A-level-further

5. A-level-further#

  1. Notes go downwards in cells; cells go upwards in sections.

  2. Cells with formulas are at top of each section.

  3. Used Languages : MMA, Sage, Python(for number calculations)

5.1. test coding#

(*Functions in MMA*)
g=9.8;
invsin[x_]:=InverseFunction[Sin][x]
N[InverseFunction[Sin][1]]
InverseFunction[Sin][1]//N  (*Wolfram 后线// *)

(*Vectors in MMA*)
xv=Subscript[x,#] & /@ Range[5]
yv=Subscript[y,#] & /@ Range[5]
xv+yv
xv yv
xv*yv
xv.yv
Norm[xv]

# Physics: Gravitation and Satellites 
G = 6.77 * 10^(-11)
Me = 5.972 * 10^(24)
radius_earth = 6378100  
force( Mass_sate, rs ) = (G * Me * Mass_sate)/(rs^2)
force1(Mass_sate, alt) = force( Mass_sate, rs = alt + radius_earth ) #Partial application, 相关但不是currying
force1(1, 0)

5.2. ★Further Y13-2#

5.2.1. Assessment 1#

x(t)=2.5*e^(-t/2)*sin(2*t)
v(t)=diff(x(t),t)
s(n(v(2)))
\[\displaystyle -0.854295151029126 \qquad \]

5.2.2. Module 3 - Work, Energy & Power#

Notes

  • Energy : There are 2 common types in physics - kinetic energy and potential energy.

    • Kinetic Energy : Energy stored in the object as motion. Formula : \(KE=\frac{1}{2}mv^2\)

    • Potential Energy : Energy stored in the object. Formula : \(PE=mgh\)

      • Elastic Potential Energy : \(PE=\frac{x^2\lambda}{2l}\)

    • Frictional Energy Lost : This is used in some problems where friction is not neglected. Formula : \(E=F_fs\)

  • Work : Work is the force times the displacement (\(Fs\)).

  • Power : Power is the rate of change of work.

g=9.8
PE=5*g*4*sin(5/36*pi);n(PE)
FE=5*g*cos(5/36*pi)*0.6*4;n(FE)

106.581795755510

5.2.3. Module 2 - Momentum & Impulse#

Notes

s(n(2.4/8.2,16))
s(n(4/7,16))

5.2.4. Module 1 - ODE Applications#

Notes

  • Hooke’s Law : When the extensions is \(x\), the natural length of the spring is \(l\) and the tension is \(T\), \(x \propto Tl\) which can also be written as \(T=\frac{\lambda x}{l}\) where \(\lambda\) is the spring constant. (\(F=kx\))

  • Properties of SHM(Simple Harmonic Motion)

    • It is periodic.

    • It oscillates around the equilibrium position.

    • The extreme points are the same distance from the equilibrium position.

    • It reaches maximum speed at the equilbrium.

    • It is stationary for an instant at the extreme points.

  • For pendulums, the 2nd order non-linear ODE as there is a \(\sin{\theta}\).

    • Because of this, it is only SHM for small \(\theta\) as when it is like that, \(\sin{\theta}\approx\theta\). After solving it, the general solution is \(\theta=A\cos{\omega t}+B\sin{\omega t}\) where \(\omega=\sqrt{\frac{g}{l}}\)

    • For angular amplitude, convert from degrees into radians by multiplying by \(\frac{\pi}{180}\).

    • For approximate period, \(k=\frac{2\pi}{\omega}\).

    • The max angular speed is \(a\omega\) in radians per second.

var('A B')
x(t)=e^t*(A*cos(2*t)+B*sin(2*t))+4/5
y(t)=e^t*(B*cos(2*t)-A*sin(2*t))+1/10
s(x(0),y(0))
\[\displaystyle A + \frac{4}{5} \qquad B + \frac{1}{10} \qquad \]
var('A B')
x(t)=e^t*(A*t+B)
y(t)=e^t*(A*t+B-A)
s(x(0),y(0))
\[\displaystyle B \qquad -A + B \qquad \]
y(t)=(3-e^(-2*t))*(sin(5*t)-cos(5*t))/12
plot(y)

var('A B')
y(t)=e^(-2*t)*(A*cos(5*t)+B*sin(5*t))+sin(5*t)/4-5/4*cos(5*t)
v(t)=diff(y(t),t)
s(y(0),v(0))

var('a b')
y(t)=a*sin(5*t)+b*cos(5*t)
v(t)=diff(y(t),t)
a(t)=diff(v(t),t)
s(a(t)+4*v(t)+29*y(t))

var('A B')
w=sqrt(7)/2
x(t)=e^(-t/2)*(A*cos(w*t)+B*sin(w*t))
v(t)=diff(x(t),t)
s(x(0),v(0))

var('A B')
x(t)=e^(-2*t)*(A*cos(t)+B*sin(t))
v(t)=diff(x,t)
s(x(0),v(0))

var('A B')
w=sqrt(171)/2
x(t)=e^(-3*t/2)*-10/w*sin(w*t)
v(t)=diff(x,t)
s(n(x(2)),n(v(2)))

var('A B')
x(t)=e^(-t/2)*(A*cos(5*t)+B*sin(5*t))
s(diff(x,t))

var('A B')
x(t)=e^(-2*t)*(A*cos(t/2)+B*sin(t/2))
s(diff(x,t))

var('A B')
x(t)=e^-t*(A*cos(2*t)+B*sin(2*t))
s(x(t),diff(x(t),t))

x(t)=0.5*cos(sqrt(160)*t)
s(n(x(0.25),16),n(x(5),16))

g=49/5
x=0.6*g/20;print(n(x,16))
x=g/160;print(x)

5.3. ★Further Y13-1#

5.3.1. Assessment 3 - Module 7 and 8#

a=vector([9, 7, 7])
b=vector([4, 7, 2])
show(a.cross_product(b))

a=vector([3, 3, 4])
b=vector([-3, 3, 4])
c=vector([0, 6, 8])
show(a.cross_product(c))

a=vector([3, -1, 0])
b=vector([7, 36, -23])
show(a.cross_product(b))

a=vector([13, -27, -21])
b=vector([1, 1, -3])
show(a.cross_product(b))
show(n(arcsin(sqrt(102^2+18^2+40^2)/(sqrt(13^2+27^2+21^2)*sqrt(1^2+1^2+3^2)))/pi*180))

a=vector([6, -2, -7])
b=vector([5, 1, -5])
show(a.cross_product(b))

a=vector([1, 0, -2])
b=vector([-4, 1, 4])
show(a.cross_product(b))

a=vector([0, -1, 3])
b=vector([5, 0, -4])
c=vector([3, 3, 6])
show(a.cross_product(b))
show(n(c.dot_product(a.cross_product(b)/sqrt(266))))

a=vector([-3, 1, 2])
b=vector([1, -1, 1])
c=vector([-1, 5, 6])
show(a.cross_product(b))
show(n(c.dot_product(a.cross_product(b))/sqrt(38)))

a=vector([3, -4, -8])
b=vector([-2, -4, 1])
c=vector([1, -2, -5])
show(a.dot_product(b.cross_product(c)))

p=var('p')
q=var('q')
x=var('x')
a=matrix([[-50, 12, 138], [-36, 10, 96], [-17, 4, 47]])
b=vector([3, 2, 1])
show(a.det())
show(a.eigenvalues())
show(expand((x-1)*(x-2)*(x-4)))
show(a.inverse()*b)

a=matrix([[3, -4, 2], [-4, -1, 6], [2, 6, -2]])
b=matrix([[1, 2, 2], [2, 1, -2], [-2, 2, -1]])
c=matrix([[9^4, 0, 0], [0, 3^4, 0], [0, 0, 6^4]])
show(a.eigenvalues())
show(a.eigenvectors_right())
show(b*c*b.inverse())

a=matrix([[6, 13], [-2, -9]])
b=matrix([[1, 13], [-1, -2]])
c=matrix([[-7^5, 0], [0, 4^5]])
show(a.eigenvalues())
show(a.eigenvectors_right())
show(b*c*b.inverse())

a=matrix([[-2, -4, 2], [-2, 1, 2], [4, 2, 5]])
b=matrix([[-26, -33, -25], [31, 42, 23], [-11, -15, -4]])
show(a.eigenvalues())
show(b.eigenvalues())
show(a.eigenvectors_right())
show(b.eigenvectors_right())

a=matrix([[1, 2], [2, -2]])
b=matrix([[3, 5], [1, -1]])
c=matrix([[7, 4], [-2, 1]])
d=matrix([[11, -24], [15, -27]])
show(a.eigenvalues())
show(b.eigenvalues())
show(c.eigenvalues())
show(d.eigenvalues())
show(a.eigenvectors_right())
show(b.eigenvectors_right())
show(c.eigenvectors_right())
show(d.eigenvectors_right())

a=vector([24, -16, 32])
b=vector([-15, -12, 9])
show(a.cross_product(b))

a=vector([-1, 4, -2])
b=vector([2, -3, -1])
c=vector([1, 2, -2])
show(b.dot_product(a.cross_product(c)))
show(b.dot_product(c.cross_product(a)))
show(c.dot_product(b.cross_product(a)))

a=vector([4, -1, -2])
b=vector([-3, 2, -5])
c=vector([2, 2, -6])
show(a.cross_product(b))
show(a.cross_product(c))
show(b.cross_product(c))

5.3.2. Module 8 - Eigenvalues and Eigenvectors#

If the corresponding eigenvalue is \(\lambda\) then for a transformation matrix \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\) $\(\begin{pmatrix}a&b\\c&d\end{pmatrix} \begin{pmatrix}x\\mx\end{pmatrix} = \lambda\begin{pmatrix}x\\mx\end{pmatrix}\)$ The calculations below are used to calculate eigenvalues and eigenvectors and it also reveals properties about the eigenvalues of the square, cube or inverse of a matrix.

The methods used are algebra and the characteristic equation.

Some notation for eigenvalues and eigenvectors are below.

The notation is used in diagonalisation.

\(\mathbf{s}_n\) stands for the \(n\) th eigenvector.

\(\mathbf{S}\) stands for \(\begin{pmatrix}\mathbf{s}_1&\cdots&\mathbf{s}_n\end{pmatrix}\) (the matrix of the eigenvectors of \(\mathrm{M}\)).

\(\Lambda\) stands for \(\begin{pmatrix}\lambda_1&0\\0&\lambda_2\end{pmatrix}\) (the diagonal matrix of the eigenvalues of \(\mathrm{M}\) in the order of the eigenvectors used for \(\mathrm{S}\)) where \(\lambda_n\) is the \(nth\) eigenvalue.

\(\mathrm{M}\) is the matrix we want to find the powers of. $\(\mathbf{MS}=\mathbf{S\Lambda}\)\( \)\(\mathbf{S^{-1}MS}=\mathbf{S^{-1}S\Lambda}\)\( \)\(\mathbf{S^{-1}MS}=\mathbf{\Lambda}\)\( When we do this we say the matrix \)\mathrm{M}$ has been reduced to diagonal form or has been diagonalised.

The matrix \(\Lambda\) is the diagonalised form of \(\mathrm{M}\)

\[\mathbf{MS}=\mathbf{S\Lambda}\]
\[\mathbf{MSS^{-1}}=\mathbf{S\Lambda S^{-1}}\]
\[\mathbf{M}=\mathbf{S\Lambda S^{-1}}\]
\[\mathbf{M}^n=\mathbf{S\Lambda}^n\mathbf{S^{-1}}\]

If \(\mathbf{D}\) is a diagonal matrix, $\(\mathbf{D}^n=\begin{pmatrix}a^n&0\\0&b^n\end{pmatrix}\)\( The Cayley-Hamilton theorem states that \)\(d_0+d_1\mathbf{M}+d_2\mathrm{M^2}-\mathrm{M^3}=0\)$

a=matrix([[1-x, 4, -1], [-1, 6-x, -1], [2, -2, 4-x]])
show(expand(a.det()))

p=var('p')
q=var('q')
a=matrix([[0, -1, 1], [6, -2, 6], [4, 1, 3]])
b=vector([1, 1, -1])
show(a.eigenvalues())
show(a.inverse()*b)
show(a^2)

k=var('k')
a=matrix([[0, 0, 2], [1, -1, 2], [1, 0, 1]])
show(a.eigenvectors_right())
b=matrix([[1, 0, -1/2], [0, 1, 0], [1, 1, 1]])
show(b.det())

m=matrix([[0, -1, 1], [6, -2, 6], [4, 1, 3]])
s=matrix([[1, 1, 0], [1, 0, 1], [-1, -1, 1]])
l=matrix([[16, 0, 0], [0, 1, 0], [0, 0, 256]])
show(m.eigenvalues())
show(m.eigenvectors_right())
show(s*l*s.inverse())

m=matrix([[9, 2], [-1, 6]])
show(m.eigenvalues())
show(m.eigenvectors_right())
m1=matrix([[1, 2], [-1, -1]])
m2=matrix([[7^5, 0], [0, 8^5]])
show(m1.inverse())
a=7^5
b=8^5
show(2*b-a)
show(2*b+2*a)
show(a-b)
show(-2*b-a)
m1*m2*m1.inverse()

The calculations below are used to calculate eigenvalues and eigenvectors and it also reveals properties about the eigenvalues of the square, cube or inverse of a matrix.

The methods used are algebra and the characteristic equation.

mt=matrix([[-1, 7, 9], [0, 1, 4], [0, 0, 3]])
show(mt.eigenvalues())
show((mt^2).eigenvalues())
show((mt^3).eigenvalues())
show(mt.inverse().eigenvalues())
mt2=matrix([[-2, -4, 2], [-2, 1, 2], [4, 2, 5]])
show(mt2.det())
show(mt2.trace())
show(mt2.eigenvalues())

The eigenvalues of a matrix is equal to the eigenvalues of the transpose of the matrix

show(expand(-((x+2)*(x-2)-1)*(x-1)+7*x+5))
f(x)=x^3-x^2-12*x
show(plot(f(x), (x, -4, 5), figsize=3))
show(f(-3))
show(f(0))
show(f(4))
mt=matrix([[1, -3, -3], [-8, 6, -3], [8, -2, 7]])
show(mt.eigenvalues())
show(mt.eigenvectors_right())
mt2=matrix([[1, -1, 0], [1, 2, 1], [-2, 1, -1]])
show(mt2.eigenvalues())
show(mt2.transpose().eigenvalues())
mt3=matrix([[-7, 4], [-5, 5]])
show(mt3.eigenvalues())
show((mt3^2).eigenvalues())
show((mt3^3).eigenvalues())
show(mt3.inverse().eigenvalues())

# Ex. 15
mt=matrix(2, 2, [[-5, 3], [6, -2]])
show(mt.eigenvalues())
mt2=matrix(3, 3, [[2, 0, 1], [-1, 2, 3], [1, 0, 2]])
show(mt2.eigenvalues())
f(x)=x^3-x^2-11*x+15
show(plot(f(x), (x, -5, 5),figsize=3))
show(n(sqrt(6)-1))
show(n(-sqrt(6)-1))
show(solve(f(x), x))

5.3.3. Module 7 - The Vector Product#

For calculating cross product, $\(\mathbf{i}\times\mathbf{i}=\mathbf{j}\times\mathbf{j}=\mathbf{k}\times\mathbf{k}=0\)\( \)\(i\times j=k\)\( \)\(j\times i=-k\)\( \)\(j\times k=i\)\( \)\(k\times j=-i\)\( \)\(k\times i=j\)\( \)\(i\times k=-j\)$

The below is working out for a formula for the cross product. $\((a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k}) \times (b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k})\)\( \)\(=a_1b_2\mathbf{k}-a_1b_3\mathbf{j}-a_2b_1\mathbf{k}+a_2b_3\mathbf{i}+a_3b_1\mathbf{j}-a_3b_2\mathbf{i}\)\( \)\(=(a_2b_3-a_3b_2)\mathbf{i}+(a_3b_1-a_1b_3)\mathbf{j}+(a_1b_2-a_2b_1)\mathbf{k}\)$

Properties 1. $\(\mathbf{a}\times\mathbf{b}=-(\mathbf{b}\times\mathbf{a})\)$ This means that the cross product is anticommutative.

\[\mathbf{a}\times\mathbf{a}=0\]
\[\mathbf{a}\times0=0\times\mathbf{a}=0\]
\[(m\mathbf{a})\times(n\mathbf{b})=mn(\mathbf{a}\times\mathbf{b})\]

For a parrallelepiped, the formulae for volume is below,

\[V=|a| |b\times c| \cos{\alpha}\]
\[V=|a| \cdot |b\times c|\]
\[V=|a\cdot(b\times c)|\]

For a tetrahedron(triangular pyramid), formula for volume is below, $\(V=\frac{1}{6}|a\cdot(b\times c)|\)$ Volumes can be calculated by scalar triple product

To find angle between planes, $\(\sin{\theta}=\frac{|n_1\times n_2|}{|n_1|*|n_2|}\)$

a=vector([-3, -3, 4])
b=vector([1, 2, 2])
c=vector([-6, 2, 5])
show(n(a.dot_product(b.cross_product(c))/sqrt(521)))
show(b.cross_product(c))

a=vector([1, -3, 2])
b=vector([1, -1, -2])
c=vector([7, -5, -7])
print('k='+str(a.dot_product(b.cross_product(c))))
d=vector([-4, 0, 5])
show(d.dot_product(b.cross_product(c)))
show(a.dot_product(d.cross_product(c)))

f(x, y)=-5*x+y+8
g(x, y)=-x-3*y-4
show(plot3d(f(x, y), (-4, 4), (-4, 4))+
plot3d(g(x, y), (-4, 4), (-4, 4)))

Calculation for calculating normal vectors and their cross product to find line of intersection

a=vector([2, 3, 2])
b=vector([4, -3, 4])
show(a.cross_product(b))

a=vector([4, -1, 3])
b=vector([3, -1, 2])
show(a.cross_product(b))
show(n(arcsin(sqrt(3)/sqrt(26*14))*180/pi))
c=vector([2, 3, 2])
d=vector([4, -3, 1])
show(c.cross_product(d))

a=vector([2, 4, -1])
b=vector([1, 1, 2])
c=vector([4, 0, 1])
a.dot_product(b.cross_product(c))

# Calc for Ex. 13
a=vector([1, -2, 3])
b=vector([2, -3, 1])
c=vector([-2, 2, -1])
show(a.cross_product(b))
show(a.cross_product(c))
d=vector([3, -4, 5])
f=vector([-5, 12, 13])
show(d.cross_product(f))

5.3.4. Assessment 2 - Module 4, 5 and 6#

show(n(-3*((-3)^2)^(1/3)))
show(n(log(9/10)+1/4))
show(n((5*log(3)-2*log(4)+2*log(2)-3/2*log(5)+arctan(2))/5))

5.3.5. Module 6 - 2nd Order ODE#

\[\frac{\mathrm{d}y}{\mathrm{d}x}+ay=0\]
\[y=Ae^{\lambda x}\]
\[y'=\lambda Ae^{\lambda x}\]

Use desolve() in SageMath to do ODE $\(2\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = 5\dfrac{\mathrm{d}y}{\mathrm{d}x}− y\)$

x = var('x'); y = function('y')(x)
show(desolve(2*diff(y, x, 2) == 5*diff(y, x) - y, y) )

x = var('x'); y = function('y')(x)
show(desolve(17*diff(y, x) == 10*diff(y, x, 2) + 3*y , y) )

5.3.6. Module 5 - Further Applications of Integration#

To find the area bounded by a parametric curve, convert \(y\) and \(\mathrm{d}x\) into terms of \(t\)

  • \[\int^{t_2}_{t_1}y\hspace{1px}\mathrm{d}x\]

For polar equations,

  • \[A=\int^{\theta_2}_{\theta_1}\frac{1}{2}r^2\mathrm{d}\theta\]

Arc Length

  • \[s=\int^{x_2}_{x_1}\sqrt{1+\begin{pmatrix}\large\frac{\mathrm{d}y}{\mathrm{d}x}\end{pmatrix}^2}\mathrm{d}x\]

Arc Length (parametric curve)

  • \[s=\int^{t_2}_{t_1}\sqrt{\large\begin{pmatrix}\frac{\mathrm{d}x}{\mathrm{d}t}\end{pmatrix}^2+\begin{pmatrix}\frac{ \mathrm{d}y}{\mathrm{d}t}\end{pmatrix}^2}\mathrm{d}t\]

Arc Length (polar curve)

  • \[s=\int^{\theta_2}_{\theta_1}\sqrt{r^2+\large\begin{pmatrix}\frac{\mathrm{d}r}{\mathrm{d}\theta}\end{pmatrix}^2 }\mathrm{d}\theta\]

For surface of revolution

  • \[S=\int_a^b2\pi y\sqrt{1+\begin{pmatrix}\frac{dy}{dx}\end{pmatrix}^2}dx\]
s(diff(2*(x-sin(x)),x))
s(diff(2*(1-cos(x)),x))
print(n(2*pi))
s(n(integral(sqrt(x^2+1)/pi,x,0,2*pi),16))
s(n(integral(sqrt(2+2*cos(x)),x,0,pi),16))
s(n(8*pi/3*(2^(3/2)-1),16))
f(x)=e^(-x/2)*(13*cos(4*x)+12*sin(4*x))
s(n(f(2*pi)))

s(diff(sec(x),x))
s(integral(sec(x),x))
print(n(ln(sqrt(3)),16))
print(n(8/27*((11/2)^(3/2)-1),16))
print(n(2*integral(sqrt(2-2*cos(x)),x,pi/2,pi),16))
print(n(8*integral(sqrt(1+x^2),x,0,2),16))
print(n(integral(sech(x),x,0,1),16))

print(n(integral(sqrt(1+9*x/4),x,0,5),30))
print(n(8/27*((49/4)^(3/2)-1),30))
print(n(integral(sqrt(1+sinh(4*x)^2),x,0,1),30))
print(n(integral(sqrt(2*(1+cos(x))),x,0,2*pi/3),30))
print(n(integral(sqrt(1+x^2),x,2,6),30))
print(integral(sqrt(1+sinh(x)^2),x,-ln(2),ln(2)))
s(expand(diff(ln(abs(sec(x))),x)^2))

s(diff(cosh(x),x))
f(x)=e^(2*x)/2-2*x-e^(-2*x)/2
g(x)=8*f
a=arcsinh(2)
b=arcsinh(1/2)
s(n(integral(4*sinh(x)^2,x,0,a)),n(f(a)-f(0)),n(g(b)-g(0)))

var('t')
a=parametric_plot((2*cosh(t),2*sinh(t)),(t,-2,2),figsize=7)
b=parametric_plot((4*cosh(t),8*sinh(t)),(t,-1.5,1.5),figsize=7)
show(a+b)

f(x)=x^3/6
s(n(f(9*pi)-f(33*pi/4)-f(7*pi)+2*f(25*pi/4)-f(11*pi/2)-f(17*pi/4)+2*f(7*pi/2)-f(11*pi/4)-f(3*pi/2)+2*f(3*pi/4)-f(0)))

var('s')
var('t')
a=parametric_plot((-2*t^2, -4*t), (t, -2, 2), figsize=5)
b=plot(-2*(x+4), (x, -8, 1), figsize=5)
show(a+b)
c=parametric_plot((2*cosh(s), 2*sinh(s)), (s, -2, 2), figsize=4)
d=parametric_plot((4*cosh(t), 8*sinh(t)), (t, -1.5, 1.5), figsize=5)
show(c+d)

t=var('t')
f(x)=x^3/6
show(parametric_plot((2*t^3, t^2), (t, 1, 2)))  # Plots
show(parametric_plot((4*cos(t), 3*sin(t)), (t, -pi/2, pi/2), figsize=3))
show(polar_plot(3*(1+cos(x)), (x, 0, 2*pi), figsize=4))
show(parametric_plot((4*cosh(t), 2*sinh(t)), (t, -2, 2), figsize=3))
show(n((e^pi-1)/4))  # Calc
show(n((3*(sqrt(3)-2)-pi)/2))
show(n(27*pi/2))
show(n(f(9*pi)-f(33*pi/4)+f(25*pi/4)-f(11*pi/2)+f(7*pi/2)-f(11*pi/4)+f(3*pi/4)))

5.3.7. Module 4 - Further Integration Methods#

The below is calculations for the definite integrals.

Trigonometric Substitutions

I_02=integral(cos(x)^2,x)
I_22=(I_02-sin(x)*cos(x)^3)/4
I_42(x)=(3*I_22-sin(x)^3*cos(x)^3)/6
s(n(I_42(pi/2)-I_42(0)))
I_13=-cos(x)^4/4
I_33(x)=(2*I_13-sin(x)^2*cos(x)^4)/6
s(I_33(pi/2)-I_33(0))
I_04=integral(cos(x)^4,x)
I_24(x)=(I_04-sin(x)*cos(x)^5)/6
s(n(I_24(pi/2)-I_24(0)))

s(integral(x*cos(x),x))
I_3(x)=x^3*sin(x)+3*x^2*cos(x)-6*x*sin(x)-6*cos(x)
s(n(I_3(pi/2)-I_3(-pi/2))) # x^3*cos(x) is an odd function
I_0(x)=sin(x)
I_2(x)=x^2*sin(x)+2*x*cos(x)-2*I_0(x)
I_4(x)=x^4*sin(x)+4*x^3*cos(x)-12*I_2(x)
I_6(x)=x^6*sin(x)+6*x^5*cos(x)-30*I_4(x)
I_8(x)=x^8*sin(x)+8*x^7*cos(x)-56*I_6(x)
s(I_8(x))
s(n(I_8(pi/2)-I_8(-pi/2)))

s(diff(tan(x),x))
print(bool((diff(tan(x),x))==sec(x)^2))
s(integral(tan(x),x))
f(x)=tan(x)^2/2-ln(abs(sec(x)))
s(n(f(pi/4)-f(0)))
s(integral(tan(x)^2,x))
g(x)=tan(x)^3/3+x-tan(x)
s(n(g(pi/4)-g(0)))
h(x)=tan(x)^5/5-g(x)
s(n(h(pi/4)-h(0)))

s(n(integral(1/((x+1)*sqrt(x^2+1)),x,0,3)))
s(n(integral(1/(2*x^2+x+2),x,0,1)))

s(n(integral(1/sqrt(x^2-x+1/2),x,1/4,1)))

s(diff(1/(x+1),x))
f(x)=ln(abs(x-1/2+sqrt(x^2-x+1/2)))
s(n((f(1)-f(1/4))/sqrt(2)))

f(x)=x^3
s(n(f(15/8)-f(0))/12)

s(n((ln(1/sqrt(2))-ln(3/sqrt(2)))/(2*sqrt(2))))
s(n(integral(1/(x^2-2),x,0,1/sqrt(2))))
f=3/((x+sqrt(3))*(x-sqrt(3))*(x-1))
s(f.partial_fraction())
s(n(3/2*((ln(sqrt(3))-ln(sqrt(3)-1))/(sqrt(3)+3)-(ln(sqrt(3))-ln(1+sqrt(3)))/(sqrt(3)-3)+ln(2))))

s(expand((x^2-x+3)*(x+1)))
s(n(2-integral((2*x+3)/(x^2-x+3),x,-1,1)))
f=4*x/(x-2)^2
s(f.partial_fraction())
g(x)=x^3/3+2*x^2+12*x+32*ln(abs(x-2))-16/(x-2)
s(n(g(1)-g(-4)))
s(expand((x^2+4*x+12)*(x-2)^2))

f=(1-2*x)/((x^2+2*x+5)*(x+2)^2)
s(f.partial_fraction())
s(integral(1/(x^2+2*x+5),x))
s(n(2/3-pi/8))

f=(1-2*x-x^2)/((x+2)^2*(x+3))
s(f.partial_fraction())
print(bool(ln(4)==2*ln(2)))
s(n(ln(3)-2*ln(2)+2/3))
s(integral(f,x,-1,1))

f=(1-2*x-x^2)/((x+2)^2*(x-3))
s(f.partial_fraction())
s(integral(1/(x+2)^2,x))
s(n(-14/25*ln(6)-1/5+11/25*ln(3)+14/25*ln(8)+1/15))

var('a')
s(integral(a/sqrt(x^2-a^2),x))
s(n(3*(ln(6+sqrt(27))-ln(3))))
s(n(5/2*(2^(2/5)-(4)^(2/5))))
s(integral(1/sqrt(a^2-x^2),x))

show(1/6*(sqrt(2)*arctan(sqrt(2)/2*x)+1/2*ln(x**2+2)+5*ln(x-2)))
show(n(integral((x^2+1)/((x^2+2)*(x-2)), (x, 3, 5))))

# Ex. 07 Improper Integrals and Partial Fractions
show(n((ln(6)-ln(2))/4))
show(plot( 1/(x^2)^1/3, (x, -2, 2), figsize=4, ymin=0, ymax=10))
show(n( 6*2^(1/3) ))
show(n(ln(3)-ln(sqrt(5))))
show(n( ( ln(3)-ln(8)+ln(6) )/5 ))
show(n( 3*ln(4)-3*ln(5)+ln(2) ))
show(n( 2*ln(3)-ln(5)/2-2/15 ))

5.3.8. Module 3 - Properties of Curves#

The cartesian and parametric equations, eccentricity, foci and directrices for various curves are below.

Parabola

  • Cartesian Equation : \(y^2=4ax\)

  • Parametric Equation : \(x=at^2, y=2at\)

  • Eccentricity : \(1\)

  • Foci : \((a, 0)\)

  • Directrices : \(x=-a\)

Ellipse

  • Cartesian Equation : \(\large\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a\gt b\)

  • Parametric Equation : \(x=a\cos{t}, y=b\sin{t}\)

  • Eccentricity : \(b^2=a^2(1-e^2)\)

  • Foci : \((\pm ae, 0)\)

  • Directrices : \(x=\large\pm\frac{a}{e}\)

Hyperbola

  • Cartesian Equation : \(\large\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)

  • Parametric Equations : \(x=a\sec{t}, y=b\tan{t}\) or \(x=a\cosh{t}, y=b\sinh{t}\)

  • Eccentricity : \(b^2=a^2(e^2-1)\)

  • Foci : \((\pm ae, 0)\)

  • Directrices : \(x=\large\pm\frac{a}{e}\)

show(plot(x/(x^2-4), (x, -5, 5), figsize=4, ymin=-10, ymax=10))
show(plot((1-x^2)/(1+x^2), (x, -7, 7), figsize=4))
show(n(sqrt(4/27)))

Below is plots and calculations for exercise 5.

t=var('t')
show(n(sqrt(1/2)))
show(n(sqrt(4/3)))
show(n(sqrt(9/7)))
show((216+36)/6)
show(84-36)
show(plot(9/x, (x, -10, 10), figsize=3, ymin=-10, ymax=10))
show(parametric_plot((cos(t)^3, sin(t)^3), (t, 0, 2*pi), figsize=3))
show(n(8*sqrt(3)))
show(n(2*sqrt(3)))
show(diff(sin(x)^3))

5.3.9. Module 2 - Further Complex Numbers#

Why complex numbers?

  • It bridges between 1D and 2D space.

  • The 1D space is just the real number line.

  • The 2D space is the cartesian plane which plots the xy co-ordinates or the argand diagram which used for plotting complex numbers.

The sum of a geometric series is $\(\frac{a(r^n-1)}{r-1}\)\( where \)a\( is the first number, \)r\( is the common ratio and \)n$ is the number of terms.

Some properties of roots of unity.

The sum of the roots of unity of the \(n\)th root add up to 0 $\(\frac{1+iz}{1-iz}=\omega^k\)\( for \)k=0, 1, 2, \cdots, n-1$

The \(n\)th roots of \(re^i\theta\) are \(\alpha, \alpha\omega, \cdots, \alpha\omega^{n-1}\),

when \(\large\omega=e^{\frac{2\pi}{n}i}\) and \(\large\alpha=r^\frac{1}{n}e^\frac{i\theta}{n}\)

For a triangle with corners \(A\), \(B\) and \(C\), where \(a\), \(b\) and \(c\) represent their complex number vectors, $\(\frac{c-a}{b-a}=ke^{i\theta}\)\( \)\(k=|\frac{c-a}{b-a}|\)\( \)\(\theta=\arg{\frac{c-a}{b-a}}\)$

show(expand((1+x)*(1+x^2))) 
angle=-arctan(1/2)/3+2*pi/3
moduli=5^(1/6)
show(n(cos(angle)*moduli))
show(n(sin(angle)*moduli))
show(144*28)

5.3.10. Module 1 - 1st Order ODE#

Below are calculations using Euler’s method and the improved Euler method

(1.148721271-1.147447)*100/1.148721271

show(n(e^0.5-0.5))

x=1
y=0
h=.1
for i in range(6):
    k1=h*(x^2-y/x)
    k2=h*((x+h)^2-(y+k1)/(x+h))
    y1=y+(k1+k2)/2
    show(i+1)
    show(n(y1, digits=7))
    x+=h
    y=y1

x=1
y=0
y1=0
h=.1
y2=0
y=h*(x^2-y/x)
y1=y2
x+=h
y2=y
for i in range(5):
    show(y)
    y=y1+2*h*(x^2-y/x)
    y1=y2
    x+=h
    y2=y
show(n(y))

x=1
y=0
h=.1
for i in range(6):
    a=x^2-y/x
    y+=a*h
    x+=h
    show(n(y))

a=1*sqrt(1)/10
b=1.1*sqrt(a+1)/10+a
c=1.2*sqrt(b+1)/10+b
d=1.3*sqrt(c+1)/10+c
f=1.4*sqrt(d+1)/10+d
g=1.5*sqrt(f+1)/10+f
h=1.6*sqrt(g+1)/10+g
show(n(a))
show(n(b))
show(n(c))
show(n(d))
show(n(f))
show(n(g))
show(n(h))

5.4. ★Further Y12-2#

5.4.1. Assessment 3#

x=var('x')
f(x)=e^(2*x)*(2*tanh(x)+sech(x)^2)
show(f(ln(2)))
g(x)=3*(e^x)^6+(e^x)^4-31*(e^x)^2+3
show(plot(g(x), (x, -2, 0.6), figsize=4))
show(n((126/7+2+1/24+1/896)/4))
show(n((6+1/2+1/160)/4))

f(x)=e^(2*x)*(2*cosh(x)+sinh(x))
show(f(ln(2)))
g(x)=sech(x)^2
show(g(ln(2)))
h(x)=e^x*(tanh(x)-sech(x)^2)/tanh(x)^2
show(h(ln(2)))

# Add circular gridlines
circular_gridlines = Graphics()
for r in [0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4]:  # Adjust radii as needed
    circular_gridlines += circle((0, 0), r, linestyle="--", color="red", alpha=0.6)

x=var('x')
show(n(sqrt(3+9)))
show(n(pi/3))
show(polar_plot(3+2*cos(4*x), (x, -pi, pi), figsize=4) + circular_gridlines)
show(polar_plot(2+2*sin(6*x), (x, 7*pi/12, 11*pi/12), figsize=4))
show(polar_plot(6*cos(x), (x, -pi, pi), figsize=4))
show(polar_plot(4.5/(1+cos(x)), (x, -pi/2, pi/2), figsize=4))
f(x)=(2*x+1)*(sech(x^2+x)^2+1/(1-(2*x+1)^2))
show(f(0))

5.4.2. Module 8 - Hyperbolic Functions#

\[\sinh{x}=\frac{e^x-e^{-x}}{2}\]
\[\cosh{x}=\frac{e^x+e^{-x}}{2}\]
\[\tanh{x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}\]
\[\mathrm{csch}\hspace{2px}x=\frac{2}{e^x-e^{-x}}\]
\[\mathrm{sech}\hspace{2px}x=\frac{2}{e^x+e^{-x}}\]
\[\mathrm{coth}\hspace{2px}x=\frac{e^x+e^{-x}}{e^x-e^{-x}}\]
\[\mathrm{arcsinh}\hspace{2px}x=\ln{(x+\sqrt{x^2+1})}\]
\[\mathrm{arccosh}\hspace{2px}x=\ln{(x+\sqrt{x^2-1})}\]

for \(x\geqslant1\) $\(\mathrm{arctanh}\hspace{2px}x=\frac{1}{2}\ln{(\frac{x+1}{x-1})}\)\( for \)-1\leqslant x \leqslant1$

\[f(x)=\sinh{x}\]
\[f'(x)=\cosh{x}\]
\[f''(x)=\sinh{x}\]
\[g(x)=\tanh{x}\]
\[g'(x)=\mathrm{sech}^2{x}\]
\[h(x)=\mathrm{cosech}{x}\]
\[h'(x)=-\mathrm{cosech}{x}\mathrm{coth}{x}\]
\[y=\mathrm{sech}{x}\]
\[y'=-\mathrm{sech}{x}\tanh{x}\]
\[\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{coth}{x}=-\mathrm{cosech}^2{x}\]
\[\int\frac{1}{\sqrt{x^2-a^2}}\mathrm{d}x=\cosh^{-1}\frac{x}{a}+c\]
\[\int\frac{1}{\sqrt{a^2+x^2}}\mathrm{d}x=\sinh^{-1}\frac{x}{a}+c\]
\[\frac{\mathrm{d}}{\mathrm{d}x}\frac{1}{a}\tanh^{-1}\frac{x}{a}=\frac{1}{a^2-x^2}\]
\[\int\tanh{x}\mathrm{d}x=\ln{(\cosh{x})}+c\]
\[\int\mathrm{coth}{x}{d}x=\ln{(sinh{x})}+c\]
# Ex. 16 Working with Hyperbolic Functions
a=sinh(1)
b=cosh(1)
f(x)=3*cosh(2*x)*cosh(3*x)-2*sinh(2*x)*sinh(3*x)
g(x)=cosh(5*x)+5*cosh(x)
show(n(e*(a+b)))
show(n((b-a)/e))
show(n(e*(a-b)/a^2))
show(n( -(a+b)/(e*a^2) ))
show(n(f(ln(2))-f(0)/5))
show(n(g(ln(2))-g(0)/10))

# Ex. 15 Hyperbolic Functions
x=var('x')
f(x)=6*x^4-17*x^3+12*x^2-7*x+6
show(arctanh(0.9))
show(n(log((3+sqrt(30))/7)))
show(n(log(1/sqrt(5))))
show(f(2))
show(f(3))
show(f(5))
show(f(1/2))
show(f(1/3))
show(f(1/5))
show(plot(f(x), (x, -3, 3), figsize=3))

# Hyperbolic plots
x=var('x')
show(plot(sinh(x), (x, -5, 5), figsize=3))
show(plot(cosh(x), (x, -5, 5), figsize=3))
show(plot(tanh(x), (x, -5, 5), figsize=3))
show(plot(csch(x), (x, -1, -0.1), figsize=3))
show(plot(sech(x), (x, -5, 5), figsize=3))
show(plot(coth(x), (x, -1, 1), figsize=3))

5.4.3. Module 7 - An Introduction to Polar Coordinates#

x=var('x')
show(n(pi/2))
show(n(pi/4))
show(n(5*pi/3))
show(n(pi-arctan(4/3)))
show(n(sqrt(3)))
show(polar_plot(2*cos(x), (x, 0, 2*pi), figsize=3))
show(polar_plot(cos(x)+sqrt(cos(x)^2+4), (x, 0, 2*pi), figsize=3))

5.4.4. Assessment 2#

show(n(ln(5)))
f(x)=x+x^3/12+x^5/80
show(n(f(4/3)))
g(x)=-cos(6*x)+6*cos(4*x)-15*cos(2*x)+10
show(n((g(pi/2)-g(pi/4))/32))
show(n(integrate(sin(x)^6, (x, pi/4, pi/2))))

x=var('x')
t=var('t')
y=var('y')
f(x)=x^8/8
f(4)-f(2)
g(x)=-1/x
show(g(7)-g(3))
h(x)=sin(x)
show(n((sin(pi/2)-sin(0))/(pi/2)))
parametric_plot((3*cos(t), 2*sin(t)), (t, 0, 2*pi))
j(t)=2*t^3/3+3*t^2/2
show(j(12)-j(6))
show(diff(x*y*(x-1), x, 2))

5.4.5. Module 6 - Complex Numbers - deMoivre’s Theorem#

deMoivre’s Theorem :

Let \(z=r(\cos{\theta}+i\sin{\theta})\).

For any integer n, $\(z^n=r^n(\cos{n\theta}+i\sin{n\theta})\)\( \)\( \)$

Let \(z=\cos{\theta}+i\sin{\theta}\).

Then using deMoivre’s Theorem : $\(z^n=\cos{n\theta}+i\sin{n\theta}\)\( \)\(z^{-n}=cos{n\theta}-i\sin{n\theta}\)\( So, \)\(z^n+z^{-n}=2\cos{n\theta}\)\( \)\(z^n-z^{-n}=2i\sin{n\theta}\)\( \)\( \)$

\[\large 1+e^{i\theta}=e^{i\frac{\theta}{2}}e^{-i\frac{\theta}{2}}+e^{i\frac{\theta}{2}}e^{i\frac{\theta}{2}}\]
\[\large =e^{i\frac{\theta}{2}}(e^{-i\frac{\theta}{2}}+e^{i\frac{\theta}{2}})\]
\[\large =2\cos{\frac{\theta}{2}}e^{i\frac{\theta}{2}}\]
# Ex. 12 Complex Exponents
64*4

# Ex. 11 deMoivre's Theorem
show(integrate(sin(x)^5, x))
show(integrate(cos(7*x), x))
4-12/35
show(integrate(cos(7*x)-cos(5*x)-3*cos(3*x)+3*cos(x), x, 0, pi/2))
f(x)=-cos(7*x)/7-cos(5*x)/5+cos(3*x)+3*cos(x)
show(f(pi/2)-f(0))
\[\cos{\left( 4 \theta \right)} = a_0 \cos^4{\theta} + a_1 \cos^3{\theta} + a_2 \cos^2{\theta} + a_3 \cos{\theta} + a_4\]

5.4.6. Module 5 - An Introduction to Maclaurin Series#

Standard Maclaurin Series Expansions

\[\mathrm{f}(x)=\mathrm{f}(0)+x\mathrm{f}'(0)+\frac{x^2}{2!}\mathrm{f}''(0)+\dots+ \frac{x^r}{r!}\mathrm{f}(r)(0)+\dots\]
\[e^x=1+x+\frac{x^2}{2!}+\dots+\frac{x^r}{r!}+\dots\]
\[\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\dots+(-1)^{r+1}\frac{x^r}{r}+\dots\]

for \(-1\lt x\leqslant1\)

\[\sin{x}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots+(-1)^r\frac{x^{2r+1}}{(2r+1)!}+\dots\]
\[\cos{x}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\dots+(-1)^r\frac{x^{2r}}{(2r)!}+\dots\]
x=var('x')
show(plot(log((1-2*x)/(1+2*x)^2), x, -20, 0.5))
\[\ln{\sqrt{\frac{1-x}{1+x}}}\]
\[\downarrow\]
\[\frac{\ln{\frac{1-x}{1+x}}}{2}\]
# Ex. 10
x=var('x')
f(x)=2*x+2*x^3/3
show((e^(3*x)-e^(-x))/e^x)
n(f(1/3))
show(diff(ln(1+x)*e^x, x))
show(diff(e^x/(1+x), x))
show(diff(e^x/(1+x)^2, x))
show(diff(arcsin(x), x))
show(diff(arcsin(x), x, 2))
show(diff(arcsin(x), x, 3))

# Ex. 09 con.
x=var('x')
y=var('y')
show(diff(log(x^2+1), x))
show(diff(log(x^2+1), x, 2))
show(diff(log(x^2+1), x, 3))
show(diff(log(x^2+1), x, 4))
show(diff(1-y*x^2, x))
show(diff(1-y*x^2, x, 2))
show(diff(1-y*x^2, x, 3))

# (* Ex. 09 con. *)
x=var('x')
show(diff(cos(x)^2, x))
show(diff(-sin(2*x), x))
show(diff(arctan(x), x))
show(diff(diff(arctan(x), x), x))
show(diff(diff(diff(arctan(x), x), x), x))

Series[Tan[x], {x, 0, 5}]
Series[Cos[x]^2, {x, 0, 4}]
Series[ArcTan[x], {x, 0, 5}]
Series[E^(x^2), {x, 0, 4}]
\[\mathrm{f}(x)=\tan(x)\]
\[\mathrm{f}^{(4)}(x)=16\tan(x)+40\tan^3(x)+24\tan^5(x)\]
\[\mathrm{f}^{(5)}(x)=16+136\tan^2(x)+240\tan^4(x)+120\tan^6(x)\]
(* Ex. 09 Maclaurin Series *)
Expand[(1-x+x^2/2-x^3/6)(1+x+x^2/2+x^3/6)]
D[Tan[x], x]  (* Tan[x]^2+1 *)af
D[Tan[x]^2, x]  (* 2Tan[x]+2Tan[x]^3 *)
D[Tan[x]^3, x]  (* 3Tan[x]^2+3Tan[x]^4 *)
D[Tan[x]^4, x]  (* 4Tan[x]^3+4Tan[x]^5 *)
D[Tan[x]^5, x]  (* 5Tan[x]^4+5Tan[x]^6 *)

5.4.7. Module 4 - Applications of Integration#

Parametric Rewriting for Volume of Revolution

For parametric equations, when t is the unit(\(a \leqslant t \leqslant b\)) :

For rotation about the \(x\) axis,

\[V_x = \pi\int^{x_b}_{x_a} y^2 \mathrm{d}x = \pi\int^{b}_{a} y^2 \frac{\mathrm{d}x} {\mathrm{d}t}\mathrm{d}t\]

For rotation about the \(y\) axis,

\[V_y=\pi\int^{y_b}_{y_a} x^2 \mathrm{d}y = \pi\int^{b}_{a} x^2 \frac{\mathrm{d}y}{\mathrm{d}t}\mathrm{d}t\]

If not parametric equations, then just use first part of formula.

(* Ex. 08 The mean of a Function *)
4/(3Pi)//N
(Sqrt[Pi]/2)^2

(* Ex. 08 The mean of a Function *)
Expand[(x^2+2)^2]
2/Sqrt[3]//N
(32/5+32/3+8)/2//N   (* Find mean values by using definite integral to calculate area *)
(E^5-27/2)/5//N
(Log[4]-Log[6]+Log[3])/2/3//N

Plot[2ArcCos[x], {x, 0, 1}]

f[t_]:=-4Sin[t]^3
Plot[f[t], {t, -3Pi, 3Pi}]

(* Ex 07 Volumes of Revolution *)
Pi(2^5-1^5)*0.6//N
Expand[(x^2+1)^2]
348Pi/5//N
62Pi/5//N
Pi^2/2//N
16Pi/3//N
Integrate[-4Sin[t]^3, {t, Pi, 0}]
Integrate[-4Sin[t]t^2, {t, 0, Pi/2}]//N
Pi*Integrate[4-x^2, {x, 0, 2}]//N
256Pi/105//N
Pi*Integrate[-2x^2/Sqrt[1-x^2], {x, 0, Pi}]//N
Pi*Integrate[(2x-x^3/2), {x, 0, 2}]//N
128Pi/21//N

(* Plot revolution *)
f[x_] := Sqrt[3 x - 2]  (*MMA 空格 是乘号 *)
RevolutionPlot3D[f[x], {x, 0, 5}, RevolutionAxis -> "x",  AspectRatio -> 1]

5.4.8. Module 3 - Further Differentiation and Integration#

Integration methods :

\[\int f'(x)g'(f(x))=g(f(x))+c\]
\[\int \frac{f'(x)}{f(x)}\hspace{1px}\mathrm{d}x=\ln{f(x)}+c\]

Trigonometric identities for integration :

\[\cos^2{x}\equiv\frac{1}{2}(1+\cos{2x})\]
\[\sin^2{x}\equiv\frac{1}{2}(1-\cos{2x})\]

\(x^2+25=169(yz+144)/(288+144x+16y-9z+xyz)\)

\(y^2+153=169(xz+16)/(288+144x+16y-9z+xyz)\)

\(z^2+160=169(xy-9)/(288+144x+16y-9z+xyz)\)

\[\hspace{10px} \dfrac{\mathrm{d}y}{\mathrm{d}x} = - \mathrm{cosec}^2 \hspace{2px} \left(\frac{x}{a} \right)\]
\[\hspace{10px} \dfrac{\mathrm{d}y}{\mathrm{d}x} = \tan{\left( \frac{x}{a} \right)} \sec{\left( \frac{x}{a}\right)}\]
\[\hspace{10px} \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{a^2}{\sqrt{1 - \left( ax \right)^2} }\]
\[\hspace{10px} \dfrac{\mathrm{d}y}{\mathrm{d}x} = -\cot{\left( \frac{x}{a} \right)} \hspace{2px} \mathrm{cosec} \hspace{2px} \left( \frac{x}{a} \right)\]
\[\hspace{10px} \dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{a^2}{\sqrt{1 - \left( ax \right)^2}}\]
\[\hspace{10px} \dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{a^2}{1 + \left( ax \right)^2}\]
(* Ass. 1 Calc (con.) *)
mat1={{1, 2, 4}, {2, 4, 5}, {0, 1, 2}};
mat2={{-3}, {3}, {6}};
Inverse[mat1].mat2//MatrixForm
D[Tan[x],x]
D[Log[u],u]
D[Cos[x]^2,x]
Integrate[Sec[x]^2,x]
Integrate[Sec[x]^2Tan[x],x]
D[Cos[3x]^5,x]
(Log[16]-Log[2])/2//N
Pi/9//N
(ArcTan[2]-ArcTan[2/Sqrt[3]])/(2Sqrt[3])//N

(* Assessment 1 Calculation *)
4-18+6-10
20-12-8+36
72-16-24+40
mat1={{12, 3, -4}, {3, 4, 12}, {4, -12, 3}};
Det[mat1]
MatrixForm[Transpose[mat1].mat1]
MatrixForm[mat1]
FactorInteger[2197]
Inverse[mat1]

N[(Pi-2)/16]
N[12-3Pi]
D[Cos[x]^3,x]

1/(2Sqrt[1-3^2])==1/Sqrt[4-4*(3^2)]
N[Pi/(3Sqrt[3])]
f[x_]:=ArcTan[(3/4)x]/4
N[f[4/3]-f[0]]
Integrate[Cot[x],x]
D[Cot[x],x]
g[x_]:=(-Csc[x/2-1]^2)/2
g[3]

D[Cot[x],x]
D[ArcCos[x],x]
Integrate[Sec[x]^2,x]
D[Sin[x]^4,x]
Integrate[1/Sqrt[4-3x^2],{x,0,1}]
D[ArcSin[Sqrt[3/4]x]/2,x]
D[ArcTan[(3/4)x]/4,x]

D[Sin[1/x],x]
D[Log[2x],x]
D[Csc[x],x]
D[ArcTan[x],x]
D[Sec[x],x]

5.4.9. Module 2 - Mathematical Induction#

For mathematical induction if \(n=k\) then \(n=k+1\) so if for \(n=1\) the claim is true then for \(\Z_+\).

For series use formula to check if \(n=k+1\) will be true.

For divisibility check if \(u_{k+1}-u_k\) is divisible by \(n\).

n=1
n=3n+4
n=3n+4
n=3n+4
f[x_]:=3^x-2
f[1]
f[2]
f[3]
f[4]

5.4.10. Module 1 - Determinant and Inverse of 3 by 3 Matrix#

mat1={{1, 0, 0}, {0, 0, 1}, {0, -1, 0}}//MatrixForm

mat1={{1, 0, 0}, {0, 0, 1}, {0, -1, 0}}//MatrixForm
mat2={{0, -1, 0}, {1, 0, 0}, {0, 0, 1}};
mat3=mat1.mat2;
mat4={{-1, 0, 0}, {0, -1, 0}, {0, 0, 1}};
mat5={{13, 53, 10}, {4, -7, -2}, {2, -8, -1}}/9;
mat6={{3}, {-2}, {4}};
mat7={{1, 1, -1}, {0, 6, 2}, {-3, 1, 1}};
MatrixForm[mat3]
MatrixForm[mat2.mat1]
Det[mat3]-Det[mat1]Det[mat2]
MatrixForm[mat4.mat4]
MatrixForm[mat5.mat6]
Det[mat7]

dist=BinomialDistribution[40,0.2];
ans=1-CDF[dist,11]

dist=NormalDistribution[12,3.5/40];
ans=CDF[dist,10.8]

dist=NormalDistribution[12,3.5];
N[CDF[dist,15]-CDF[dist,10]]

dist=NormalDistribution[];
quan1=Quantile[dist,0.05]
quan2=Quantile[dist,0.75]

mat1={{1, 1, 1}, {x, y, z}, {x^2, y^2, z^2}}//MatrixForm;
mat2={{p, q, r}, {p^2, q^2, r^2}, {qr, pr, pq}};
Det[mat1]
Simplify[Det[mat2]]
12*2-53*4-3*5*10+3*53-2*10+4*12*5
3*4-3*3-2*5*4-2*2

5.5. ★Further Y12-1#

5.5.1. Module 7 - Vectors 1#

ArcCos[0.44141]*180/Pi

44+29
29*4-30
3*29-21

N[Sqrt[(73/29)^2+(86/29)^2+(66/29)^2]]

N[Sqrt[(27/10)^2+(9/10)^2]]

Sqrt[75]
N[7*5Sqrt[3]/14]

N[ArcCos[11/14]*180/Pi]

vec1={{2}, {3}, {2}};
vec2={{6}, {-2}, {1}};
vec1-vec2*3/2 

(-2/7)^2+(8/7)^2+(18/7)^2

Angle between 2 vectors

\(\cos{\theta}=\large\frac{a_1b_1+a_2b_2}{|\mathbf{a}||\mathbf{b}|}\)

Scalar Product/Dot Product

\(\mathbf{a}\mathbf{b}=ab\cos{\theta}\)

\(\mathbf{a}\mathbf{b}=a_1b_1+a_2b_2\)

5.5.2. Module 3 and 6 - Further Algebra and Functions 1 and 2#

mat1={{1, -1, 2}, {2, 1, 1}, {2, -2, 1}};
mat2=Inverse[mat1];
mat3={{9}, {0}, {3}};
mat4=mat2.mat3;
f[x_]:=x(x+1)/2
g[x_]:=x(x+1)(2x+1)/6
h[x_]:=2g[x]+3f[x]
j[x_]:=x^2-17x+22
k[x_]:=x(x+1)^2
MatrixForm[mat2]
MatrixForm[mat4]
36*7
f[50]-f[20]
g[100]-g[90]
f[50]-f[15]+5*35
h[50]-h[25]
j[1]
j[3]
j[-2]
k[30]-k[20]

f[x_]:=2x(x+1)
g[x_]:=x(x+1)(2x+1)+2x
h[x_]:=x(x+1)(x+2)(x+3)
i[x_]:=x(x-1)(x+1)(x+2)
f[20]-f[9]
g[10]-g[4]
h[1]
h[2]
h[3]
i[1]
i[2]
i[3]
16*4*9

35*4.2
35*0.42*0.58

dist=BinomialDistribution[35,0.42];
1-CDF[dist,17]

dist=NormalDistribution[34,4.5];
ans=1-CDF[dist,36.2]

dist=NormalDistribution[];
ans=1.5Quantile[dist,0.95]
10-ans

dist=NormalDistribution[1.78,.23];
N[CDF[dist,1.82]-CDF[dist,1.82]]
N[CDF[dist,1.9]-CDF[dist,1.7]]^2

45-17-15-4
70-31-17-15
N[101/240]
240-101-7-9
123/3
N[41/80]
240-101-31-17-7-15-4-9
101+31+7
N[139/195]
69.2/40
Sqrt[2.81/40]
1.78-0.23*2

dist=BinomialDistribution[20,0.3];
N[PDF[dist,1]]
N[CDF[dist,3]]
1-N[CDF[dist,9]]

f[x_]:=x^3+4x^2-x+2
g[x_]:=5x^2-9x+22
h[x_]:=x^2+2x+9
f[2]
g[4]
h[4]

f[x_]:=4x(4x+1)(8x^2-1)
g[x_]:=2(x-1)(2x-1)(8(x-1)^2-1)
Expand[f[x]-g[x]]

n=4;
f[x_]:=x(x+1)/2
g[x_]:=x(x+1)(2x+1)/6
h[x_]:=x^2(x+1)^2/4
n(n+1)(8n-5)/6
Simplify[Expand[2h[4x]-6g[4x]]]

n=20;
n1=15;
n2=4;
a=40;
b=19;
25*51
n(n+1)(2n+1)/6
15*31+60-5*9-18
(n1(n1+1)(2n1+1)-n2(n2+1)(2n2+1))/6
2(25*51-1.5*50+9*1.5-5*9)
(a(a+1)(2a+1)-b(b+1)(2b+1))*5/6-84
Expand[(x+1)(x+2)(x-3)]
4*5*9

n=Pi/6;
f[x_]:=Sin[2x]-x
g[x_]:=f'[x]
g[x]
n=N[n-f[n]/g[n]]

dist=BinomialDistribution[25,0.7];
N[1-CDF[dist,20]]

lower=N[Quantile[NormalDistribution[],0.6]]
upper=N[Quantile[NormalDistribution[],0.8]]
sd=0.2/(upper-lower)
5.9-lower*sd

dist=NormalDistribution[24500,5200];
ans=1-N[CDF[dist, 26730]]

Expand[(2a-1)(2b-1)+(2a-1)(2c-1)+(2b-1)(2c-1)]

Simplify[Divide[f[z],z-3]]

f[z_]:=z^3-z^2+11z-51;
Plot[f[z],{z,1,5}]

Suppose \(ax^2+bx+c=0\) has roots \(x=\alpha\) and \(x=\beta\)

Then \(\alpha+\beta=-\large\frac{b}{a}\) and \(\alpha\beta=\large\frac{c}{a}\)

Suppose \(ax^3+bx^2+cx+d=0\) has roots \(x=\alpha\), \(x=\beta\) and \(x=\gamma\)

Then \(\alpha+\beta+\gamma=\large\frac{-b}{a}\), \(\alpha\beta+\alpha\gamma+\beta\gamma=\large\frac{c}{a}\), \(\alpha\beta\gamma=-\large\frac{d}{a}\), \(\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\alpha\gamma+\beta\gamma)\) and \(\large\frac{1}{\alpha}+\large\frac{1}{\beta}+\large\frac{1}{\gamma}=\large \frac{\alpha\beta+\alpha\gamma+\beta\gamma}{\alpha\beta\gamma}\)

Suppose \(ax^4+bx^3+cx^2+dx+e\) has roots \(x=\alpha\), \(x=\beta\), \(x=\gamma\) and \(x=\delta\)

Then \(\Sigma\alpha=-\large\frac{b}{a}\), \(\Sigma\alpha\beta=\large\frac{c}{a}\), \(\Sigma\alpha\beta\gamma=-\large\frac{d}{a}\) and \(\alpha\beta\gamma\delta=\large \frac{e}{a}\)

b+c+d

a=3;
b=2a;
c=a-2;
d=1/a;
N[b*c+c*d+b*d]

1+12+18-81

Expand[(3a+1)(3b+1)(3c+1)]

Expand[(3a+1)(3b+1)+(3a+1)(3c+1)+(3b+1)(3c+1)]

-3/4-6+16

1.5^3-3*3*1.5+3*1

Expand[(a+b+c)^3]

1-3/2+6/2-2/2

1-(a+b+c)+(ab+ac+bc)-abc=1+1.5+3+1

Expand[(1-a)(1-b)(1-c)]

5.5.3. Modules 2 and 5 - Matrices 1 and 2#

Matrices for transformations:

Reflection in the x axis:

\[\begin{split}\begin{pmatrix}1&&0\\0&&-1\end{pmatrix}\end{split}\]

Reflection in the y axis:

\[\begin{split}\begin{pmatrix}1&&0\\0&&1\end{pmatrix}\end{split}\]

Reflection in the line y=x axis:

\[\begin{split}\begin{pmatrix} 0 && 1 \\ 1 && 0 \end{pmatrix}\end{split}\]

Anticlockwise rotation of theta about the origin:

$$\begin{pmatrix}\cos{\theta}&&-\sin{\theta}\ \sin{\theta}&&\cos{\theta}\end{pmatrix}

k=5;
mat1={{3,1,2},{1, -2, 1}, {-1, 4, 1}};
mat2={{k+1, 8, -4}, {-3, k-4, 7}, {3, -11, 2k+3}};
mat3={{3, -2, 2}, {2, 3, -1}, {1, 3, 1}};
mat4={{2, 3, -4}, {1, -1, -2}, {5, 2, 3}};
mat5={{2, 3, -1}, {1, -2, -3}, {1, 5, 2}};
mat6=Inverse[mat4];
mat7={{11},{8},{7}};
mat8=mat6.mat7;
Det[mat1]
Det[mat3]
MatrixForm[mat2]
MatrixForm[mat6]
Det[mat5]
MatrixForm[mat8]

4*27-7*33
3*27+5*33
6*3-7*1+5*1
6*1+7*2-5*4
6*2-7*1-5*1
2*3-5*1-1
6*16-7*13+5*15
2*16-5*13-15
13^2-15*7-16*2
6*3-2*8-4*22

f[x_]:=x^3-2x^2-5x+6;
g[x_]:=x-1
Simplify[f[x]/g[x]]

p={{1,-2},{3,-2}};
q={{-2,1},{1, -2}};
r={{-1, -2}, {-3, 4}};
MatrixForm[p.q]
MatrixForm[r.q]
MatrixForm[p.r]

Det[mat1]

mat1={{3,-8}, {-5, 7}};
MatrixForm[mat1.unitsquare]

mat1={{-7,-2}, {4, -2}};
MatrixForm[mat1.unitsquare]

mat1={{5,-1},{-3,1}};
MatrixForm[mat1.unitsquare]

unitsquare={{0,0,1,1},{0,1,1,0}};
mat1={{2,2},{-1,4}};
MatrixForm[mat1.unitsquare]

mat1={{4,2},{3,6}}
Det[mat1]

Identity matrices have 1’s on the lead diagonal that stretches from the top-left corner to the bottom-right corner.

For example, below is 3 by 3 identity matrix:

\[\begin{split}\mathbf{I} = \begin{pmatrix} 1 && 0&&0 \\ 0 && 1&&0\\0&&0&&1 \end{pmatrix}\end{split}\]
(* Ex 05 q 7 *)
mat1={{-3,2},{-8,5}};
mat2={{x},{2x+1}};
MatrixForm[mat1.mat2]

mat1={{3,7},{2,4}};
mat2={{2,3},{-1,2}};
mat5={{3,6,1},{-4,-2,5}};
mat7=IdentityMatrix[3];
mat6=mat1.mat5;
mat3=mat1*mat2;
mat4=mat1.mat2;
MatrixForm[mat3]
MatrixForm[Transpose[mat1]]
MatrixForm[mat6]
MatrixForm[mat7]

mat1={{1,0},{0,1}};
mat2={{-1,0},{-1,0}};
MatrixForm[mat1.mat2]

5.5.4. Module 1 and 4 - Complex Numbers 1 (Introduction) Complex Numbers 2 (Complex Plane)#

ArcTan[0.5]
N[ArcTan[2]]

N[3E^(3I)]

Expand[(a+b*I)^3]

N[2E^I]

0.1415*-2.4034-1.7954*1.995
0.1415*1.7954+1.9950*-2.4034

N[Cos[-1]]
N[Sin[-1]]
N[Cos[-2]]
N[Sin[-2]]
N[Cos[2]]
N[Sin[2]]
2Cos[1.5]
2Sin[1.5]
3Cos[2.5]
3Sin[2.5]

(1.081^2+1.683^2)(-0.909^2-0.416^2)

(3+4I)(3-4I)

f[z_]:=(2z-2I)/(1+I)-3+I
NSolve[f[z],z]

Pi-ArcTan[1.5]
Contents