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Math 142B function
创建日期：2024-05-30 14:47:58
所属分类：数学mathematics
标签： Math 142B
function
Math 142B
March 18, 2023
Final Examination v.A
(Total Points: 50)
Name:
PID:
Instructions
1. Write your Name and PID in the spaces provided above.
2. Complete the Excel with Integrity Pledge on the last page.
3. Make sure your Name is on every page you turn in.
4. No calculators, tablets, phones, or other electronic devices are allowed during this exam.
5. Put away ANY devices that can be used for communication or can access the Internet.
6. You may use one handwritten page of notes, but no books or other assistance during this exam.
7. Read each question carefully and answer each question completely.
8. Write your solutions clearly in the spaces provided.
9. Show all of your work. No credit will be given for unsupported answers, even if correct.
0.(2 points) Carefully read and complete the instructions at the top of this exam sheet and any additional
instructions written on the chalkboard during the exam.
v.A (page 2 of 9) Name:
1.(6 points) Find
lim
x→0
[
1
x
− 1
x2
log (1 + x)
]
.
Fully justify your answer.
v.A (page 3 of 9) Name:
2.(6 points) Find
lim
x→+∞
(
1− 1
x
)x
.
Fully justify your answer.
v.A (page 4 of 9) Name:
3.(6 points) Use power series to find the value of the numerical series
∞∑
n=0
1
(n+ 1) · 2n .
v.A (page 5 of 9) Name:
4.(6 points) Determine whether the following improper integral converges or diverges:∫ π
2
0
1
cos2(x) sin2(x)
dx.
v.A (page 6 of 9) Name:
5.(8 points) Suppose f : R→ R is an infinitely differentiable function such that∣∣f (k)(x)∣∣ ≤Mk+1 k! for all x ∈ R and k ∈ N
for some M > 0 where the bound is attained at x = 0 for infinitely many indices.
Recall that the Taylor series for f centered at x = 0 has the form
∞∑
k=0
f (k)(0)
k!
xk.
(a) Find the radius of convergence R ≥ 0 of the Taylor series of f centered at 0 (being an
instance of a power series).
(b) Prove that the Taylor series converges to f on (−R,R) and that the convergence is uniform
on any [−r, r] ⊂ (−R,R).
v.A (page 7 of 9) Name:
6.(8 points) Not every uniformly convergent sequence of differentiable functions (fn) defined on R converges
to a differentiable function f on R.
Consider the sequence of continuously differentiable functions (gn) defined on R by
gn(x) =
1
n
arctan(n2x).
(a) Show that (gn) converges uniformly on R to a continuous function g defined on R and
determine a formula for g(x).
(b) Show that the sequence (g′n(0)) is unbounded. Briefly explain why this implies that (g′n)
does not converge pointwise to g′ on R. [Don’t forget that arctan′(y) = 1
1+y2
.]
v.A (page 8 of 9) Name:
7.(8 points) Write the letter of the sequence in the right column in the space next to the matching description
in the left column. No justification is required. Yes, there is a sequence that will not be matched.
Each correctly placed letter (including a correctly unplaced letter) will earn one point.
A sequence of continuous functions
{fn} whose pointwise limit function f
is not continuous
A.
fn : R→ R
fn(x) =
n∑
k=0
1
k!
xk
A sequence of continuous functions
{fn} whose pointwise limit function f
is continuous but
∫ 1
0
f ̸= lim
n→∞
∫ 1
0
fn
B.
fn : [0, 1]→ R
fn(x) =
{
1 if x = k2n , k ∈ Z
0 otherwise
A sequence of polynomial functions
{fn} whose pointwise limit function f
is analytic (that is, represented by a
power series)
C.
fn : [0, 1]→ R
fn(x) = x
n
A sequence of integrable functions
{fn} whose pointwise limit function f
is not integrable
D.
fn : R→ R
fn(x) =
1
n arctan
(
n2x
)
A sequence of infinitely differentiable
functions {fn} whose uniform limit
function f is not differentiable
E.
fn : [0, 1]→ R
fn(x) =

n2x if 0 ≤ x < 1n
2n− n2x if 1n ≤ x < 2n
0 if 2n ≤ x ≤ 1
A sequence of bounded functions {fn}
whose pointwise limit function f is
unbounded
F.
fn : (−1, 1)→ R
fn(x) =
√
x2 + 1n
A sequence of noncontinuous
functions {fn} whose uniform limit
function f is continuous
G.
fn : [0, 1]→ R
fn(x) =
x
x2 + 1n
H.
fn : [−1, 1]→ R
fn(x) =
{
− 1n if − 1 ≤ x < 0
1
n if 0 ≤ x ≤ 1
Math 142B Excel with Integrity Pledge
The Excel with Integrity pledge affirms the UC San Diego commitment to excel with integrity
both on and off campus, in academic, professional, and research endeavors.
According to the International Center for Academic Integrity, academic integrity means
having the courage to act in ways that are honest, fair, responsible, respectful & trustworthy
even when it is difficult. Creating work with integrity is important because otherwise we
are misrepresenting our knowledge and abilities and the University is falsely certifying our
accomplishments. And when this happens, the UCSD degree loses its value and we’ve all
wasted our time and talents!
Name: PID:
Excel with Integrity Pledge
I am fair to my classmates and instructors by not using any unauthorized aids.
I respect myself and my university by upholding educational and evaluative goals.
I am honest in my representation of myself and of my work.
I accept responsibility for ensuring my actions are in accord with academic integrity.
I show that I am trustworthy even when no one is watching.
Affirm your adherence to this pledge by writing the following statement in the space below:
I Excel with Integrity.
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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A 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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 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