Reference no: EM131097893
E19: Numerical Methods for Engineering Applications Spring 2016 - HOMEWORK 9
1. Second-order forward difference approximation to the first derivative
Derive the approximation
f'(x) ≈ 1/2h(-3f(x) + 4f(x+h) - f(x+2h))
and show that it has asymptotic error of O(h2).
2. Dual numbers and partial derivatives
Extend the dual.py code we wrote in class to to use dual numbers to automatically evaluate both partial derivatives of the following function f(x, y):
def f(x, y): return -2.0*x * exp(-(x**2 + y**2)/(4.0))
You should evaluate the function and its derivatives at (x, y) = (0.5, 1.5).
Here are two possible strategies:
- Easier: Call the function f twice, each time with one Dual object as an argument, and one normal floating point number.
- Harder: Extend the Dual class so that self.eps is a flat numpy array of length two where the infinitessimal parts correspond to the partials in x and y.
In either case, you will need to provide a definition of the exp function which handles dual numbers correctly (see sqrt and cos in dual.py for examples).
Write a test function to show that your program computes the correct derivatives by comparing to the numerical central difference approximations for the first derivative. Submit a printout of your program and its output.
3. Integrating quadratics and cubics with Simpson's rule
It turns out that for some functions, the adaptive Simspson's integrator we looked at in class doesn't need to do too much work.
a. Why does the code always hit the base case (i.e. does not recurse) when the function f(x) being integrated is any quadratic function?
b. Furthermore, why does the code always hit the base case when the function f(x) being integrated is a cubic function?
4. A new twist on 2D integration
I extended the 2D integral function we worked on in class to create simpsons2d_mystery.
a. What is this function good for? What does it provide that simpsons2d does not?
b. Using traditional calculus notation, write down the 2D integral which is assigned to the mystery variable at the bottom of the program (you need not solve the integral analytically, but it might help you figure out what's going on here).
Attachment:- Assignment.rar
Average inflation rates tend to be higher in countries
: Average inflation rates tend to be higher in countries that have relatively weak judicial systems and property rights (remember chapter 1 and the importance of secure property rights?). Why do you think this is the case? Would you expect these countr..
|
Write the results statistical
: What are the F critical values for A effect, B effect, and the interaction - Which effects are significant? Write the results statistical
|
What is a market failure
: What is a market failure? It refers to a breakdown in a market economy because of widespread corruption in government.
|
Original amount of the mortgage loan
: A mortgage loan is repaid with annual installment payments payable at the end of each year for 30 years. Each subsequent payment is 2% higher than the previous one. The interest rate charged on the loan balance is:
|
Provide a definition of the exp function
: E19: Numerical Methods for Engineering Applications Spring 2016 - HOMEWORK 9. In either case, you will need to provide a definition of the exp function which handles dual numbers correctly (see sqrt and cos in dual.py for examples)
|
Respectable economic motive for selling of state-owned firms
: Explain the possible problems for consumers of each of the following ways of dealing with a natural monopoly: Why do most economists argue that the money raised by selling off state-owned firms is not an especially good argument for selling them? Wha..
|
New contract with the federal government
: Congratulations! Your company just won a lucrative new contract with the Federal Government to supply widgets for a total value of $50 million. If you perform well and on time on this job it will open the door to hundreds of millions more of futur..
|
Create the folders and files that appear in the file layout
: Create the folders and files that appear in the attached file layout. Define at least two styles in your CSS file. These styles are completely up to you. You can change the body font, or create a header class, or format something in the footer.
|
Problem regarding the career opportunities
: Write a 2-3 page paper. In your paper include the following: Discuss positions that are available for people with a knowledge of finance? Provide 3 examples of occupations that also use finance. Give a brief description of each.
|