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MATH226 function
创建日期：2024-05-31 15:59:31
所属分类：数学mathematics
标签： MATH226
function

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MATH226 Problem Sheet 3

• This problem sheet is for practice only; solutions will not be assessed, and do
not need to be submitted to Canvas.
• However, similar questions may appear in the assessments.
3.1 The sine function has the Taylor series expansion
sin x =
∞∑
j=0
(−1)j x
2j+1
(2j + 1)! .
(a) Write a Maple procedure which computes this sum up to a finite number
of terms using a do loop. Do not use an upper limit for the index; use
the condition
if ( abs( t ) <= eps * abs( s ) ) then
break :
end if :
where s is the current value of the sum, t is the next term to be added
and eps = 0.5 ∗ 10−10. Use your procedure to compute sin x for five
values of x, and compare these to values obtained using Maple’s built
in sine function.
(b) In your sine procedure, insert a print statement to show how many terms
have been summed when the break statement is activated. Now use
your procedure to compute sin x for x = 0.1, x = pi−0.1, x = 2pi+ 0.1
and x = 3pi − 0.1. Don’t forget to use evalf to obtain a numerical
approximation to pi. What do you notice? Why does this happen?
(c) In view of your answer to part (b), write a Maple procedure which takes
as its argument a real number x and returns as its result a value x0
such that
− pi2 ≤ x0 ≤
pi
2 and sin x0 = sin x.
Use this to improve the efficiency of your sine procedure, and repeat the
four computations from part (b) to show that this has indeed worked.
Hint: first obtain a value in the interval (−pi, pi], then use symmetry to
find x0.
3.2 (a) Write a Maple procedure that takes as its argument a real number x
and computes the sum
F (x) = 48
pi4
∞∑
j=1
(−1)j+1
j4
cos(jpix), −1 ≤ x ≤ 1.
using a do loop. Do not use an upper limit for the index; use the
condition
if ( abs( t ) <= eps * abs( s ) ) then
break :
end if :
where s is the current value of the sum, t is the next term to be added
and eps = 0.5 ∗ 10−10.
(b) Plot a graph of F (x) for −1 ≤ x ≤ 1. Then plot the function
g(x) = x4 − 2x2 + 7/15 on the same axes. What do you notice?
Use plot( F , -1 .. 1 ) to create the graph of F .
(c) Check the accuracy of your procedure at x = 0.4, 0.5 and 0.6. What
do you notice? Why do you think this happens, and how should the
procedure be changed to prevent it?
3.3 Write a procedure which takes a student number (a nine digit positive integer
with no leading zeros) as its argument and returns an array containing the
individual digits in their original order. That is, the result of
dismantle_student_number( 123456789 )
should be
[
1 2 3 4 5 6 7 8 9
]
. Test the procedure using your own student
number, and two other nine digit natural numbers.
Hint: you can use the mod command to obtain the last digit of an integer;
for example
1023 mod 10
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MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 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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 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