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authorSebastiano Tronto <sebastiano.tronto@gmail.com>2021-05-25 17:10:49 +0200
committerSebastiano Tronto <sebastiano.tronto@gmail.com>2021-05-25 17:10:49 +0200
commitd6c61d988bfa4255baf9cdae42db59ebee38363f (patch)
tree118ff3c2424e735149c145524965a4a337e50beb /src/Homework4
parent46eef66b1e1571c77dc828d7e950b129b4c8bfd0 (diff)
downloadmathsoftware-d6c61d988bfa4255baf9cdae42db59ebee38363f.tar.gz
mathsoftware-d6c61d988bfa4255baf9cdae42db59ebee38363f.zip
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-rw-r--r--src/Homework4/.ipynb_checkpoints/Homework4-checkpoint.ipynb172
-rw-r--r--src/Homework4/.ipynb_checkpoints/Homework4-solutions-checkpoint.ipynb61
-rw-r--r--src/Homework4/.ipynb_checkpoints/Untitled-checkpoint.ipynb6
-rw-r--r--src/Homework4/Homework4.aux20
-rw-r--r--src/Homework4/Homework4.ipynb172
-rw-r--r--src/Homework4/Homework4.log953
-rw-r--r--src/Homework4/Homework4.out0
-rw-r--r--src/Homework4/Homework4.pdfbin0 -> 116266 bytes
-rw-r--r--src/Homework4/Homework4.tex530
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diff --git a/src/Homework4/.ipynb_checkpoints/Homework4-checkpoint.ipynb b/src/Homework4/.ipynb_checkpoints/Homework4-checkpoint.ipynb
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index 0000000..47f63cf
--- /dev/null
+++ b/src/Homework4/.ipynb_checkpoints/Homework4-checkpoint.ipynb
@@ -0,0 +1,172 @@
1{
2 "cells": [
3 {
4 "cell_type": "markdown",
5 "metadata": {},
6 "source": [
7 "*For this exercise you should have received this text in .ipynb format. Complete the exercises by modifying this file, and submit the modified version*\n",
8 "\n",
9 "**Deadline:** Sunday, June 6."
10 ]
11 },
12 {
13 "cell_type": "markdown",
14 "metadata": {},
15 "source": [
16 "**Exercise 1 (6 points)**\n",
17 "\n",
18 "Use Sage to find the intersection points *in the real plane* (that is, only those points such that *both* coordinates are real numbers) of the following pairs of geometric objects:\n",
19 "\n",
20 "* The circle of equation $x^2 + y^2 = 4$ and the ellipse of equation $\\left(\\frac x2\\right)^2 + (2y)^2 = 4$.\n",
21 "* The circle of equation $x^2 + y^2 = 4$ and the ellipse of equation $\\left(\\frac x2-2\\right)^2 + (2y)^2 = 4$.\n",
22 "* The curve of equation $y^2 = x^3 -x +1$ and the horizontal line $y=10$.\n",
23 "* The $x$-axis and the graph of the function $f(x)=\\log(x) - e^{-x}$. *Hint: $f(x)$ has only one real zero.*"
24 ]
25 },
26 {
27 "cell_type": "code",
28 "execution_count": null,
29 "metadata": {},
30 "outputs": [],
31 "source": []
32 },
33 {
34 "cell_type": "markdown",
35 "metadata": {},
36 "source": [
37 "**Exercise 2 (6 points)**\n",
38 "\n",
39 "(a) Use Sage to compute\n",
40 "* the derivative\n",
41 "* a primite (i.e. integral)\n",
42 "* the power series expansion around $0$ up to order $4$\n",
43 "\n",
44 "of the following functions:\n",
45 "* $f(x) = e^x$\n",
46 "* $f(x) = \\sin(x)$\n",
47 "* $f(x) = \\cos(x)$\n",
48 "* $f(x) = \\tan(x)$\n",
49 "* $f(x) = \\log(1+x)$\n",
50 "* $f(x) = \\sqrt[3]{1+x}$\n",
51 "\n",
52 "(b) Use Sage to get the Latex code that represents the objects you computed above.\n",
53 "\n",
54 "(c) Arrange the results of the previous points in a table in Latex. The table should have 4 columns (function, derivative, integral, series) and one row for each of the functions above. *Note: when including Latex in a Markdown cell in Jupyter you will not receive any warning if you make mistakes; instead the Latex will simply not be rendered and it will appear as plain text. If you have troubles making this work you can send me a separate .tex (and .pdf) file.*"
55 ]
56 },
57 {
58 "cell_type": "code",
59 "execution_count": null,
60 "metadata": {},
61 "outputs": [],
62 "source": [
63 "# Compute derivatives etc..."
64 ]
65 },
66 {
67 "cell_type": "code",
68 "execution_count": null,
69 "metadata": {},
70 "outputs": [],
71 "source": [
72 "# Compute Latex code"
73 ]
74 },
75 {
76 "cell_type": "markdown",
77 "metadata": {},
78 "source": [
79 "(Write your table here)"
80 ]
81 },
82 {
83 "cell_type": "markdown",
84 "metadata": {},
85 "source": [
86 "**Exercise 3 (4 points)**\n",
87 "\n",
88 "The equation\n",
89 "\\begin{align*}\n",
90 "y^2+x^{16}=1\n",
91 "\\end{align*}\n",
92 "determines a closed curve in $\\mathbb R^2$ that looks like a rounded square. Determine the area of that shape, giving both an exact value (which might depend on some functions that Sage knows, but you don't) and an approximate value."
93 ]
94 },
95 {
96 "cell_type": "code",
97 "execution_count": null,
98 "metadata": {},
99 "outputs": [],
100 "source": []
101 },
102 {
103 "cell_type": "markdown",
104 "metadata": {},
105 "source": [
106 "**Exercise 4 (12 points)**\n",
107 "\n",
108 "A team of biologists is monitoring the population of river shrimps in the Alzette. At first they thought that the size $P(t)$ of their population on day $t$ would satisfy the differential equation $P'(t)=P(t)/10$. However this does not work well with the data they have collected, so they now believe that the population of shrimps follows the formula $P'(t)=P(t)/10-b$ for some value of $b$ between 1 and 100. They need your help here.\n",
109 "\n",
110 "(a) Using Sage, find a solution for the differential equation with initial conditions\n",
111 "\\begin{align*}\n",
112 "\\begin{cases}\n",
113 "P'(t)&=\\frac{P(t)}{10}-b\\\\\n",
114 "P(1)&=1000\n",
115 "\\end{cases}\n",
116 "\\end{align*}\n",
117 "where $b$ is a generic constant.\n",
118 "\n",
119 "(b) The list `data` in the cell below contains the actual number of shrimps that was measured every day from day $1$ (the $0$ at the beginning is meaningless, but it will help to keep it there). Plot in one single picture, possibly using different colors for each:\n",
120 "* The data as a bar chart.\n",
121 "* A curve that interpolates the data, using one of the methods shown in class.\n",
122 "* The solution of the differential equation for $b=0$.\n",
123 "* The solution of the differential equation for a value of $b$ of your choice ($1\\leq b\\leq 100$) that fits the data better than $b=0$. *(For this last point there is no right or wrong choice, just pick one that looks good)*"
124 ]
125 },
126 {
127 "cell_type": "code",
128 "execution_count": null,
129 "metadata": {},
130 "outputs": [],
131 "source": [
132 "data = [0, 1000, 1123, 1223, 1190, 1432, 1553, 1709, 1826, 1980, 2146, 2172, 2383, 2588, 2822, 3401, 3330, 4157, 3994, 4995, 5392, 5910, 6468, 7128, 7325, 7984, 9634, 10473, 11761, 12777]\n"
133 ]
134 },
135 {
136 "cell_type": "markdown",
137 "metadata": {},
138 "source": [
139 "**Grading**\n",
140 "\n",
141 "This homework assignment is worth $28$ ($24+4$) points, distributed as described above.\n",
142 "\n",
143 "Your final grade for the course will be the total of points you obtained (notice that the maximum is $20+20+16+28=84$) divided by $4$, rounded to the nearest integer. More precisely\n",
144 "\n",
145 "\\begin{align*}\n",
146 "\\operatorname{grade} = \\operatorname{min}\\left(20, \\left\\lfloor \\frac{\\operatorname{total}}{4} + 0.5\\right\\rfloor\\right)\n",
147 "\\end{align*}"
148 ]
149 }
150 ],
151 "metadata": {
152 "kernelspec": {
153 "display_name": "SageMath 9.2",
154 "language": "sage",
155 "name": "sagemath"
156 },
157 "language_info": {
158 "codemirror_mode": {
159 "name": "ipython",
160 "version": 3
161 },
162 "file_extension": ".py",
163 "mimetype": "text/x-python",
164 "name": "python",
165 "nbconvert_exporter": "python",
166 "pygments_lexer": "ipython3",
167 "version": "3.8.5"
168 }
169 },
170 "nbformat": 4,
171 "nbformat_minor": 4
172}
diff --git a/src/Homework4/.ipynb_checkpoints/Homework4-solutions-checkpoint.ipynb b/src/Homework4/.ipynb_checkpoints/Homework4-solutions-checkpoint.ipynb
new file mode 100644
index 0000000..f8617fd
--- /dev/null
+++ b/src/Homework4/.ipynb_checkpoints/Homework4-solutions-checkpoint.ipynb
@@ -0,0 +1,61 @@
1{
2 "cells": [
3 {
4 "cell_type": "code",
5 "execution_count": 40,
6 "metadata": {},
7 "outputs": [
8 {
9 "name": "stdout",
10 "output_type": "stream",
11 "text": [
12 "f(x) \t Derivative \t Integral \t Series expansion\n",
13 "e^x \t e^x \t e^x \t 1 + 1*x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + Order(x^5)\n",
14 "sin(x) \t cos(x) \t -cos(x) \t 1*x + (-1/6)*x^3 + Order(x^5)\n",
15 "cos(x) \t -sin(x) \t sin(x) \t 1 + (-1/2)*x^2 + 1/24*x^4 + Order(x^5)\n",
16 "tan(x) \t tan(x)^2 + 1 \t log(sec(x)) \t 1*x + 1/3*x^3 + Order(x^5)\n",
17 "log(x + 1) \t 1/(x + 1) \t (x + 1)*log(x + 1) - x - 1 \t 1*x + (-1/2)*x^2 + 1/3*x^3 + (-1/4)*x^4 + Order(x^5)\n",
18 "(x + 1)^a \t a*(x + 1)^(a - 1) \t (x + 1)^(a + 1)/(a + 1) \t 1 + (a)*x + (1/2*a^2 - 1/2*a)*x^2 + (1/6*a^3 - 1/2*a^2 + 1/3*a)*x^3 + (1/24*a^4 - 1/4*a^3 + 11/24*a^2 - 1/4*a)*x^4 + Order(x^5)\n"
19 ]
20 }
21 ],
22 "source": [
23 "print(\"f(x)\", \"\\t\", \"Derivative\", \"\\t\", \"Integral\", \"\\t\", \"Series expansion\")\n",
24 "var('a')\n",
25 "f(x) = (1+x)^a\n",
26 "assume(a>0)\n",
27 "functions = [e^x, sin, cos, tan, log(1+x), f]\n",
28 "for g in functions:\n",
29 " print(g(x), \"\\t\", derivative(g(x), x), \"\\t\", integral(g(x),x), \"\\t\", g(x).series(x==0,5))"
30 ]
31 },
32 {
33 "cell_type": "code",
34 "execution_count": null,
35 "metadata": {},
36 "outputs": [],
37 "source": []
38 }
39 ],
40 "metadata": {
41 "kernelspec": {
42 "display_name": "SageMath 9.2",
43 "language": "sage",
44 "name": "sagemath"
45 },
46 "language_info": {
47 "codemirror_mode": {
48 "name": "ipython",
49 "version": 3
50 },
51 "file_extension": ".py",
52 "mimetype": "text/x-python",
53 "name": "python",
54 "nbconvert_exporter": "python",
55 "pygments_lexer": "ipython3",
56 "version": "3.8.5"
57 }
58 },
59 "nbformat": 4,
60 "nbformat_minor": 4
61}
diff --git a/src/Homework4/.ipynb_checkpoints/Untitled-checkpoint.ipynb b/src/Homework4/.ipynb_checkpoints/Untitled-checkpoint.ipynb
new file mode 100644
index 0000000..7fec515
--- /dev/null
+++ b/src/Homework4/.ipynb_checkpoints/Untitled-checkpoint.ipynb
@@ -0,0 +1,6 @@
1{
2 "cells": [],
3 "metadata": {},
4 "nbformat": 4,
5 "nbformat_minor": 4
6}
diff --git a/src/Homework4/Homework4.aux b/src/Homework4/Homework4.aux
new file mode 100644
index 0000000..a74227c
--- /dev/null
+++ b/src/Homework4/Homework4.aux
@@ -0,0 +1,20 @@
1\relax
2\providecommand\hyper@newdestlabel[2]{}
3\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
4\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
5\global\let\oldcontentsline\contentsline
6\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
7\global\let\oldnewlabel\newlabel
8\gdef\newlabel#1#2{\newlabelxx{#1}#2}
9\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
10\AtEndDocument{\ifx\hyper@anchor\@undefined
11\let\contentsline\oldcontentsline
12\let\newlabel\oldnewlabel
13\fi}
14\fi}
15\global\let\hyper@last\relax
16\gdef\HyperFirstAtBeginDocument#1{#1}
17\providecommand\HyField@AuxAddToFields[1]{}
18\providecommand\HyField@AuxAddToCoFields[2]{}
19\providecommand \oddpage@label [2]{}
20\gdef \@abspage@last{2}
diff --git a/src/Homework4/Homework4.ipynb b/src/Homework4/Homework4.ipynb
new file mode 100644
index 0000000..99481b9
--- /dev/null
+++ b/src/Homework4/Homework4.ipynb
@@ -0,0 +1,172 @@
1{
2 "cells": [
3 {
4 "cell_type": "markdown",
5 "metadata": {},
6 "source": [
7 "*For this exercise you should have received this text in .ipynb format. Complete the exercises by modifying this file, and submit the modified version*\n",
8 "\n",
9 "**Deadline:** Sunday, June 6."
10 ]
11 },
12 {
13 "cell_type": "markdown",
14 "metadata": {},
15 "source": [
16 "**Exercise 1 (6 points)**\n",
17 "\n",
18 "Use Sage to find the intersection points *in the real plane* (that is, only those points such that *both* coordinates are real numbers) of the following pairs of geometric objects:\n",
19 "\n",
20 "* The circle of equation $x^2 + y^2 = 4$ and the ellipse of equation $\\left(\\frac x2\\right)^2 + (2y)^2 = 4$.\n",
21 "* The circle of equation $x^2 + y^2 = 4$ and the ellipse of equation $\\left(\\frac x2-2\\right)^2 + (2y)^2 = 4$.\n",
22 "* The curve of equation $y^2 = x^3 -x +1$ and the horizontal line $y=10$.\n",
23 "* The $x$-axis and the graph of the function $f(x)=\\log(x) - e^{-x}$. *Hint: $f(x)$ has only one real zero.*"
24 ]
25 },
26 {
27 "cell_type": "code",
28 "execution_count": null,
29 "metadata": {},
30 "outputs": [],
31 "source": []
32 },
33 {
34 "cell_type": "markdown",
35 "metadata": {},
36 "source": [
37 "**Exercise 2 (6 points)**\n",
38 "\n",
39 "(a) Use Sage to compute\n",
40 "* the derivative\n",
41 "* a primite (i.e. integral)\n",
42 "* the power series expansion around $0$ up to order $4$\n",
43 "\n",
44 "of the following functions:\n",
45 "* $f(x) = e^x$\n",
46 "* $f(x) = \\sin(x)$\n",
47 "* $f(x) = \\cos(x)$\n",
48 "* $f(x) = \\tan(x)$\n",
49 "* $f(x) = \\log(1+x)$\n",
50 "* $f(x) = \\sqrt[3]{1+x}$\n",
51 "\n",
52 "(b) Use Sage to get the Latex code that represents the objects you computed above.\n",
53 "\n",
54 "(c) Arrange the results of the previous points in a table in Latex. The table should have 4 columns (function, derivative, integral, series) and one row for each of the functions above. *Note: when including Latex in a Markdown cell in Jupyter you will not receive any warning if you make mistakes; instead the Latex will simply not be rendered and it will appear as plain text. If you have troubles making this work you can send me a separate .tex (and .pdf) file.*"
55 ]
56 },
57 {
58 "cell_type": "code",
59 "execution_count": null,
60 "metadata": {},
61 "outputs": [],
62 "source": [
63 "# Compute derivatives etc..."
64 ]
65 },
66 {
67 "cell_type": "code",
68 "execution_count": null,
69 "metadata": {},
70 "outputs": [],
71 "source": [
72 "# Compute Latex code"
73 ]
74 },
75 {
76 "cell_type": "markdown",
77 "metadata": {},
78 "source": [
79 "(Write your table here)"
80 ]
81 },
82 {
83 "cell_type": "markdown",
84 "metadata": {},
85 "source": [
86 "**Exercise 3 (4 points)**\n",
87 "\n",
88 "The equation\n",
89 "\\begin{align*}\n",
90 "y^2+x^{16}=1\n",
91 "\\end{align*}\n",
92 "determines a closed curve in $\\mathbb R^2$ that looks like a rounded square. Determine the area of that shape, giving both an exact value (which might depend on some functions that Sage knows, but you don't) and an approximate value."
93 ]
94 },
95 {
96 "cell_type": "code",
97 "execution_count": null,
98 "metadata": {},
99 "outputs": [],
100 "source": []
101 },
102 {
103 "cell_type": "markdown",
104 "metadata": {},
105 "source": [
106 "**Exercise 4 (12 points)**\n",
107 "\n",
108 "A team of biologists is monitoring the population of river shrimps in the Alzette. At first they thought that the size $P(t)$ of their population on day $t$ would satisfy the differential equation $P'(t)=P(t)/10$. However this does not work well with the data they have collected, so they now believe that the population of shrimps follows the formula $P'(t)=P(t)/10-b$ for some value of $b$ between 1 and 100. They need your help here.\n",
109 "\n",
110 "(a) Using Sage, find a solution for the differential equation with initial conditions\n",
111 "\\begin{align*}\n",
112 "\\begin{cases}\n",
113 "P'(t)&=\\frac{P(t)}{10}-b\\\\\n",
114 "P(1)&=1000\n",
115 "\\end{cases}\n",
116 "\\end{align*}\n",
117 "where $b$ is a generic constant.\n",
118 "\n",
119 "(b) The list `data` in the cell below contains the actual number of shrimps that was measured every day from day $1$ (the $0$ at the beginning is meaningless, but it will help to keep it there). Plot in one single picture, possibly using different colors for each:\n",
120 "* The data as a bar chart.\n",
121 "* A curve that interpolates the data, using one of the methods shown in class.\n",
122 "* The solution of the differential equation for $b=0$.\n",
123 "* The solution of the differential equation for a value of $b$ of your choice ($1\\leq b\\leq 100$) that fits the data better than $b=0$. *(For this last point there is no right or wrong choice, just pick one that looks good)*"
124 ]
125 },
126 {
127 "cell_type": "code",
128 "execution_count": null,
129 "metadata": {},
130 "outputs": [],
131 "source": [
132 "data = [0, 1000, 1123, 1223, 1190, 1432, 1553, 1709, 1826, 1980, 2146, 2172, 2383, 2588, 2822, 3401, 3330, 4157, 3994, 4995, 5392, 5910, 6468, 7128, 7325, 7984, 9634, 10473, 11761, 12777]\n"
133 ]
134 },
135 {
136 "cell_type": "markdown",
137 "metadata": {},
138 "source": [
139 "**Grading**\n",
140 "\n",
141 "This homework assignment is worth $28$ ($24+4$) points, distributed as described above.\n",
142 "\n",
143 "Your final grade for the course will be the total of points you obtained (notice that the maximum is $20+20+16+28=84$) divided by $4$, rounded to the nearest integer. More precisely\n",
144 "\n",
145 "\\begin{align*}\n",
146 "\\operatorname{grade} = \\operatorname{min}\\left(20, \\left\\lfloor \\frac{\\operatorname{total}}{4} + 0.5\\right\\rfloor\\right)\n",
147 "\\end{align*}"
148 ]
149 }
150 ],
151 "metadata": {
152 "kernelspec": {
153 "display_name": "SageMath 9.0",
154 "language": "sage",
155 "name": "sagemath"
156 },
157 "language_info": {
158 "codemirror_mode": {
159 "name": "ipython",
160 "version": 3
161 },
162 "file_extension": ".py",
163 "mimetype": "text/x-python",
164 "name": "python",
165 "nbconvert_exporter": "python",
166 "pygments_lexer": "ipython3",
167 "version": "3.8.5"
168 }
169 },
170 "nbformat": 4,
171 "nbformat_minor": 4
172}
diff --git a/src/Homework4/Homework4.log b/src/Homework4/Homework4.log
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2entering extended mode
3 \write18 enabled.
4 %&-line parsing enabled.
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1\documentclass[11pt]{article}
2
3 \usepackage[breakable]{tcolorbox}
4 \usepackage{parskip} % Stop auto-indenting (to mimic markdown behaviour)
5
6 \usepackage{iftex}
7 \ifPDFTeX
8 \usepackage[T1]{fontenc}
9 \usepackage{mathpazo}
10 \else
11 \usepackage{fontspec}
12 \fi
13
14 % Basic figure setup, for now with no caption control since it's done
15 % automatically by Pandoc (which extracts ![](path) syntax from Markdown).
16 \usepackage{graphicx}
17 % Maintain compatibility with old templates. Remove in nbconvert 6.0
18 \let\Oldincludegraphics\includegraphics
19 % Ensure that by default, figures have no caption (until we provide a
20 % proper Figure object with a Caption API and a way to capture that
21 % in the conversion process - todo).
22 \usepackage{caption}
23 \DeclareCaptionFormat{nocaption}{}
24 \captionsetup{format=nocaption,aboveskip=0pt,belowskip=0pt}
25
26 \usepackage[Export]{adjustbox} % Used to constrain images to a maximum size
27 \adjustboxset{max size={0.9\linewidth}{0.9\paperheight}}
28 \usepackage{float}
29 \floatplacement{figure}{H} % forces figures to be placed at the correct location
30 \usepackage{xcolor} % Allow colors to be defined
31 \usepackage{enumerate} % Needed for markdown enumerations to work
32 \usepackage{geometry} % Used to adjust the document margins
33 \usepackage{amsmath} % Equations
34 \usepackage{amssymb} % Equations
35 \usepackage{textcomp} % defines textquotesingle
36 % Hack from http://tex.stackexchange.com/a/47451/13684:
37 \AtBeginDocument{%
38 \def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
39 }
40 \usepackage{upquote} % Upright quotes for verbatim code
41 \usepackage{eurosym} % defines \euro
42 \usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
43 \usepackage{fancyvrb} % verbatim replacement that allows latex
44 \usepackage{grffile} % extends the file name processing of package graphics
45 % to support a larger range
46 \makeatletter % fix for grffile with XeLaTeX
47 \def\Gread@@xetex#1{%
48 \IfFileExists{"\Gin@base".bb}%
49 {\Gread@eps{\Gin@base.bb}}%
50 {\Gread@@xetex@aux#1}%
51 }
52 \makeatother
53
54 % The hyperref package gives us a pdf with properly built
55 % internal navigation ('pdf bookmarks' for the table of contents,
56 % internal cross-reference links, web links for URLs, etc.)
57 \usepackage{hyperref}
58 % The default LaTeX title has an obnoxious amount of whitespace. By default,
59 % titling removes some of it. It also provides customization options.
60 \usepackage{titling}
61 \usepackage{longtable} % longtable support required by pandoc >1.10
62 \usepackage{booktabs} % table support for pandoc > 1.12.2
63 \usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
64 \usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
65 % normalem makes italics be italics, not underlines
66 \usepackage{mathrsfs}
67
68
69
70 % Colors for the hyperref package
71 \definecolor{urlcolor}{rgb}{0,.145,.698}
72 \definecolor{linkcolor}{rgb}{.71,0.21,0.01}
73 \definecolor{citecolor}{rgb}{.12,.54,.11}
74
75 % ANSI colors
76 \definecolor{ansi-black}{HTML}{3E424D}
77 \definecolor{ansi-black-intense}{HTML}{282C36}
78 \definecolor{ansi-red}{HTML}{E75C58}
79 \definecolor{ansi-red-intense}{HTML}{B22B31}
80 \definecolor{ansi-green}{HTML}{00A250}
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82 \definecolor{ansi-yellow}{HTML}{DDB62B}
83 \definecolor{ansi-yellow-intense}{HTML}{B27D12}
84 \definecolor{ansi-blue}{HTML}{208FFB}
85 \definecolor{ansi-blue-intense}{HTML}{0065CA}
86 \definecolor{ansi-magenta}{HTML}{D160C4}
87 \definecolor{ansi-magenta-intense}{HTML}{A03196}
88 \definecolor{ansi-cyan}{HTML}{60C6C8}
89 \definecolor{ansi-cyan-intense}{HTML}{258F8F}
90 \definecolor{ansi-white}{HTML}{C5C1B4}
91 \definecolor{ansi-white-intense}{HTML}{A1A6B2}
92 \definecolor{ansi-default-inverse-fg}{HTML}{FFFFFF}
93 \definecolor{ansi-default-inverse-bg}{HTML}{000000}
94
95 % commands and environments needed by pandoc snippets
96 % extracted from the output of `pandoc -s`
97 \providecommand{\tightlist}{%
98 \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
99 \DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
100 % Add ',fontsize=\small' for more characters per line
101 \newenvironment{Shaded}{}{}
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103 \newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}}
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115 \newcommand{\NormalTok}[1]{{#1}}
116
117 % Additional commands for more recent versions of Pandoc
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134 \newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
135
136
137 % Define a nice break command that doesn't care if a line doesn't already
138 % exist.
139 \def\br{\hspace*{\fill} \\* }
140 % Math Jax compatibility definitions
141 \def\gt{>}
142 \def\lt{<}
143 \let\Oldtex\TeX
144 \let\Oldlatex\LaTeX
145 \renewcommand{\TeX}{\textrm{\Oldtex}}
146 \renewcommand{\LaTeX}{\textrm{\Oldlatex}}
147 % Document parameters
148 % Document title
149 \title{Mathematical Software - Homework 4}
150 \date{\textbf{Deadline:} Sunday, June 6}
151
152
153
154
155
156% Pygments definitions
157\makeatletter
158\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax%
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235
236\def\PYZbs{\char`\\}
237\def\PYZus{\char`\_}
238\def\PYZob{\char`\{}
239\def\PYZcb{\char`\}}
240\def\PYZca{\char`\^}
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242\def\PYZlt{\char`\<}
243\def\PYZgt{\char`\>}
244\def\PYZsh{\char`\#}
245\def\PYZpc{\char`\%}
246\def\PYZdl{\char`\$}
247\def\PYZhy{\char`\-}
248\def\PYZsq{\char`\'}
249\def\PYZdq{\char`\"}
250\def\PYZti{\char`\~}
251% for compatibility with earlier versions
252\def\PYZat{@}
253\def\PYZlb{[}
254\def\PYZrb{]}
255\makeatother
256
257
258 % For linebreaks inside Verbatim environment from package fancyvrb.
259 \makeatletter
260 \newbox\Wrappedcontinuationbox
261 \newbox\Wrappedvisiblespacebox
262 \newcommand*\Wrappedvisiblespace {\textcolor{red}{\textvisiblespace}}
263 \newcommand*\Wrappedcontinuationsymbol {\textcolor{red}{\llap{\tiny$\m@th\hookrightarrow$}}}
264 \newcommand*\Wrappedcontinuationindent {3ex }
265 \newcommand*\Wrappedafterbreak {\kern\Wrappedcontinuationindent\copy\Wrappedcontinuationbox}
266 % Take advantage of the already applied Pygments mark-up to insert
267 % potential linebreaks for TeX processing.
268 % {, <, #, %, $, ' and ": go to next line.
269 % _, }, ^, &, >, - and ~: stay at end of broken line.
270 % Use of \textquotesingle for straight quote.
271 \newcommand*\Wrappedbreaksatspecials {%
272 \def\PYGZus{\discretionary{\char`\_}{\Wrappedafterbreak}{\char`\_}}%
273 \def\PYGZob{\discretionary{}{\Wrappedafterbreak\char`\{}{\char`\{}}%
274 \def\PYGZcb{\discretionary{\char`\}}{\Wrappedafterbreak}{\char`\}}}%
275 \def\PYGZca{\discretionary{\char`\^}{\Wrappedafterbreak}{\char`\^}}%
276 \def\PYGZam{\discretionary{\char`\&}{\Wrappedafterbreak}{\char`\&}}%
277 \def\PYGZlt{\discretionary{}{\Wrappedafterbreak\char`\<}{\char`\<}}%
278 \def\PYGZgt{\discretionary{\char`\>}{\Wrappedafterbreak}{\char`\>}}%
279 \def\PYGZsh{\discretionary{}{\Wrappedafterbreak\char`\#}{\char`\#}}%
280 \def\PYGZpc{\discretionary{}{\Wrappedafterbreak\char`\%}{\char`\%}}%
281 \def\PYGZdl{\discretionary{}{\Wrappedafterbreak\char`\$}{\char`\$}}%
282 \def\PYGZhy{\discretionary{\char`\-}{\Wrappedafterbreak}{\char`\-}}%
283 \def\PYGZsq{\discretionary{}{\Wrappedafterbreak\textquotesingle}{\textquotesingle}}%
284 \def\PYGZdq{\discretionary{}{\Wrappedafterbreak\char`\"}{\char`\"}}%
285 \def\PYGZti{\discretionary{\char`\~}{\Wrappedafterbreak}{\char`\~}}%
286 }
287 % Some characters . , ; ? ! / are not pygmentized.
288 % This macro makes them "active" and they will insert potential linebreaks
289 \newcommand*\Wrappedbreaksatpunct {%
290 \lccode`\~`\.\lowercase{\def~}{\discretionary{\hbox{\char`\.}}{\Wrappedafterbreak}{\hbox{\char`\.}}}%
291 \lccode`\~`\,\lowercase{\def~}{\discretionary{\hbox{\char`\,}}{\Wrappedafterbreak}{\hbox{\char`\,}}}%
292 \lccode`\~`\;\lowercase{\def~}{\discretionary{\hbox{\char`\;}}{\Wrappedafterbreak}{\hbox{\char`\;}}}%
293 \lccode`\~`\:\lowercase{\def~}{\discretionary{\hbox{\char`\:}}{\Wrappedafterbreak}{\hbox{\char`\:}}}%
294 \lccode`\~`\?\lowercase{\def~}{\discretionary{\hbox{\char`\?}}{\Wrappedafterbreak}{\hbox{\char`\?}}}%
295 \lccode`\~`\!\lowercase{\def~}{\discretionary{\hbox{\char`\!}}{\Wrappedafterbreak}{\hbox{\char`\!}}}%
296 \lccode`\~`\/\lowercase{\def~}{\discretionary{\hbox{\char`\/}}{\Wrappedafterbreak}{\hbox{\char`\/}}}%
297 \catcode`\.\active
298 \catcode`\,\active
299 \catcode`\;\active
300 \catcode`\:\active
301 \catcode`\?\active
302 \catcode`\!\active
303 \catcode`\/\active
304 \lccode`\~`\~
305 }
306 \makeatother
307
308 \let\OriginalVerbatim=\Verbatim
309 \makeatletter
310 \renewcommand{\Verbatim}[1][1]{%
311 %\parskip\z@skip
312 \sbox\Wrappedcontinuationbox {\Wrappedcontinuationsymbol}%
313 \sbox\Wrappedvisiblespacebox {\FV@SetupFont\Wrappedvisiblespace}%
314 \def\FancyVerbFormatLine ##1{\hsize\linewidth
315 \vtop{\raggedright\hyphenpenalty\z@\exhyphenpenalty\z@
316 \doublehyphendemerits\z@\finalhyphendemerits\z@
317 \strut ##1\strut}%
318 }%
319 % If the linebreak is at a space, the latter will be displayed as visible
320 % space at end of first line, and a continuation symbol starts next line.
321 % Stretch/shrink are however usually zero for typewriter font.
322 \def\FV@Space {%
323 \nobreak\hskip\z@ plus\fontdimen3\font minus\fontdimen4\font
324 \discretionary{\copy\Wrappedvisiblespacebox}{\Wrappedafterbreak}
325 {\kern\fontdimen2\font}%
326 }%
327
328 % Allow breaks at special characters using \PYG... macros.
329 \Wrappedbreaksatspecials
330 % Breaks at punctuation characters . , ; ? ! and / need catcode=\active
331 \OriginalVerbatim[#1,codes*=\Wrappedbreaksatpunct]%
332 }
333 \makeatother
334
335 % Exact colors from NB
336 \definecolor{incolor}{HTML}{303F9F}
337 \definecolor{outcolor}{HTML}{D84315}
338 \definecolor{cellborder}{HTML}{CFCFCF}
339 \definecolor{cellbackground}{HTML}{F7F7F7}
340
341 % prompt
342 \makeatletter
343 \newcommand{\boxspacing}{\kern\kvtcb@left@rule\kern\kvtcb@boxsep}
344 \makeatother
345 \newcommand{\prompt}[4]{
346 \ttfamily\llap{{\color{#2}[#3]:\hspace{3pt}#4}}\vspace{-\baselineskip}
347 }
348
349
350
351 % Prevent overflowing lines due to hard-to-break entities
352 \sloppy
353 % Setup hyperref package
354 \hypersetup{
355 breaklinks=true, % so long urls are correctly broken across lines
356 colorlinks=true,
357 urlcolor=urlcolor,
358 linkcolor=linkcolor,
359 citecolor=citecolor,
360 }
361 % Slightly bigger margins than the latex defaults
362
363 \geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
364
365
366
367\begin{document}
368
369 \maketitle
370
371
372
373
374 \emph{For this exercise you should have received this text in .ipynb
375format. Complete the exercises by modifying this file, and submit the
376modified version}
377
378 \textbf{Exercise 1 (6 points)}
379
380Use Sage to find the intersection points \emph{in the real plane} (that
381is, only those points such that \emph{both} coordinates are real
382numbers) of the following pairs of geometric objects:
383
384\begin{itemize}
385\tightlist
386\item
387 The circle of equation \(x^2 + y^2 = 4\) and the ellipse of equation
388 \(\left(\frac x2\right)^2 + (2y)^2 = 4\).
389\item
390 The circle of equation \(x^2 + y^2 = 4\) and the ellipse of equation
391 \(\left(\frac x2-2\right)^2 + (2y)^2 = 4\).
392\item
393 The curve of equation \(y^2 = x^3 -x +1\) and the horizontal line
394 \(y=10\).
395\item
396 The \(x\)-axis and the graph of the function
397 \(f(x)=\log(x) - e^{-x}\). \emph{Hint: \(f(x)\) has only one real
398 zero.}
399\end{itemize}
400
401
402\vspace{0.5cm}
403 \textbf{Exercise 2 (6 points)}
404
405\begin{enumerate}
406\def\labelenumi{(\alph{enumi})}
407\tightlist
408\item
409 Use Sage to compute
410\end{enumerate}
411
412\begin{itemize}
413\tightlist
414\item
415 the derivative
416\item
417 a primite (i.e.~integral)
418\item
419 the power series expansion around \(0\) up to order \(4\)
420\end{itemize}
421
422of the following functions:
423\begin{itemize}
424\item \(f(x) = e^x\)
425\item \(f(x) = \sin(x)\)
426\item \(f(x) = \cos(x)\)
427\item \(f(x) = \tan(x)\)
428\item \(f(x) = \log(1+x)\)
429\item \(f(x) = \sqrt[3]{1+x}\)
430\end{itemize}
431
432\begin{enumerate}
433\def\labelenumi{(\alph{enumi})}
434\setcounter{enumi}{1}
435\item
436 Use Sage to get the Latex code that represents the objects you
437 computed above.
438\item
439 Arrange the results of the previous points in a table in Latex. The
440 table should have 4 columns (function, derivative, integral, series)
441 and one row for each of the functions above. \emph{Note: when
442 including Latex in a Markdown cell in Jupyter you will not receive any
443 warning if you make mistakes; instead the Latex will simply not be
444 rendered and it will appear as plain text. If you have troubles making
445 this work you can send me a separate .tex (and .pdf) file.}
446\end{enumerate}
447
448
449\vspace{0.5cm}
450 \textbf{Exercise 3 (4 points)}
451
452The equation \begin{align*}
453y^2+x^{16}=1
454\end{align*} determines a closed curve in \(\mathbb R^2\) that looks
455like a rounded square. Determine the area of that shape, giving both an
456exact value (which might depend on some functions that Sage knows, but
457you don't) and an approximate value.
458
459
460\vspace{0.5cm}
461 \textbf{Exercise 4 (12 points)}
462
463A team of biologists is monitoring the population of river shrimps in
464the Alzette. At first they thought that the size \(P(t)\) of their
465population on day \(t\) would satisfy the differential equation
466\(P'(t)=P(t)/10\). However this does not work well with the data they
467have collected, so they now believe that the population of shrimps
468follows the formula \(P'(t)=P(t)/10-b\) for some value of \(b\) between
4691 and 100. They need your help here.
470
471\begin{enumerate}
472\def\labelenumi{(\alph{enumi})}
473\item
474 Using Sage, find a solution for the differential equation with initial
475 conditions \begin{align*}
476 \begin{cases}
477 P'(t)&=\frac{P(t)}{10}-b\\
478 P(1)&=1000
479 \end{cases}
480 \end{align*} where \(b\) is a generic constant.
481\item
482 The list \texttt{data} in the cell below contains the actual number of
483 shrimps that was measured every day from day \(1\) (the \(0\) at the
484 beginning is meaningless, but it will help to keep it there). Plot in
485 one single picture, possibly using different colors for each:
486\end{enumerate}
487
488\begin{itemize}
489\tightlist
490\item
491 The data as a bar chart.
492\item
493 A curve that interpolates the data, using one of the methods shown in
494 class.
495\item
496 The solution of the differential equation for \(b=0\).
497\item
498 The solution of the differential equation for a value of \(b\) of your
499 choice (\(1\leq b\leq 100\)) that fits the data better than \(b=0\).
500 \emph{(For this last point there is no right or wrong choice, just
501 pick one that looks good)}
502\end{itemize}
503
504 \begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
505\prompt{In}{incolor}{ }{\boxspacing}
506\begin{Verbatim}[commandchars=\\\{\}]
507\PY{n}{data} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1000}\PY{p}{,} \PY{l+m+mi}{1123}\PY{p}{,} \PY{l+m+mi}{1223}\PY{p}{,} \PY{l+m+mi}{1190}\PY{p}{,} \PY{l+m+mi}{1432}\PY{p}{,} \PY{l+m+mi}{1553}\PY{p}{,} \PY{l+m+mi}{1709}\PY{p}{,} \PY{l+m+mi}{1826}\PY{p}{,} \PY{l+m+mi}{1980}\PY{p}{,} \PY{l+m+mi}{2146}\PY{p}{,} \PY{l+m+mi}{2172}\PY{p}{,} \PY{l+m+mi}{2383}\PY{p}{,} \PY{l+m+mi}{2588}\PY{p}{,} \PY{l+m+mi}{2822}\PY{p}{,} \PY{l+m+mi}{3401}\PY{p}{,} \PY{l+m+mi}{3330}\PY{p}{,} \PY{l+m+mi}{4157}\PY{p}{,} \PY{l+m+mi}{3994}\PY{p}{,} \PY{l+m+mi}{4995}\PY{p}{,} \PY{l+m+mi}{5392}\PY{p}{,} \PY{l+m+mi}{5910}\PY{p}{,} \PY{l+m+mi}{6468}\PY{p}{,} \PY{l+m+mi}{7128}\PY{p}{,} \PY{l+m+mi}{7325}\PY{p}{,} \PY{l+m+mi}{7984}\PY{p}{,} \PY{l+m+mi}{9634}\PY{p}{,} \PY{l+m+mi}{10473}\PY{p}{,} \PY{l+m+mi}{11761}\PY{p}{,} \PY{l+m+mi}{12777}\PY{p}{]}
508\end{Verbatim}
509\end{tcolorbox}
510
511\vspace{0.5cm}
512 \textbf{Grading}
513
514This homework assignment is worth \(28\) (\(24+4\)) points, distributed
515as described above.
516
517Your final grade for the course will be the total of points you obtained
518(notice that the maximum is \(20+20+16+28=84\)) divided by \(4\),
519rounded to the nearest integer. More precisely
520
521\begin{align*}
522\operatorname{grade} = \operatorname{min}\left(20, \left\lfloor \frac{\operatorname{total}}{4} + 0.5\right\rfloor\right)
523\end{align*}
524
525
526 % Add a bibliography block to the postdoc
527
528
529
530\end{document}

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