Lecture 45: Fourth Review

The question on the fourth review focus on

  1. function definitions, differentiation, integration;

  2. plotting in two and three dimensions;

  3. solving linear, differential, polynomial equations, linear programming.

The material corresponds to the second review, which prepared for the second midterm exam.

Calculus

  1. Write a function to make polynomials in a system. The k-th polynomial in the system is

    \[f_k(x_1,x_2, \ldots, x_n) = x_k + \sum_{i=1}^{n-k} x_i x_{k+i}, \quad k=1,2,\ldots,n.\]

    For example, for \(n=8\), the polynomials are

    x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 + x6*x7 + x7*x8 + x1
x1*x3 + x2*x4 + x3*x5 + x4*x6 + x5*x7 + x6*x8 + x2
x1*x4 + x2*x5 + x3*x6 + x4*x7 + x5*x8 + x3
x1*x5 + x2*x6 + x3*x7 + x4*x8 + x4
x1*x6 + x2*x7 + x3*x8 + x5
x1*x7 + x2*x8 + x6
x1*x8 + x7
x8

  2. Define a piecewise function int_inv_cub which as function of the end point \(b\) always returns the correct value of \({\displaystyle \int_{-1}^b \frac{1}{x^3} dx}\).

  3. The arc length of continuous function \(f(x)\) over an interval \([a,b]\) can be defined as \({\displaystyle \int_a^t \sqrt{1+[f'(x)]^2}}\).

    1. Compute the arc length of the positive half of the unit circle, i.e.: \(f(x) = \sqrt{1-x^2}\) (answer \(= \pi\)).

    2. Create a function (call it arc_length) in \(t\) which returns a 10-digit floating-point approximation of the arc length of the positive half of the circle, for \(x \in [0,t]\).

  4. Consider the recurrence relation

    \[h(n) = 5 h(n-1) - 6 h(n-2), \quad {\rm for} \ n \geq 2, \quad {\rm with} \ h(0) = 1 \ {\rm and} \ h(1) = -2.\]

    Answer the following the questions.

    1. The generating function \({\displaystyle g(x) = \frac{1-7x}{1-5x + 6x^2}}\) defines \(h(n)\) as the coefficient with \(x^n\) in the Taylor expansion of \(g(x)\). Use \(g(x)\) to define \(h\) as a function (call it t) of \(n\) which gives the value of \(h(n)\).

    2. Write a function to compute \(h(n)\), directly using the recurrence relation from above. Make sure your function can compute \(h(120)\). Compare with the result of (a).

  5. The Legendre polynomials are defined by

    \[P_0(x) = 1, \quad P_1(x) = x, \quad P_n(x) = \frac{2n-1}{n} x P_{n-1}(x) - \frac{n-1}{n} P_{n-2}(x), \ {\rm for} \ n \geq 2.\]

    Write a efficient recursive function legendre to compute \(P_n(x)\). The function legendre takes on input the degree \(n\) and the variable \(x\).

    Compare the output of your legendre (50, \(x\) ) with the legendre_P (50, \(x\) ).

  6. Consider the point \(P = (1,1)\) on the curve defined by \(xy - 2 x + 1 = 0\).

    Compute the slope of the tangent line to the curve at \(P\) in two ways:

    1. with implicit differentiation,

    2. with a Taylor series.

Plotting and Solving Equations

  1. Suppose we want to plot the curve \(x^4 + x^2 y^2 - y^2 = 0\) for \(x\) and \(y\) both between \(-1\) and \(+1\).

    1. Sampling this curve as given in rectangular coordinates, how many samples do we need to take from the curve to obtain a nice plot?

    2. Convert the curve into polar coordinates and plot. Give all commands used to obtain the plot. How many samples of the curve are needed here?

  2. Solve \(x^2 a^2 - 2x^2 a - 3 x^2 - x a^2 + 4 x a - 3 x + a^2 + 2 a - 15\) for \(x\) for all values of the parameter \(a\).

    Be as complete as possible in your description of the solution.

  3. Find the point with real coordinates on the curve \(xy - 2 x + 3 = 0\) closest to the origin.

  4. Consider the system

    \[\begin{split}\left\{ \begin{array}{rcl} x^2 - 2 y^2 - 1 & = & 0 \\ x y - 2 x - 3 & = & 0. \\ \end{array} \right.\end{split}\]

    How many real solutions does this system have?

  5. Consider \(y'' + 6 y' + 13 y = 0\), with \(y(\pi/2) = -2\) and \(y'(\pi/2) = 8\).

    1. Find an exact solution to this initial value problem and use this to create a function \(f\) which returns a numerical 10-digit floating-point approximation of the solution.

    2. Solve this initial value problem numerically. Compare the solution with the value for \(y(2)\) and also with \(f(2)\) obtained in (a).

  6. A 5-by-5 variable Toeplitz matrix has the following form:

    [t0 t1 t2 t3 t4]
[t8 t0 t1 t2 t3]
[t7 t8 t0 t1 t2]
[t6 t7 t8 t0 t1]
[t5 t6 t7 t8 t0]

    for the symbols in the list [t0, t1, t2, t3, t4, t5, t6, t7, t8].

    For general dimension \(n\), the \((i,j)\)-th element of the Toepliz matrix \(T\) is

    \[\begin{split}T_{(i,j)} = \left\{ \begin{array}{lcl} j - i & {\rm if} & j \geq i \\ j - i + 2 n - 1 & {\rm if} & j < i. \end{array} \right.\end{split}\]

    Give the command(s) to define a variable Toeplitz matrix, for any dimension \(n\).

  7. Maximize \(x_1 + x_2\) subject to \(-x_1 + 2 x_2 \leq 8\), \(4 x_1 - 3 x_2 \leq 8\), \(2 x_1 + x_2 \leq 14\), \(x_1 \geq 0\), and \(x_2 \geq 0\).

    Write the commands to define this problem and then solve it. What are the values of \(x_1\) and \(x_2\) at the optimal solution?