# Lecture 45: Fourth Review¶

The question on the fourth review focus on

function definitions, differentiation, integration;

plotting in two and three dimensions;

solving linear, differential, polynomial equations, linear programming.

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

## Calculus¶

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

Define a piecewise function

`int_inv_cub`

which as function of the end point \(b\) always returns the correct value of \({\displaystyle \int_{-1}^b \frac{1}{x^3} dx}\).The arc length of continuous function \(f(x)\) over an interval \([a,b]\) can be defined as \({\displaystyle \int_a^t \sqrt{1+[f'(x)]^2}}\).

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

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 \(x \in [0,t]\).

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.

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 \(h(n)\).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).

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, \(x\) ).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:

with implicit differentiation,

with a Taylor series.

## Plotting and Solving Equations¶

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

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

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?

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.

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

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?

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

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.

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

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 \(n\).

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?