Question / Answer Booklet
THE UNIVERSITY OF AUCKLAND
SUMMER SCHOOL 2021 Campus: City
MATHEMATICAL MODELLING 2 Term Test
Copyright By PowCoder代写 加微信 powcoder
(Time allowed: 90 minutes)
Answer all questions.
The answer for each question should be written in the space provided. If you
run out of space, extra space is available at the back of this booklet.
An appendices booklet containing information for the Data Analysis section and a formula sheet is provided.
This is a restricted book lite test. This means that you may bring in a single piece of A4 paper that is written or typed on both sides, NO attachments; glued, stapled or otherwise attached. This will be checked prior to the start of your test.
Calculators are not permitted.
Question 1 2 3 4 5 6 7 Total
Outof 10 4 5 5 10 6 10 50
Question / Answer Booklet
SECTION A – Ordinary Differential Equations Question 1 (10 marks)
Consider the ordinary differential equation:
y′′ +4y′ +8y = 16t2 +4
subject to the initial conditions:
y(0)=1 , y′(0)=2
(a) Find the complementary function for the unforced response, yc. Write this in a trigono-
metric form if appropriate. (4 marks)
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(b) Find the particular integral for the forced response, yp. (3 marks)
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Question / Answer Booklet
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(c) Find the total solution to the ODE. (3 marks)
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Question / Answer Booklet Question 2 (4 marks)
Consider the ordinary differential equation:
y′ =(t−y)2
subject to the initial condition y(0) = 0. Perform one iteration of the improved Euler method for a time step of ∆t = 1. (4 marks)
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Question / Answer Booklet Question 3 (5 marks)
Use the method of Laplace transforms to solve the ordinary differential equation: y′′ + 9y = 3 δ (t − 2)
y(0)=0 , y′(0)=0 (5marks)
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Question / Answer Booklet
Question 4 (5 marks)
Identify the location of all five of the stationary points of:
f(x,y)= 2 −2x2y−xy2 +4xy+1
You do not need to classify these stationary points.
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Question / Answer Booklet Question 5 (10 marks)
(a) Consider the double integral:
with the following region of integration, R:
Evaluate this double integral using polar coordinates.
sinx2 +y2 dxdy
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Question / Answer Booklet
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(b) Consider the double integral:
f (x, y) dx dy
Sketch the region of integration and re-write the integral after changing the order of integration. You do not need to evaluate the integral. (3 marks)
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Question / Answer Booklet
SECTION C – Data Analysis Question 6 (6 marks)
(a) Comment on the validity of the following claim:
A two-sample t-test on the difference in average ride-share waiting times between two platforms returned a p-value of 0.82. This gives us certainty
that the null hypothesis must be true.
The following questions relate to the analysis given in Appendix A: Orange Juice – One-sample Analysis.
(b) Briefly comment on the exploratory analysis presented.
(c) Interpret the p-value resulting from the t.test() run on this dataset.
Question / Answer Booklet Question 7 (10 marks)
The following questions relate to the analysis given in Appendix B: Orange Juice – Regression Analysis.
(a) Write down the regression equation fitted in oj.lm. (2 marks)
(b) With reference to an appropriate plot, comment on whether oj.lm satisfies the equality of variance assumption. (1 mark)
WARNING: Marks will be deducted for answers that are not specifically asked for in the questions. Please keep answers concise. Do NOT do brain dumps.
(c) Assuming that the relevant assumptions are satisfied, write a brief Executive Sum- mary of the main conclusions you would draw from this analysis. (3 marks)
Question / Answer Booklet
(d) Give predictions for the sweetness of an individual batch of orange juice, where the pectin content was:
(i) 100mg/L (ii) 300mg/L
and comment briefly on the validity of each prediction. (2 marks)
Question 7 continues overleaf
Question / Answer Booklet
(e) The orange juice manufacturer has collected more data on the chemical composition of batches of orange juice, and the coresponding subjective sweetness score as evaluated by professionals. A machine learning model trained on this data is to be used to predict how consumers would perceive the sweetness of each batch of orange juice. State a concern you might have with this model. (1 mark)
(f) A colleague suggests that fitting a quartic term (i.e. a x4 term) would improve the model fit and hence result in better predictions in the future. Evaluate briefly whether this is a good idea. (1 mark)
Question / Answer Booklet
Spare answer page
Please use the blank space below if you run out of space inside the answer booklet. Clearly state the number of the question to which your work here relates.
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Question / Answer Booklet
Spare answer page
Please use the blank space below if you run out of space inside the answer booklet. Clearly state the number of the question to which your work here relates.
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ENGSCI 211
THE UNIVERSITY OF AUCKLAND
SUMMER SCHOOL 2021 Campus: City
ENGINEERING SCIENCE Mathematical Modelling 2
TEST APPENDICES
APPENDIX A: Orange Juice – One-sample Analysis APPENDIX B: Orange Juice – Regression Analysis Formula Sheet
Page 2 4 7
Page 1 of 8
APPENDIX ENGSCI 211 APPENDIX A: Orange Juice – One-sample Analysis
The following description applies for the data used in both Appendix A and B.
A manufacturer of orange juice has developed a subjective scale for evaluating the sweetness of their orange juice. It is of interest to determine the relationship, if any, between the amount of pectin (a gelling agent found in fruit) detected in the orange juice, and how sweet the orange juice was found to be.
Sweetness subjective sweetness score, as evaluated by professional orange juice tasters Pectin amount of pectin detected in the juice, in mg/L
Reference:
McClave, J.T. et al. (2000). A First Course in Business Statistics, 8e. Upper Saddle River, NJ: .
library(s20x)
oj.df = read.table(“OJuice.txt”, header = TRUE)
boxplot(oj.df$Pectin)
normcheck(oj.df$Pectin)
Theoretical Quantiles
oj.df$Pectin
Page 2 of 8
Sample Quantiles
250 300 350 400
ENGSCI 211
normcheck(log(oj.df$Pectin))
Theoretical Quantiles
t.test(log(oj.df$Pectin), mu = log(240))
log(oj.df$Pectin)
## One Sample t-test
## data: log(oj.df$Pectin)
## t = 1.5438, df = 23, p-value = 0.1363
## alternative hypothesis: true mean is not equal to 5.480639
## 95 percent confidence interval:
## 5.462502 5.605470
## sample estimates:
## mean of x
## 5.533986
ci = t.test(log(oj.df$Pectin), mu = log(240))$conf.int
## [1] 235.6865 271.9098
## attr(,”conf.level”)
## [1] 0.95
Page 3 of 8
Sample Quantiles
APPENDIX ENGSCI 211 APPENDIX B: Orange Juice – Regression Analysis plot(Sweetness ~ Pectin, data = oj.df)
oj.lm = lm(Sweetness ~ Pectin, data = oj.df)
modcheck(oj.lm)
Fitted values
Theoretical Quantiles
Cook’s Distance plot
Residuals from lm(Sweetness ~ Pectin)
Page 4 of 8
Cook’s distance
0.00 0.05 0.10 0.15 0.20 0.25
Sample Quantiles
Residuals Sweetness
5.2 5.4 5.6 5.8 6.0
ENGSCI 211
summary(oj.lm)
## lm(formula = Sweetness ~ Pectin, data = oj.df)
## Residuals:
## Min 1Q Median 3Q Max
## -0.54373 -0.11039 0.06089 0.13432 0.34638
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.2520679 0.2366220 26.422 <2e-16 ***
## Pectin -0.0023106 0.0009049 -2.554 0.0181 *
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Residual standard error: 0.215 on 22 degrees of freedom
## Multiple R-squared: 0.2286, Adjusted R-squared: 0.1936
## F-statistic: 6.52 on 1 and 22 DF, p-value: 0.01811
confint(oj.lm)
## 2.5 % 97.5 %
## (Intercept) 5.761344017 6.742791801
## Pectin -0.004187233 -0.000434019
summary(oj.df$Pectin)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 210.0 226.2 243.5 257.0 265.0 410.0
pred.df = data.frame(Pectin = c(100, 300))
predict(oj.lm, pred.df, interval = "confidence")
## fit lwr upr
## 1 6.021005 5.712715 6.329295
## 2 5.558880 5.437193 5.680568
predict(oj.lm, pred.df, interval = "prediction")
## fit lwr upr
## 1 6.021005 5.478925 6.563085
## 2 5.558880 5.096694 6.021066
Page 5 of 8
APPENDIX ENGSCI 211
This page has been intentionally left blank for your rough working. It will not be marked.
Page 6 of 8
ENGSCI 211 Formula Sheet
Version: Semester One 2018
Trigonometric identities
Univariate calculus
sin2 A + cos2 A = 1 sec2A=1+tan2A cosec2A=1+cot2A
sin A + sin B = 2 sin A + B cos A − B 22
derivative f′(x)g(x) + f(x)g′(x)
antiderivative xn+1 ,n ̸= −1
ln |f (x)|
ln |sec x|
ln|cosecx−cotx|
= ln tan x 2
ln|secx+tanx| =lntan(π4 +x2) ln |sin x|
1 lna+x, |x|a 2a x+a
1 arctan x aa
function sinA−sinB=2cosA+BsinA−B f(x)g(x)
f′(x)g(x)−f(x)g′(x) , g ̸= 0
2 2 f(x) A+B A−B g(x) cos A + cos B = 2 cos 2 cos 2 xn sin(A ± B)=sinAcosB±cosAsinB 1
ln |x| 1c ecx
ecx 1 f′(x)
cos(A ± B) = cos A cos B ∓ sin A sin B tan(A±B)= tanA±tanB
cos A − cos B = 2 sin A + B sin B − A x
2 2 ln|x| x1
Partial fractions
1 ∓ tanAtanB sin 2A = 2 sin A cos A
cos 2A = cos2 A − sin2 A = 2 cos2 A − 1
= 1 − 2 sin2 A
log x 1 cosx a xlna
2tanA 1 − tan2 A
(x−a)(x−b)(x−c)… x−a x−b x−c
2 sin A cos B = sin(A + B) + sin(A − B) 2 cos A cos B = cos(A + B) + cos(A − B) 2 sin A sin B = cos(A − B) − cos(A + B)
ecx ax sin x cosx tan x cosecx sec x arcsin x arccosx arctan x
cecx axlna, a>0 cos x −sinx sec2 x
tan x cosecx
eix +e−ix 2
eix −e−ix 2i
secx −cosecxcotx cotx
eix =cosx+isinx e−ix =cosx−isinx
= A + B + C +…
1+x2 Integration by Parts:
a2 −x2 √ 1
ln(x+√x2 +a2)
secxtanx 1 a2 −x2
x2 −a2 1 a2+x2
− 1−x2 1√1
arcsin x, |x| < a a
(x − a)(x − b)3
p(x) (x−a)(x2 +bx+c)
Geometric series
n−1 ark=a1−rn
= A + B1 + B2
x − a x − b (x − b)2
= A + Bx+C
+ B3 (x − b)3
lnx+√x2−a2, |x|>a
Numerical methods for ODEs
EulerMethod: f =f(t ,y )
⋆ E IE fn+fn⋆+1
ark=a 1 ,|r|<1 k=0 1−r
u dx dx = u v −
n n n n+1 n n
=y +∆tf ImprovedEulerMethod: fn+1 =f(tn+1,yn+1) yn+1 =yn+∆t 2
Page 1 of 2
ENGSCI 211 Formula Sheet
Version: Semester One 2018
Laplace transforms
L{f(t)}=F(s)= 0t |ω|
sinx x− 3! + 5! − 7! +… x2 x4 x6
cosx 1−2! +4! −6! +… x3 x5 x7
sinhx x+ 3! + 5! + 7! +…
x2 x4 x6 coshx 1+ + + +…
(1 + x)n 1 + nx + n(n−1) x2 + . . . 2!
operation s-domain t-domain 1 t-derivative 2nd t-derivative nth t-derivative time integral periodic function convolution
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