IT103: Algebra
Module 5: Polynomial and Rational Functions
2024-11-25
Grades vs Attendance
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Periodicity of a pendulum
\[\eqalign{T &=& 2\pi \sqrt{\frac{L}{g}}\\
\frac{T}{2\pi} &=& \sqrt{\frac{L}{g}}\\
\frac{T^2}{2^2\pi^2} &=& \frac{L}{g}\\
g &=& \frac{4\pi^2L}{T^2}\\
}\]
Log
\[\eqalign{10^0&=&1;& \log_{10}(1)&=&0\\
10^1&=&10;& \log_{10}(10)&=&1\\
10^2&=&100;& \log_{10}(100)&=&2\\
10^3&=&1000;& \log_{10}(1000)&=&3\\
}\]
Changing the base
\[\eqalign{2^6&=&64;& \log_2(64) &=&6\\
4^3&=&64;& \log_4(64) &=&3\\
8^2&=&64;& \log_8(64) &=&2\\
16^{1.5}&=&64;& \log_{16}(64) &=&1.5\\
32^{1.2}&=&64;& \log_{32}(64) &=&1.2\\
}\]
Graphics
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Graphics log
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Multiplication of numbers with exponents
\[\small\eqalign{100 \times 1000 &=& 100000;&10^2 \times 10^3= 10^5\\
2\times 3&=&6;& 10^{0.30103}\times 10^{0.47712}= 10^{0.77815}\\
20\times 30&=&600;& 10^{1.30103}\times 10^{1.47712}= 10^{2.77815}\\
}\]
Exercises
\[\left(\frac{x}{y}\right)^3(x\cdot y)^2\] \[\frac{-35x^2y^4}{5x^6y^{-8}}\] \[\sqrt{3(-3x^2y^4)^3}\]
Dataset
a b c
25.70331052 0.09737727 4.25726243
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Surge Function
\[y=20x^{3}e^{-0.5x}\]
Logistic function
\[y=20x^{3}e^{-0.5x}\]
S curve
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Inverted function
\[\eqalign{P &=& 25.6 e^{0.03t}\\
\frac{P}{25.6} &=& e^{0.03t}\\
\ln\left(\frac{P}{25.6}\right) &=& 0.03t\\
\frac{\ln\left(\frac{P}{25.6}\right)}{0.03} &=& t\\
}\]
Mystery series 1
\[\matrix{sequence&1,&\underbrace{}&3,&\underbrace{}&7,&\underbrace{}&15,&\underbrace{}&31\\
difference&&2&&4&&8&&16&\\}\]
\[\eqalign{direct:& a_n &=&{2^n} -1\\
recursive:&a_{n+1} &=& a_n + 2\cdot (a_{n} - a_{n-1})\\}\]
Mystery series
Series data
\[\small 12, 27, 58, 111, 192, 307, 462, 663, 916, 1227\] Table of difference
\[\begin{matrix}series & dif_1 & dif_2 & dif_3 \\
12 & 15 & 16 & 6\\
27 & 31 & 22 & 6\\
58 & 53 & 28 & 6\\
111 & 81 & 344 &6\\
192 & 115 & 40 & 6 \\
307 & 155 & 46 &6 \\
462 &201 &52 &6 & \\
663 & 253 & 58 & \\
916 & 311 & & \\
1227& & & \\
\end{matrix}\]
Suggested model:
\[y = ax^3 + bx^2 + cx + d\]
Linear algebra
\[\left[\begin{matrix}
1 & 1 &1&1\\
8 & 4& 2&1\\
27 & 9 &3&1\\
64 &16 & 4 &1\\
\end{matrix}\right] \left[\begin{matrix}a \\ b \\c \\ d \\\end{matrix}\right]=\left[\begin{matrix}
12\\ 27\\ 58\\ 111\\
\end{matrix}\right]\]
Linear solution
\[\left[\begin{matrix}
1 & 0 &0&0\\
0 & 1& 0&0\\
0 & 0 &1&0\\
0 &0 & 0 &1\\
\end{matrix}\right]\left[\begin{matrix}a \\ b \\c \\ d \\\end{matrix}\right]=\left[\begin{matrix}
1\\ 2\\ 2\\ 7\\
\end{matrix}\right]\]
Suggested solution
\[y = x^3 + 2x^2 + 2x + 7\]
Comparison to series to calculations
\[original = \small 12, 27, 58, 111, 192, 307, 462, 663, 916, 1227\]
\[calculated =\small 12, 27, 58, 111, 192, 307, 462, 663, 916, 1227, 1602, 2047, 2568, 3171, 3862, 4647, 5532, 6523, 7626, 8847\]
Another example
Data of the Series
\[\left[-6, 0, 10, 24, 42, 64, 90, 120, 154, 192\right]\]
Difference Table
\[\begin{matrix}
series & dif & dif_2\\
-6 & 6 & 4\\
0 & 10 & 4\\
10 & 14 & 4\\
24 & 18 & 4\\
42 & 22 & 4\\
64 & 26 & 4\\
90 & 30 & 4\\
120 & 34 & 4\\
154 & 38 & NA\\
192 & NA & NA\\
\end{matrix}\]
Suggested model
\[y = ax^2 + bx + c\]
Linear model
\[\left[\begin{matrix}
1 &1 &1\\
4 &2 &1\\
9 &3 &1\\
\end{matrix}\right] \left[\begin{matrix}
a\\
b\\
c\\
\end{matrix}\right] = \left[\begin{matrix}
-6\\
0\\
10\\
\end{matrix}\right]\]
Gaussian elimination and back substitution
Initial Data
\[\begin{matrix}
1 & 1 & 1 & -6 \\
4 & 2 & 1 & 0 \\
9 & 3 & 1 & 10 \\
\end{matrix}\]
Swap R1 and R3
\[\begin{matrix}
9 & 3 & 1 & 10\\
4 & 2 & 1 & 0 \\
1 & 1 & 1 & -6 \\
\end{matrix}\]
R1 * 1/9
\[\begin{matrix}
1 & 1/3 & 1/9 & 10/9 \\
4 & 2 & 1 & 0 \\
1 & 1 & 1 & -6 \\
\end{matrix}\]
R2-(R1*4)
\[\begin{matrix}
1 & 1/3 & 1/9 & 10/9 \\
0 & 2/3 & 5/9 & -40/9\\
1 & 1 & 1 & -6 \\
\end{matrix}\]
R3 - R1
\[\begin{matrix}
1 & 1/3 & 1/9 & 10/9 \\
0 & 2/3 & 5/9 & -40/9\\
0 & 2/3 & 8/9 & -64/9\\
\end{matrix}\]
R2 * 3/2
\[\begin{matrix}
1 & 1/3 & 1/9 & 10/9 \\
0 & 1 & 5/6 & -20/3 \\
0 & 2/3 & 8/9 & -64/9\\
\end{matrix}\]
R1-(R2 * 1/3)
\[\begin{matrix}
1 & 0 & -1/6 & 10/3 \\
0 & 1 & 5/6 & -20/3 \\
0 & 2/3 & 8/9 & -64/9 \\
\end{matrix}\]
R3 - (R2* 2/3)
\[\begin{matrix}
1 & 0 & -1/6 & 10/3 \\
0 & 1 & 5/6 & -20/3 \\
0 & 0 & 1/3 & -8/3 \\
\end{matrix}\]
R3 * 3
\[\begin{matrix}
1 & 0 & -1/6 & 10/3 \\
0 & 1 & 5/6 & -20/3 \\
0 & 0 & 1 & -8\\ \end{matrix}\]
R1 +(R3 * 1/6)
\[\begin{matrix}
1 & 0 & 0 & 2 \\
0 & 1 & 5/6 & -20/3 \\
0 & 0 & 1 & -8\\
\end{matrix}\]
R2 - (R3 * 5/6)
\[\begin{matrix}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & -8\\
\end{matrix}\]
Suggested formula
\[y = 2x^2 -8\]
Comparison
\[original = \tiny\left[-6, 0, 10, 24, 42, 64, 90, 120, 154, 192\right]\] \[calculated=\tiny\left[ -6, 0, 10, 24, 42, 64, 90, 120, 154, 192, 234, 280, 330, 384, 442, 504, 570, 640, 714, 792\right]\]
Adding elements of a algebric series
\[\small\eqalign{s &=&a_1 + (a_1+d) + (a_1+2d) +...+(a_n-2d)+(a_n-d)+(a_n)\\
&=&a_n+ (a_n-d) + (a_n-2d) +...+(a_1+2d)+(a_1+d)+(a_1)\\
2s &=&n(a_1+a_n)\\
s &=& \frac{n(a_1 +a_n)}{2}\\
}\]
Sum of squares
\[s=1 + 4+ 9 + 16 + 25 + 36 + 49 + 64=204\] \[s=(n)(n+1)(2n+1)/6=8\cdot9(2\cdot 8+1)=204\]
Calculating the value of e
\[e = 1+ \frac{1}{1!}+ \frac{1}{2!}+ \frac{1}{3!}+ \frac{1}{4!} + \dots\]
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Orders of magnitude
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