Introduction to MATLAB Exercise 2

Symbolic calculations

1. Find the roots of the polynomial \(x^5+x^4+1=0\) in three different ways:
   • Define the anonymous function ( fun1 = @(x) ...), using fzero
   • Define the vector of the coefficients of the polynomial (do not miss those zeros!), using roots
   • Solve it symbolically using solve
   You can convert symbolic variable to floating point numbers using double.
2. Validate the following indefinite integral symbolically, either using int to integrate or diff to differentiate the right hand side. (Remember in MATLAB the natural logarithm is log and the inverse tangent function is atan). You may also have to simplify your answer, to confirm the results clearly.
   • \(\displaystyle \int \frac{dx}{(a^2+x^2)^{3/2}} = \frac{x}{a^2\sqrt{a^2+x^2}}\)
   • \(\displaystyle \int \ln(ax+b)dx = \left(x+\frac{b}{a}\right)\ln(ax+b)-x\)
   • \(\displaystyle \int \ln(x^2+a^2)dx = x\ln(x^2+a^2)+2a\arctan\frac{x}{a}-2x\)
3. Evaluate the following definite integrals symbolically (using int): \begin{equation} \int_{-\infty}^\infty \frac{\sin x}{x} dx,\qquad \int_0^1 \frac{1}{1+x^2} dx,\qquad \int_0^{\infty} e^{-x^2} dx,\qquad \int_0^1 \frac{\ln(x+1)}{x^2+1}dx,\qquad \int_0^1 \frac{x^{11}-1}{\ln x} dx. \end{equation}
4. Use symsum (symbolic sum) to validate the following sums (use inf for \(\infty\)) \begin{equation} \sum_{n=1}^\infty \frac{1}{3^nn} = \log\frac{3}{2},\qquad \sum_{n=1}^\infty \frac{n}{3^n} = \frac{3}{4},\qquad \sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}. \end{equation}
5. Use symprod to evaluate the following infinite product (notice the starting value of \(n\) in each expressions):
   • You should get a simple answer for the following two: \begin{equation} \prod_{n=3}^\infty \left(1-\frac{4}{n^2}\right),\qquad \prod_{n=1}^\infty \left(\frac{4n^2}{4n^2-1}\right) . \end{equation}
   • You should get a complicated expression for the following two: \begin{equation} \prod_{n=2}^\infty \left(1-\frac{1}{n^4}\right),\qquad \prod_{n=2}^\infty \frac{n^3-1}{n^3+1}. \end{equation} You should simplify the output simplify by specifying how many steps (the default step is just one), because the output could still be complicated. When using simplify you may have to tell matlab to try more steps before giving up.
   • Validate the following two infinite products by showing that both sides agree with several digits (MATLAB fails to simplify the output even using simplify with many steps of simplification). you can use double to convert symbolic numbers to floating numbers, and use vpa if you want to show more digits. \begin{equation} \prod_{n=2}^\infty \left(1-\frac{1}{n^3}\right) =\frac{\cosh (\pi\sqrt{3}/2)}{3\pi},\qquad \prod_{n=1}^\infty \left(1+\frac{1}{n^4}\right) =\frac{\cosh (\pi\sqrt{2}) - \cos (\pi \sqrt{2})}{2\pi^2}. \end{equation}

Graphics

1. Plot more heart curves written in different formats (choose the right plotting functions):
   • \( r(\theta)=1-\sin \theta\) using either polar or the parametric form in Cartesian coordinate using plot. You have to generate a vector for the angle \(\theta\), say from \(0\) to \(2\pi\), and then the radius \( r(\theta)=1-\sin \theta\).
   • Parametric form in Cartesian coordinate using plot: \( x=\ln |t|\sin (t) \cos (t) ,\quad y=|t|^{0.3}\sqrt{\cos(t)}\) for \( t\in [-1,1]\). Choose the vector for the parameters to be t = [2.^(-20:-4) 0.1:0.01:1];t = [-t(end:-1:1) t];. (Try to see what happens when you choose t = -1:0.01:1; or t = linspace(-1,1,101)).
   • The implicit form using meshgrid and contour (choose the range of the grid from -10 to 10 in both direction first and then refine it if you like) \begin{equation} x^2 + \left[ y - \frac{2(x^2+|x|-6)}{3(x^2+|x|+2)} \right]^2 = 36. \end{equation}

       
     x = linspace(-8,8,101);  % vector for x-coordinates 
     y = linspace(-8,8,101);  % vector for y-coordiantes
     [X,Y] = meshgrid(x,y);   % Get the grid points, matrices for (X,Y)
     % you have to specify the contour of one single level, and have to repeat the value in the command
     % "contour" like [0,0] for the zero contour
     

   • In polar coordinates: \begin{equation} r(\theta) = 2 - 2\sin\theta + \sin\theta \frac{\sqrt{|\cos\theta|}}{1.4+\sin\theta}. \end{equation}
   Hint: use abs for the absolute value, and choose the proper range of the axis for the implicit plot.
2. The equation for a hypocycloid is given by \begin{equation} x = r(k-1)\cos \theta + r\cos ((k-1)\theta),\qquad y = r(k-1)\sin \theta - r\sin ((k-1)\theta). \end{equation} Plot the curve for \(r=1.0,k=3.0\).
   
   You may have to add axis equal to make the aspect ration equal.
3. In this exercise, you are going to plot the function \( (x-1)^{13}\) on the interval \([0.9,1.1]\) in two ways: (1) evaluate \( (x-1)^{13} \) directly; (2) evaluate its expansion (in polynomial) \(x^{13}-13x^{12}+\cdots 13x-1\). You can convert the symbolic exression of \( (x-1)^{13}\) into polynomial using sym2poly (you will get a vector for the coefficients of the polynomial, instead of calculating them by yourself), and then evaluate the polynomial using polyval. The function is plotted more accurately in which way, when evaluated as \((x-1)^{13}\) or as the polynomial expansion?

     
   syms z;
   p = sym2poly((z-1)^13);  % get the coefficients for the polynomial (z-1)^13
   x = linspace(0.9,1.1,51);
   % plot the data from polynomial evaluation, using the style "*" or "."