplify the following expression:

arcsin(sin(3))

Solution:

PLEASE TURN OVER

CompreisUm zoutally byfactor 2y tan 2 1 I y tan att 1

Translate leftReflect u by lai y axis

y tan Z x I C y tan utz I

Reflectin a axis

u Translatedown by 3

y fun 2 al t I 7 y fan Z al z

In C z tv

sin 3 Sin T 3

t y it 3 arcs in Siu 13 IT 3

Math 1A Midterm 1 (002), Page 3 of 5

3. Calculate (using the limit laws) the following limits. If a limit does not exist determineif it is ∞, −∞ or neither.

(a) (15 points)lim

x→π/2−x sec(x)

Solution:

(b) (10 points)

limx→∞

ln(x3 + 1

x4 + x+ 1)

Solution:

PLEASE TURN OVER

y Cascais

Xax Sec x 2

Casca

U'mx

soLimse Tz Cnsc c

Liar cos Cx Otx s

deg x 3 1 3 C 4 degCx4txtion x 3 1 I c

OA 4 at 1

Cim Tu se a

c Ot

Lim Yu f3 1

c a 4a

Math 1A Midterm 1 (002), Page 4 of 5

4. (25 points) Let f(x) =

"1− 2x if x ≤ 2

−3 if x > 2.

Prove, using ϵ, δ methods, that f is continuous at x = 2.

Solution:

PLEASE TURN OVER

Alin im float 7 z 37C 32

Fix E o

T l 2x l 2x 3 x Z

LzChoose S E

3 2 Z

3 y 314 2 1 12 413

11

C Ix Z c 21 21 c Ka Glee l Za C z cx

x C 2

c S2 1 x 3 C

OC Ix z c E v x Zz

3 C z o C ET

Hence liar text 712 3always true soc 72nothing to check

Math 1A Midterm 1 (002), Page 5 of 5

5. (25 points) Does there exist a tangent line to the curve

y =2− x

x− 1

which contains the point (1, 0)? Carefully justify your answer. If you calculate a deriva-tive do so using the limit definition. You do not need to use ϵ, δ methods.

Solution:

END OF EXAM

7 Cx Z K

2C I7 Cath 7 a

2 a h z a7 a Lim Liar ath 1 a l

o h u oh

Z a h a I 2 a Cat h lLimu o h a h c Ca l

Lim Ya f a hayth ya 2h 1 2 1 4h 70 h Cath 1 Ca i

I ILimu o Cath 1 Ca i a 1,2

Equation A tangent at a a y II x al

too ou tangent II It di al

aaI a 2 1

a 3

There is precisely one tangat line contaig1,01 It is at a 3