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1 | 1 | { |
2 | | - "name": "ME1 Stress Analysis Tutorial Sheet 1. Moments", |
3 | | - "year": "2023-24", |
| 2 | + "name": "Mathematical Analysis (1st Year) - Problem Sheet 3", |
| 3 | + "year": "0", |
4 | 4 | "questions": [ |
5 | 5 | { |
6 | | - "question_content": "## Problem 1.1\n\nA 20 N force is applied to the end of a 0.4 m lever.\n\n\nCalculate the magnitude of the moment that this force produces about the point O , and state the direction of the rotational effect.", |
7 | | - "solution_content": "[6.9 Nm, clockwise]", |
| 6 | + "question_content": "1. This question concerns the definition of the least upper bound and greatest lower bound for sets of numbers and the difference between this notion and that of largest or smallest element of the set. To remind yourself of the difference between least upper bound and largest element, consider the set of real numbers $x$ such that $x<6$. This set has no largest element, but its least upper bound is 6 (which does not live in the set). If instead of $x<6$ we had $x \\leq 6$, then the set does have a largest element, namely 6 , which is again the least upper bound (but now does live in the set).\n\nFor each of the sets described below, write down the largest element of the set, the smallest element of the set, the least upper bound and the greatest lower bound (if they exist - they don't always).\n(i) The set of all even positive integers.\n(ii) The set of all rational numbers $r$ satisfying $0 \\leq r<1$.\n(iii) The set of all values of $x$ for which $x=\\sin t$ for some real $t$.\n(iv) The set of all numbers of the form $3^{-m}+5^{-n}$ for any positive integers $m, n$.\n(v) The set of all real numbers satisfying $-1 \\leq \\tan x \\leq 1$.\n(vi) The set of numbers of the form $1+1 / n$, where $n$ is any positive integer.\n(vii) The set of positive integers $n$ satisfying $n^{2} \\leq 10$.\n(viii) The set of real numbers $y$ of the form\n\n$$\ny=\\frac{(2 x+5)}{(x+1)}\n$$\n\nwhere $x>0$.", |
| 7 | + "solution_content": "1.(i) $\\quad S=\\{2,4,6,8, \\ldots\\}$\nmin $=2$, no max, glb. $=2$, to lub.\n(ii) $S=\\{r \\mid r \\in \\mathbb{Q}, 0 \\leqslant r<1\\}$ mur $=0, g l b=0$, no max, l.u.b. $=1$\n(iii) $S=\\{x \\mid x=\\sin t, t \\in \\mathbb{R}\\}$\n$\\min =g l \\cdot b=-1, \\quad \\max =l \\cdot u \\cdot b=+1$\n(iv) $\\quad S=\\left\\{\\left.\\frac{1}{3^{m}}+\\frac{1}{5^{n}} \\right\\rvert\\, m, n \\in \\mathbb{N}\\right\\}$\nno min, but $g \\cdot l \\cdot b=0$ since $\\frac{1}{3^{m}}+\\frac{1}{5} n>0$\n\n$$\n\\max =\\frac{1}{3}+\\frac{1}{5}=l \\cdot u \\cdot b\n$$\n\n(v) $\\quad S=\\{x \\mid x \\in \\mathbb{R},-1 \\leqslant \\tan x \\leqslant 1\\}$\nno min, no max, no g.l.b., no l.u.b\n(vi) $S=\\left\\{\\left.1+\\frac{1}{n} \\right\\rvert\\, n \\in \\mathbb{N}\\right\\}$\nno $\\min , g \\cdot l \\cdot b \\cdot=1, \\max =l \\cdot u \\cdot b .=2$\n(ii) $S=\\left\\{n \\mid n^{2} \\leqslant 10, n \\in \\mathbb{N}\\right\\}$\nmin $=g \\cdot l \\cdot b=1, \\max =l \\cdot u \\cdot b .=3$.\n(vii) $S=\\left\\{y \\left\\lvert\\, y=\\frac{(2 x+5)}{(x+1)}\\right., x \\in \\mathbb{R}, x>0\\right\\}$\n\n$l \\cdot u \\cdot b=5, \\quad g \\cdot l \\cdot b=2$, no min since $x<\\infty$ and no max since $x>0 \\quad(y=5$ achemil at $x=0$ ).", |
8 | 8 | "images": [ |
9 | | - "0_2025_07_29_62fff12b7847de9f2d97g-2.jpg" |
| 9 | + "0_2025_08_04_5ea1367dbcee79ac4f33g-4.jpg" |
10 | 10 | ] |
11 | 11 | }, |
12 | 12 | { |
13 | | - "question_content": "## Problem 1.2\n\nA mechanic must tighten a bolt to 45 N cm .\n(a) If the mechanic's wrench is 15 cm long, what force must be applied by the mechanic perpendicular to the end of the wrench?\n(b) The maximum force that a mechanic can exert with one arm is 500 N . If the mechanic must close a valve that requires a torque of 150 N m , how long must the moment arm be?\n[3 N; 0.3 m ].", |
14 | | - "solution_content": "", |
| 13 | + "question_content": "2. Let $A_{n}=\\left\\{a_{1}, a_{2}, \\cdots a_{n}\\right\\}$ be a finite set with $\\left|A_{n}\\right|=n$. Write down the complete list of subsets of $A_{n}$ for the cases $n=1,2,3$. Use induction to show that the total number of possible subsets of $A_{n}$ is $2^{n}$. [Hint: Write a set of $n$ elements as the union of a set containing\njust one of its elements together with a set of $n-1$ elements. Also note that any set has both itself and the empty set as subsets.]", |
| 14 | + "solution_content": "2. $n=1 \\quad A=\\{a\\}$\n\n2 subsets, $A$ and $\\phi$.\n\n$$\nn=2 \\quad A=\\left\\{a_{1}, a_{2}\\right\\}\n$$\n\n4 subsets: $A, \\phi,\\left\\{a_{1}\\right\\},\\left\\{a_{2}\\right\\}$\n\n$$\nn=3 \\quad A=\\left\\{a_{1}, a_{2}, a_{3}\\right\\}\n$$\n\n8 subsets: $A, \\Phi,\\left\\{a_{1}\\right\\},\\left\\{a_{2}\\right\\},\\left\\{a_{3}\\right\\},\\left\\{a_{11}, a_{2}\\right\\},\\left\\{a_{1}, a_{3}\\right\\},\\left\\{a_{2}, a_{3}\\right\\}$.\n\nGeneral $n$\n\n$$\n\\begin{aligned}\nA_{n} & =\\left\\{a_{1}, a_{2}, \\ldots a_{n}\\right\\} \\\\\n& =\\left\\{a_{1}, a_{2}, \\ldots a_{n-1}\\right\\} \\cup\\left\\{a_{n}\\right\\} \\\\\n& =A_{n-1} \\cup\\left\\{a_{n}\\right\\}\n\\end{aligned}\n$$\n\nThe subsets of $A_{n}$ all have its form $B_{n-1} \\cup \\phi(=B)$ or $B_{n-1} \\cup\\left\\{a_{n}\\right\\}$\nwhere $B_{n-1}$ ranges aver all the $2^{n-1}$ subsets of $A_{n-1} \\ldots A_{n}$ therefore hav\n\n$$\n2^{n-1}+2^{n-1}=2^{n} \\text { subsets. }\n$$\n\nI.e adoling one mare alement to the ret multiples the number of subsets by $Z$.", |
15 | 15 | "images": [] |
16 | 16 | }, |
17 | 17 | { |
18 | | - "question_content": "A 25 N planter is hung from a bracket. Calculate the moments applied by the 25 N about the mounting points A and B .\n\n[ $\\mathrm{M}_{\\mathrm{A}}=\\mathrm{M}_{\\mathrm{B}}=70.71 \\mathrm{~N} \\mathrm{~m}$, clockwise.]\n\n## Problem 1.4\n", |
19 | | - "solution_content": "A child attempts to drive a toy car and applies a force at three separate instants as shown below.", |
20 | | - "images": [ |
21 | | - "1_2025_07_29_62fff12b7847de9f2d97g-2.jpg" |
22 | | - ] |
| 18 | + "question_content": "3. Show that the function\n\n$$\nf(n, m)=(2 n-1) \\times 2^{m-1}\n$$\n\ndefines a bijection from $\\mathbf{N} \\times \\mathbf{N}$ to $\\mathbf{N}$ (where $\\mathbf{N}$ denotes the set of natural numbers). Hence show that the rationals are countable.", |
| 19 | + "solution_content": "3.\n\n$$\n\\begin{aligned}\nf(n, m) & =(2 n-1) \\times 2^{m-1} \\text { defines a map } \\\\\nf: \\mathbb{N} \\times \\mathbb{N} & \\longrightarrow \\mathbb{N} \\\\\n& (n, m) \\longrightarrow(2 n-1) \\times 2^{m-1} \\quad n, m=1,2,3, \\ldots\n\\end{aligned}\n$$\n\nThe map is surjecture becaure every point in may be exprened in the form $(2 n-1) \\times 2^{m-1}$ To 2ee this note first that $m=1$ gries $(2 n-1)$, all the odd numbers. Second, every even number may be unitten as $(2 n-1) \\times 2^{n-1}$ for rame $m, n$.\n\nNow cheel for injective.\nSuppose $\\left(2 n^{\\prime}-1\\right) \\times 2^{m \\prime-1}=(2 n-1) \\times 2^{m-1}$\nIf $m>m^{\\prime}, \\quad \\underbrace{\\left(2 n^{\\prime}-1\\right)}_{\\text {odd }}=\\underbrace{(2 n-1) \\times 2^{m-m^{\\prime}}}_{\\text {odd }}$\nThis is inconsistent unters $m=m$ !\n( Samilarly for $m<m^{\\prime}$ ).\n\n$$\n\\Rightarrow 2 n^{\\prime}-1=2 n-1 \\Rightarrow n^{\\prime}=n .\n$$\n\nSo $\\left(2 n^{\\prime}-1\\right) \\times 2^{m^{\\prime}-1}=(2 n-1) \\times 2^{m-1} \\Rightarrow n=n^{\\prime}, m=m^{\\prime}$.\nThis means may is also injective.\nSurjecture + injecture = bijective -\n\nSince the rationals $n / m$ comesponel to $N \\times N$ if follows that they are courtable.", |
| 20 | + "images": [] |
23 | 21 | }, |
24 | 22 | { |
25 | | - "question_content": "\n(b)\n\n(c)\n\nIf the magnitude of the force exerted by the child in all cases is 5 N and the radius of the steering wheel is 10 cm , calculate the moment applied to the steering wheel.\n[0.5 N m anti-clockwise; 0.5 N m clockwise; zero].\n\n## Problem 1.5\n\nA 4000 N force $F=4000 \\mathrm{~N}$ is applied at the trailer hitch shown below.", |
26 | | - "solution_content": "", |
27 | | - "images": [ |
28 | | - "4_2025_07_29_62fff12b7847de9f2d97g-3.jpg", |
29 | | - "5_2025_07_29_62fff12b7847de9f2d97g-3.jpg", |
30 | | - "3_2025_07_29_62fff12b7847de9f2d97g-3.jpg" |
31 | | - ] |
| 23 | + "question_content": "4. Consider the function defined by\n\n$$\n\\begin{gathered}\ny=3 x \\quad \\text { for } \\quad x \\leq \\frac{1}{2} \\\\\ny=3(1-x) \\quad \\text { for } \\quad x \\geq \\frac{1}{2}\n\\end{gathered}\n$$\n\nShow that this function maps the Cantor set into itself. [Hint: Use the ternary representation and see what happens under the above maps].", |
| 24 | + "solution_content": "$-4$.\n\n$$\ny= \\begin{cases}3 x & x \\leqslant 1 / 2 \\\\ 3(1-x) & x \\geqslant 1 / 2\\end{cases}\n$$\n\nRecall etat elements $x \\in \\subset$ (cantrset) have ter temay verrenentation\n\n$$\nx=0 \\cdot a_{1} a_{2} a_{3} \\ldots \\quad a_{i}=0 \\text { or } 2\n$$\n\nFor $x \\leqslant 1 / 2$, elements of $C$ lie in $[0,1 / 3]$ and are of ete form\n\n$$\nx=0.0 a_{2} a_{3} a_{4} \\ldots . .\n$$\n\nso $3 x=0 . a_{2} a_{3} a_{4} \\ldots$ clearly in $C$.\nFor $x \\geqslant 1 / 2$, elements of $C$ lie $i$, $[2 / 3, F 1]$ and are of the form\n\n$$\n\\begin{aligned}\nx= & 0.2 a_{2} a_{3} a_{4} \\ldots . \\quad a_{i}=0 \\text { or } 2 \\\\\n(1-x)= & 0.22222 \\ldots \\\\\n& -0.2 a_{2} a_{3} a_{4} \\ldots \\\\\n= & 0.0(\\underbrace{\\left(2-a_{2}\\right)\\left(2-a_{3}\\right)\\left(2-a_{4}\\right)}_{a_{l} a b_{y}=0 \\text { or } 2 \\text { y } a_{i}=2 \\text { or } 0} \\ldots . \\\\\n3(1-x)= & 0 .\\left(2-a_{2}\\right)\\left(2-a_{3}\\right) \\ldots .\n\\end{aligned}\n$$\n", |
| 25 | + "images": [] |
32 | 26 | }, |
33 | 27 | { |
34 | | - "question_content": "$\\left[\\mathrm{M}_{\\mathrm{A}}=600 \\mathrm{~N} \\mathrm{~m}\\right.$ anti-clockwise, $\\mathrm{M}_{\\mathrm{B}}=600 \\mathrm{~N} \\mathrm{~m}$ clockwise; $\\mathrm{M}_{\\mathrm{A}}=1039 \\mathrm{~N} \\mathrm{~m}$ anti- clockwise, $\\mathrm{M}_{\\mathrm{B}}$ $=0 \\mathrm{l}$.\n", |
35 | | - "solution_content": "", |
| 28 | + "question_content": "5. Let $A_{i}$ be an infinite but countable set, for $i=1,2,3 \\cdots$. Show that the infinite union\n\n$$\nA=\\bigcup_{i=1}^{\\infty} A_{i}=A_{1} \\cup A_{2} \\cup A_{3} \\cdots\n$$\n\nis countable. [Hint: write out the first few sets using the notation $A_{1 j}, A_{2 j} \\cdots$ where $j$ denotes the elements of each set, and use what you already know about types of set which are countable].\n6. Consider maps $f$ from $\\mathbb{N}$ to $T$ with $|T|=2$. Prove that the set of all possible maps from $\\mathbb{N} \\rightarrow T$ is uncountable. (Hint: Find a way to use the binary representation to label all the possible choices of $f$ ).", |
| 29 | + "solution_content": "5. \n\n$A_{1 i}=\\left\\{A_{11}, A_{12}, A_{13}, \\ldots\\right\\}$\n$A_{2 i}=\\left\\{A_{21}, A_{22}, A_{23}, \\ldots\\right\\}$\n$A_{3 i}=\\left\\{A_{31}, A_{32}, A_{33} \\ldots\\right\\}$\nHeme elements of $A=\\bigcup_{i=1} A_{i}$\ncansit of element $A_{i j}$ where $i, j \\in \\mathbb{N}$, ie $A_{i j} \\in \\mathbb{N} \\times \\mathbb{N}$ which is countable.\n\n6. \n\nThis is the answer for question 6", |
36 | 30 | "images": [] |
37 | 31 | } |
38 | 32 | ] |
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