Problem 7
Consider a quadratic function \(f(x ; a, b, c)=a x^{2}+b x+c .\) Make a class Quadratic for representing \(f\), where \(a, b\), and \(c\) are attributes, and the methods are 1\. value for computing a value of \(f\) at a point \(x\), 2\. table for writing out a table of \(x\) and \(f\) values for \(n x\) values in the interval \([L, R]\) 3\. roots for computing the two roots. Name of program file: Quadratic.py.
Problem 8
To elongate a spring a distance \(x\), one needs to pull the spring with a force \(k x\). The parameter \(k\) is known as the spring constant. The corresponding potential energy in the spring is \(\frac{1}{2} k x^{2}\). Make a class for springs. Let the constructor store \(k\) as a class attribute, and implement the methods force \((x)\) and energy \((x)\) for evaluating the force and the potential energy, respectively. The following function prints a table of function values for an arbitrary mathematical function \(f(x) .\) Demonstrate that you can send the force and energy methods as the \(f\) argument to table. def table(f, a, b, n, heading=’’): """Write out f(x) for x in [a,b] with steps h=(b-a)/n.""" print heading h = (b-a)/float(n) for i in range(n+1): x = a + i*h print ’function value = %10.4f at x = %g’ % (f(x), x) Name of program file: Spring.py.
Problem 10
Make a class that can only do one thing: print a writes "Hello, World!" to the screen, when a is an instance of the class. Name of program file: HelloWorld.py.
Problem 14
A widely used formula for numerical differentiation of a function \(f(x)\) takes the form $$ f^{\prime}(x) \approx \frac{f(x+h)-f(x-h)}{2 h} $$ This formula usually gives more accurate derivatives than ( \(7.1)\) because it applies a centered, rather than a one-sided, difference. The goal of this exercise is to use the formula (7.7) to automatically differentiate a mathematical function \(f(x)\) implemented as a Python function \(f(x)\). More precisely, the following code should work: def f(x): return 0.25*x**4 df = Central(f) # make function-like object df # df(x) computes the derivative of f(x) approximately: for x in (1, 5, 10): df_value = df(x) # approx value of derivative of f at point x exact = x**3 # exact value of derivative print "f’(%d)=%g (error=%.2E)" % (x, df_value, exact-df_value) Implement class Central and test that the code above works. Include an optional argument \(\mathrm{h}\) to the constructor in class Central so that one can specify the value of \(h\) in the approximation (7.7). Apply class Central to produce a table of the derivatives and the associated approximation errors for \(f(x)=\ln x, x=10\), and \(h=\) \(0.5,0.1,10^{-3}, 10^{-5}, 10^{-7}, 10^{-9}, 10^{-11} .\) Collect class Central and the two applications of the class in the same file, but organize the file as a module so that class Central can be imported in other files. Name of program file: Central.py.
Problem 38
The company PROD produces two different products, \(\mathrm{P}_{1}\) and \(\mathrm{P}_{2}\), based on three different raw materials, \(\mathrm{M}_{1}, \mathrm{M}_{2}\) og \(\mathrm{M}_{3}\). The following table shows how much of each raw material \(\mathrm{M}_{i}\) that is required to produce a single unit of each product \(\mathrm{P}_{j}\) : For instance, to produce one unit of \(\mathrm{P}_{2}\) one needs 1 unit of \(\mathrm{M}_{1}, 3\) units of \(\mathrm{M}_{2}\) and 4 units of \(\mathrm{M}_{3}\). Furthermore, PROD has available 100 , 80 and 150 units of material \(\mathrm{M}_{1}, \mathrm{M}_{2}\) and \(\mathrm{M}_{3}\) respectively (for the time period considered). The revenue per produced unit of product \(\mathrm{P}_{1}\) is \(150 \mathrm{NOK}\), and for one unit of \(\mathrm{P}_{2}\) it is \(175 \mathrm{NOK}\). On the other hand the raw materials \(\mathrm{M}_{1}, \mathrm{M}_{2}\) and \(\mathrm{M}_{3}\) cost 10,17 and \(25 \mathrm{NOK}\) per unit, respectively. The question is: How much should PROD produce of each product? We here assume that PROD wants to maximize its net revenue (which is revenue minus costs). a) Let \(x\) and \(y\) be the number of units produced of product \(P_{1}\) and \(\mathrm{P}_{2}\), respectively. Explain why the total revenue \(f(x, y)\) is given by \(f(x, y)=150 x-(10 \cdot 2+17 \cdot 5) x+175 y-(10 \cdot 1+17 \cdot 3+25 \cdot 4) y\) and simplify this expression. The function \(f(x, y)\) is linear in \(x\) and \(y\) (check that you know what linearity means). b) Explain why PROD's problem may be stated mathematically as follows: maximize \(f(x, y)\) subject to $$ \begin{array}{r} 2 x+y \leq 100 \\ 5 x+3 y \leq 80 \\ 4 y \leq 150 \\ x \geq 0, y \geq 0 \end{array} $$ This is an example of a linear optimization problem. c) The production \((x, y)\) may be considered as a point in the plane. Illustrate geometrically the set \(T\) of all such points that satisfy the constraints in model (7.8). Every point in this set is called a feasible point. (Hint: For every inequality determine first the straight line obtained by replacing the inequality by equality. Then, find the points satisfying the inequality (a halfplane), and finally, intersect these halfplanes.) d) Make a program optimization1.py for drawing the straight lines defined by the inequalities. Each line can be written as \(a x+b y=c\). Let the program read each line from the command line as a list of the \(a, b\), and \(c\) values. In the present case the command-line arguments will be $$ [2,1,100],[5,3,80],[0,4,150], \quad[1,0,0], \quad[0,1,0] \text {, } $$ e) Let \(\alpha\) be a positive number and consider the level set of the function \(f\), defined as the set $$ L_{\alpha}=\\{(x, y) \in T: f(x, y)=\alpha\\} $$ This set consists of all feasible points having the same net revenue \(\alpha\). Extend the program with two new command-line arguments holding \(p\) and \(q\) for a function \(f(x, y)=p x+q y\). Use this information to compute the level set lines \(y=\alpha / q-p x / q\), and plot the level set lines for some different values of \(\alpha\) (use the \(\alpha\) value in the legend for each line). f) Use what you saw in e) to solve the problem (7.8) geometrically.
