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MAST10005 Calculus

创建日期：2024-05-14 15:56:56

所属分类：数学mathematics

标签： MAST10005

Calculus

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Semester 2 Assessment, 2019

School Mathematics and Statistics
MAST10005 Calculus 1
Writing time: 3 hours
Reading time: 15 minutes
This is NOT an open book exam
This paper consists of 6 pages (including this page)
Authorised Materials
• Mobile phones, smart watches and internet or communication devices are forbidden.
• Calculators, tablet devices or computers must not be used.
• No handwritten or print materials may be brought into the exam venue.
Instructions to Students
• You must NOT remove this question paper at the conclusion of the examination.
• All questions may be attempted. All answers should be appropriately justified.
• Number the questions and question parts clearly. Start each question on a new page.
• Use the left pages for rough working. Write material that you wish to be marked on the
right pages only.
• A limited formula sheet is provided on the back of this cover page. You may remove this
cover page in order to use the formula sheet if you wish.
• This exam paper consists of 6 pages in total and there are 12 questions.
• There are 12 questions with marks as shown. The total number of marks available is 109.
Instructions to Invigilators
• Students must NOT remove this question paper at the conclusion of the examination.
• Each candidate should be issued with an examination booklet, and with further booklets
as needed.
This paper may be held in the Baillieu Library
MAST10005 Semester 2, 2019
Useful Formulae
Pythagorean identity
cos2(x) + sin2(x) = 1
Compound angle formulae
sin(x+ y) = sin(x) cos(y) + cos(x) sin(y)
cos(x+ y) = cos(x) cos(y)− sin(x) sin(y)
tan(x+ y) =
tan(x) + tan(y)
1− tan(x) tan(y)
Derivatives of inverse trigonometric functions
arcsin′(x) =
1√
1− x2
arccos′(x) = − 1√
1− x2
arctan′(x) =
1
1 + x2
Antiderivatives from inverse trigonometric functions∫
1√
s2 − x2 dx = arcsin
(x
s
)
+ C∫ −1√
s2 − x2 dx = arccos
(x
s
)
+ C∫
1
s2 + x2
dx =
1
s
arctan
(x
s
)
+ C
where s is a positive constant, and C is an arbitrary constant of integration.
Complex exponential formulae
eiθ = cos(θ) + i sin(θ)
cos(θ) =
1
2
(
eiθ + e−iθ
)
sin(θ) =
1
2i
(
eiθ − e−iθ)
Vector projections
• The vector projection of v onto u is v‖ = (uˆ · v)uˆ = ku, where k ∈ R is the unique solution
of u · (v − ku) = 0.
• The vector component of v perpendicular to u is v⊥ = v − v‖.
Complex roots
The n-th roots of w = seiφ are s
1
n ei(
1
n
(φ+2kpi)) for k = 0, 1, . . . , n− 1.
Changes in speed
Provided r′(t) 6= 0, the speed function ‖r′(t)‖ is decreasing when r′(t) · r′′(t) < 0 and
increasing when r′(t) · r′′(t) > 0.
Page 2 of 6 pages
MAST10005 Semester 2, 2019
Two properties of complex numbers
(I) |zw| = |z||w| (II) zz¯ = |z|2
Question 1 (9 marks) Let
A = {z ∈ C : zz¯ < |z|}, B = {z ∈ C : |z − iz| <
√
2}.
(a) Prove that |z − iz| = √2 |z| for every z ∈ C.
(b) Prove carefully that A ⊆ B. You may use the result of (a).
(c) Is B ⊆ A? If your answer is yes, prove it. If your answer is no, give a counterexample.
(d) Express the set
D =
{
x ∈ R
∣∣∣∣ 1− xx2 + x+ 1 < 2
}
as a union of intervals.
Question 2 (7 marks)
(a) Evaluate the following in Cartesian form x+ iy where x and y are real:
(i)
(6− 2i)(4− 7i)
2− i
(ii)
(−
√
3 + i)6
(b) Find the following real numbers:
(i)
Arg
(
−
√
3 + i
)
(ii) ∣∣∣∣(6− i)2(4− 3i)1 + 6i
∣∣∣∣
Question 3 (7 marks)
(a) Sketch the set A = {z ∈ C | Arg(z) ∈ (pi4 , 3pi4 ]}.
(b) Multiply a general z = reiθ ∈ C by i = ei pi2 . Hence describe the geometric effect of
multiplying a complex number by i.
(c) Use your answers to (a) and (b) to sketch the set B = {w ∈ C | iw ∈ A} (on a new
diagram).
Page 3 of 6 pages
MAST10005 Semester 2, 2019
Question 4 (8 marks)
(a) Find the set of solutions of
z6 = 32(1− i
√
3)
Express your answers in exponential polar form.
(b) Find the set S = {z ∈ C | z4 + 5z2 + 4 = 0}. Express your answers in Cartesian form.
(c) Use your answer to (b) express P (z) = z4 + 5z2 + 4 as product of two quadratic factors
with real coefficients.
Question 5 (7 marks) Consider the function
f(x) = arcsin(arccos(x))
(a) Find the implied domain of f . You will need to express your answer in terms of cosine.
(b) Find the implied range of f .
Question 6 (7 marks)
(a) Find the angle θ between u = (
√
3, 0, 0) and v = (1, 0,
√
3).
(b) For A = (1, 1, 1), B = (3, 3,−1) and Q = (3,−1,−1):
(i) Find the point P on the line L passing through A and B that is closest to Q.
(ii) Find the distance from Q to the line L.
Question 7 (12 marks) Consider the curve
C = {(x, y) ∈ R2 | x2 − xy + 2y2 = 7}
(a) Find all intercepts of C.
(b) Use implicit differentiation to find a formula for dydx in terms of x, y.
(c) Find any points in C where the tangent line is horizontal.
(d) Find any points in C where the tangent line is vertical.
(e) Use (a) to (d) to sketch C.
Page 4 of 6 pages
MAST10005 Semester 2, 2019
Question 8 (8 marks) The function r : R −→ R2 defined by
r(t) = et cos(t)i+ et sin(t)j
is a type of parametric curve known as a logarithmic spiral.
(a) Calculate the velocity r′(t) at time t.
(b) Calculate the speed ‖r′(t)‖ at time t.
(c) Calculate the acceleration r′′(t) at time t.
(d) Find the cosine of the angle θ between r(t) and r′(t) at time t and hence find the angle
between r(t) and r′(t).
(e) Calculate r′(t) · r′′(t) and hence explain why the speed is always increasing.
Question 9 (13 marks) Consider the function given by formula
f(x) = 2x
√
1− x2
(a) Find the implied domain of f .
(b) Find the x and y intercepts of f . Express your answer as a set.
(c) Find the stationary points of f . Express your answer as a set.
(d) Find the intervals on which f is concave up and the intervals on which f is concave down.
(e) Find the inflection points of f . Express your answer as a set.
(f) What does the answer to (d) tell us about the stationary points found in (c)?
(g) Use the answers to previous parts of this question to sketch the graph of f .
Question 10 (9 marks)
(a) Verify that −e−x(x+ 1) is an antiderivative of f(x) = xe−x.
(b) Use integration by substitution and your answer to (a) to evaluate∫ pi
2
0
sin(x)e− sin(x) cos(x) dx.
It may help to express the integrand in terms of f(x) aa defined in (a). Explain carefully
which rules of integration you use.
(c) Evaluate the integral ∫ e√pi
1
sin(log2(x)) log(x)
x
dx.
Explain carefully which rules of integration you use.
(d) Find the area enclosed between the curves y = sin(x), y = cos(x) and the and the hori-
zontal axis on [0, pi]. A sketch is always helpful.
Page 5 of 6 pages
MAST10005 Semester 2, 2019
Question 11 (11 marks) Evaluate the following antiderivatives.
(a)
∫
(2x− 3)2(x+ 2) 32 dx. [Hint: A linear substitution will work here.]
(b)
∫
1
4x2 + 8x+ 5
dx.
(c)
∫
cos2(x) sin5(x) dx.
Question 12 (11 marks) Consider the initial value problem
dy
dx
= (y2 − 1) cos(x), y = 0 when x = 0.
(a) What are the constant solutions of the differential equation?
(b) Explain briefly why the range of the solution of the initial value problem is a subset of
(−1, 1).
(c) Use partial fractions to express
1
y2 − 1 as a sum of two fractions with linear denominators.
(d) Find a formula for y by solving the initial value problem. You will need to use your answer
to (c). The result from (b) will allow you to remove all absolute value signs when solving
to find a formula for y. You may use this even if you haven’t completed part (b).
End of Exam—Total Available Marks = 109

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25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 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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 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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