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MTH1010 Assignment
创建日期：2024-05-13 15:58:34
所属分类：数学mathematics
标签： MTH1010
Assignment

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MTH1010 Assignment 

Mathematics

Task MTH1010 Assignment 2. This assignment mark will count 5% towards your final MTH1010 mark.
Objectives When you have completed this assignment you should have developed the following skills:
• Understanding and working with polynomial functions;
• Understanding and working with limits and limits of functions;
• Understanding and working with rational functions;
• Understanding and working with exponential and logarithmic expressions;
• Using exponential and/or logarithmic functions in applied settings;
• Consulting in groups but writing up final drafts individually.
Instructions Before you start this assignment
• Read your lecture notes up to and including logarithmic functions and the relevant sections in
Bolster Academy.
• Read through the tutorial exercises you have completed as an aid to doing the assignment
problems.
Attempting the assignment Keep in mind the following:
• Write neatly.
• Show all steps and working.
• Use words to briefly describe/explain steps in your working and diagrams to illustrate.
• Most marks are given to correct working and explanations.
• 10% of the total assignment marks are given to mathematical communication. Please read the
Mathematics Assignment Writing Guide on the MTH1010 Moodle page.
• Final answers are only worth one mark each.
Submission Be aware that:
• Assignment 2is due on Friday 16th September 23:55. It should be placed in the online
submission tool provided in Moodle.
• Extension to an assignments due date, on reasonable grounds, must be sort from the Unit
Coordinator before the due date, if possible.
• Assignments submitted after the due date should be submitted directly to the Unit
Coordinator .
• Assignments submitted after the due date, without Unit Coordinator granted extension, will be
penalised for lateness by 10% of the assignment total per calendar day.
Page: 1 of 3
MTH1010 Assignment 2 School of Mathematics Semester 2 2022
1. Using only algebra and the limit laws, evaluate the following limits, or if the limit does not exist,
explain why.
(a) i. lim
x−→1
(
7x3 − 5x+ 3√x
x+ 1
)
.
ii. lim
x−→∞
(
5x− 2
4x+ 7
)
.
iii. lim
x−→−3
( |x+ 3|
x2 + x− 6
)
.
iv. lim
x−→4
√
25− x2 − 3
x2 − 16
)
.
Note: Your working should clearly show only one limit law per line. Furthermore, you should
clearly explain any other vital steps in your working which is not a limit law.
(b) Consider the function
f(x) =
{
0 if x is an integer
1 if x is not an integer
i. Sketch the graph of y = f(x) for x ∈ [−4, 4].
ii. Find lim
x−→0
(f(x)) or explain why it does not exist.
2. Consider the rational function
F(x) =
8x2 + 20x− 12
4x2 + 4x− 3 .
(a) Factorise the numerator, f(x) = 8x2 + 20x− 12, as much as possible.
(b) Factorise the denominator, g(x) = 4x2 + 4x− 3, as much as possible.
(c) Write the factorised form of F , and then identify the Cartesian coordinates of any holes for
y = F (x).
(d) Find Cartesian coordinates of the x- and y-intercepts for y = F (x).
(e) Apply polynomial long division to write the function F(x) in asymptotic form.
(f) Given your answer in (e) state the nature and equations of any asymptotes for y = F (x).
(g) State the domain and range of F .
(h) Sketch the graph of y = F (x). Make sure all holes, axes-intercepts and asymptotes are clearly
labelled.
3. Algebraically solve the following equations for x (where x is a real number). Leave your answers in
exact form, that is, do not use a calculator. Show all working and clearly explain key steps in your
working.
(a) 323x+1 = 128.
(b) log5(4x− 3) = log3(9).
(c) 9
(
1
2
)x
= 4 + 2
(
1
4
)x
.
(d) 3
(
8−x
)
=
1
2
(
72x+1
)
. (Write your final answer in terms of the natural logarithm function).
(e) 2 log2x+1(2x+ 4)− log2x+1(4) = 2.
Page: 2 of 3
MTH1010 Assignment 2 School of Mathematics Semester 2 2022
4. A new cafe opens in a country town. The manager of the cafe decides not to invest in advertising
because the coffee is so good that he is confident that word will spread of the new cafe and its
wonderful coffee. A marketing survey has shown that the percentage of the population in the country
town who will hear of this new cafe is governed by the logistic model
P (t) =
A
B + e−kt
where P (t) represents the percentage of the town’s population who hear about the cafe t days after
the cafe opens. The constants A, B and k are to be determined from observations found by the
market survey.
The market survey of this word-of-mouth campaign finds that
• On the day the cafe opened (at t = 0) 5% of the town’s population had heard about the cafe.
• After one week (t = 7) the word about the cafe has spread to 10% of the town’s population.
• Eventually, that is, after a very long time, everyone in the town will have heard about the cafe.
(a) Given the three pieces of information provided by the marketing survey of this advertisement
campaign, find the exact values of the constants A, B and k. Write the equation describing
how many people will have heard about the cafe at time t days after the cafe opened.
(b) Approximately what percentage of the town knew about the cafe 2 weeks after the cafe opened?
(c) Approximately how many days after the cafe opened had half the population heard about the
cafe?
(d) Sketch the graph of P (t) =
A
B + e−kt
. Make sure you label the axes, any asymptotes and
axes-intercepts.
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EMESTER1 MAS8383 ECO 512 STAT 8310 CMN3102 MA577 CMPT 225 COMP0045 Methods Equations MATH 113 Math 3283W ICS-33 EN4302/ENT603 BIA B350F MA318 MTHM506 MECH 5315M ECONOMICS MAS8382 COM 2004 PX3142 XJEL 3662 STAT 231 MGT B399F PX4131 COM2108 30AM MSCI521 ECONM1015 ACFI304 GR5241 MATH10101 ECON 2070 COMS31900 EFIM20011 CSE 250B ACSE-5 STAT4207 SOFT2412 ELEC 331 AY2020 MSIN0209 FIT2094 CS 241L ST404 CS 411 3D CA3 INF4002 COMP3506 SIT718 CS 188 CS 170 MATH 138 CSE 251A CMT118 DATA5207 CMPT307 CMPT459 T1 PGEE11108 COSC 3P32 ME 163 MATH96007 7QQMM203 EE6227 BA 574 STA 4234 ECO206Y1Y ECO2008 MT2300 MATH 11158 CS 527 MACM 316 CSE474 2DI70 ISE 525 COMP3411 EMAT30007 7CCSMBDT MFIN704 MFIN704W21 MfIB ECE 438 CS 436 F36 CS5344 FE 621A FE621 TBA 612 ECM3151 ECMM422 MF1 MSIN0107 ESSMENT CS 5854 MFIT 5008 MIPS R4000 ENG5274 BU52018 ENG2079 BIOA02 STA2570 F0 INF553 STAT 440 STAT3008 COMP9417 COMP6451 6COM1042 BU52006 ST303 HW-1 CISC-235 CS 510 MATH3161 ECN 140 ECE391 Comp 3613 CVEN 9625 PRG455 UESTION 1 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