Chapter 3

Compute the area of an arbitrary triangle. An arbitrary triangle can be described by the coordinates of its three vertices: \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)\), numbered in a counterclockwise direction. The area of the triangle is given by the formula $$ A=\frac{1}{2}\left|x_{2} y_{3}-x_{3} y_{2}-x_{1} y_{3}+x_{3} y_{1}+x_{1} y_{2}-x_{2} y_{1}\right| $$ Write a function area(vertices) that returns the area of a triangle whose vertices are specified by the argument vertices, which is a nested list of the vertex coordinates. For example, vertices can be \([[0,0],[1,0],[0,2]]\) if the three corners of the triangle have coordinates \((0,0),(1,0)\), and \((0,2)\). Test the area function on a triangle with known area. Name of program file: area_triangle.py.

Compute the length of a path. Some object is moving along a path in the plane. At \(n\) points of time we have recorded the corresponding \((x, y)\) positions of the object: \(\left(x_{0}, y_{0}\right),\left(x_{1}, y_{2}\right), \ldots,\left(x_{n-1}, y_{n-1}\right) .\) The total length \(L\) of the path from \(\left(x_{0}, y_{0}\right)\) to \(\left(x_{n-1}, y_{n-1}\right)\) is the sum of all the individual line segments \(\left(\left(x_{i-1}, y_{i-1}\right)\right.\) to \(\left.\left(x_{i}, y_{i}\right), i=1, \ldots, n-1\right)\) : $$ L=\sum_{i=1}^{n-1} \sqrt{\left(x_{i}-x_{i-1}\right)^{2}+\left(y_{i}-y_{i-1}\right)^{2}} $$ Make a function pathlength \((x, y)\) for computing \(L\) according to the formula. The arguments \(\mathrm{x}\) and \(\mathrm{y}\) hold all the \(x_{0}, \ldots, x_{n-1}\) and \(y_{0}, \ldots, y_{n-1}\) coordinates, respectively. Test the function on a triangular path with the four points \((1,1),(2,1),(1,2)\), and \((1,1) .\) Name of program file: pathlength.py.

Approximate a function by a sum of sines. We consider the piecewise constant function $$ f(t)= \begin{cases}1, & 0

Write a function for numerical integration. An approximation to the integral of a function \(f(x)\) over an interval \([a, b]\) can found by first approximating \(f(x)\) by the straight line that goes through the end points \((a, f(a))\) and \((b, f(b))\), and then finding the area between the straight line and the \(x\) axis (which is the area of a trapezoid). Derive the formula for this area. Make a function integrate1 \((f, a, b)\) returning the value of the formula when \(f\) is a Python function \(f(x)\) implementing \(f(x)\), and a and \(b\) are the integration limits. Use the integrate1 function to compute the following integrals: \(\int_{0}^{\ln 3} e^{x} d x, \int_{0}^{\pi} \cos x d x, \int_{0}^{\pi} \sin x d x\), and \(\int_{0}^{\pi / 2} \sin x d x\), In each case, write out the error, i.e., the difference between the exact integral and the approximation computed by integrate1. Make rough sketches on a piece of paper of the trapezoid for each integral in order to understand how the method behaves in the different cases. Name of program file: int1_f.py.

Implement the factorial function. The factorial of \(n\), written as \(n !\), is defined as $$ n !=n(n-1)(n-2) \cdots 2 \cdot 1 $$ with the special cases $$ 1 !=1, \quad 0 !=1 $$ For example, \(4 !=4 \cdot 3 \cdot 2 \cdot 1=24\), and \(2 !=2 \cdot 1=2\). Write a function fact \((n)\) that returns \(n !\). Return 1 immediately if \(x\) is 1 or 0 , otherwise use a loop to compute \(n !\). Name of program file: fact.py. Remark. You can import a ready-made factorial function by >> from math import factorial >>> factorial (4) 24

Compute velocity and acceleration from position data; one dimension. Let \(x(t)\) be the position of an object moving along the \(x\) axis. The velocity \(v(t)\) and acceleration \(a(t)\) can be approximately computed by the formulas $$ v(t) \approx \frac{x(t+\Delta t)-x(t-\Delta t)}{2 \Delta t}, \quad a(t) \approx \frac{x(t+\Delta t)-2 x(t)+x(t-\Delta t)}{\Delta t^{2}} $$ where \(\Delta t\) is a small time interval. As \(\Delta t \rightarrow 0\), the above formulas approach the first and second derivative of \(x(t)\), which coincide with the well-known definitions of velocity and acceleration. Write a function kinematics \((x, t, d t=1 E-4)\) for computing \(x, v\), and \(a\) time \(t\), using the above formulas for \(v\) and \(a\) with \(\Delta t\) corresponding to dt. Let the function return \(x, v\), and \(a\). Test the function with the position function \(x(t)=e^{-(t-4)^{2}}\) and the time point \(t=5\) (use \(\left.\Delta t=10^{-5}\right)\). Name of program: kinematics 1.py.

Compute velocity and acceleration from position data; two dimensions. An object moves a long a path in the \(x y\) plane such that at time \(t\) the object is located at the point \((x(t), y(t))\). The velocity vector in the plane, at time \(t\), can be approximated as $$ v(t) \approx\left(\frac{x(t+\Delta t)-x(t-\Delta t)}{2 \Delta t}, \frac{y(t+\Delta t)-y(t-\Delta t)}{2 \Delta t}\right) $$ The acceleration vector in the plane, at time \(t\), can be approximated as $$ a(t) \approx\left(\frac{x(t+\Delta t)-2 x(t)+x(t-\Delta t)}{\Delta t^{2}}, \frac{y(t+\Delta t)-2 y(t)+y(t-\Delta t)}{\Delta t^{2}}\right) $$ Here, \(\Delta t\) is a small time interval. As \(\Delta t \rightarrow 0\), we have the limits \(v(t)=\left(x^{\prime}(t), y^{\prime}(t)\right)\) and \(a(t)=\left(x^{\prime \prime}(t), y^{\prime \prime}(t)\right)\) Make a function kinematics \((x, y, t, d t=1 E-4)\) for computing the velocity and acceleration of the object according to the formulas above (t corresponds to \(t\), and dt corresponds to \(\Delta t\) ). The function should return three 2 -tuples holding the position, the velocity, and the acceleration vector, all at time \(t\). Test the function for the motion along a circle with radius \(R\) and absolute velocity \(R \omega: x(t)=R \cos \omega t\) and \(y(t)=R \sin \omega t\). Compute the velocity and acceleration for \(t=1\) using \(R=1, \omega=2 \pi\), and \(\Delta t=10^{-5}\). Name of program: kinematics2.py.

Find the max and min values of a function. Write a function maxmin \((f, a, b, n=1000)\) that returns the maximum and minimum values of a mathematical function \(f(x)\) (evaluated at n points) in the interval between a and b. The following test program from math import cos, pi print maxmin \((\cos ,-\mathrm{pi} / 2,2 * \mathrm{pi}, 100001)\) should write out \((1.0,-1.0)\). The maxmin function can compute a set of \(n\) uniformly spaced coordinates between a and b stored in a list \(x\), then compute \(f\) at the points in \(x\) and store the values in another list y. The Python functions max (y) and min \((y)\) return the maximum and minimum values in the list \(\mathrm{y}\), respectively. Note that this is a "brute force" method for computing the extrema of a function - in contrast to the standard approach where one computes \(f^{\prime}(x)\) and solves \(f^{\prime}(x)=0\), and examines the end points \(f(a)\) and \(f(b)\), to find exact extreme points and values. Name of program file: func_maxmin.py.

Find the max/min elements in a list. Given a list a, the max function in Python's standard library computes the largest element in a: max(a). Similarly, min(a) returns the smallest element in a. The purpose of this exercise is to write your own max and min function. Use the following technique: Initialize a variable max_elem by the first element in the list, then visit all the remaining elements (a [1:]), compare each element to max_elem, and if greater, make max_elem refer to that element. Use a similar technique to compute the minimum element. Collect the two pieces of code in functions. Name of program file: maxmin_list. py.

Express a step function as a Python function. The following "step" function is known as the Heaviside function and is widely used in mathematics: $$ H(x)=\left\\{\begin{array}{l} 0, x<0 \\ 1, x \geq 0 \end{array}\right. $$ Write a Python function \(\mathrm{H}(\mathrm{x})\) that computes \(H(x)\). Name of program file: Heaviside.py.