Computational Materials Science

Computational Olympics I 2

Computational Olympics I

For this assignment, you must answer as many of the questions below as possible in the time allotted. You must work individually. You may use hand calculations, software, calculators, etc. to complete this assignment. Note, some section are arranged in increasing levels of difficulty. If a problem seems too lengthy, it may be advantageous to move on to another problem. Show hand calculations and/or write the name of software used (with primary commands) on separate paper. You may either sketch graphical solutions

Computational Olympics I 2

I. Working with Data

1. Find the slope of the line given by the following data points

0, 0

12, 24

48 96

2. Find the slope of the line given by the following data points

10, 30

20, 90

50, 270

3. Estimate the slope of the best fit line given by the following table of x,y data.

x / y
1.129 / -10.4488
1.986 / 95.86262
4.120 / 494.5282
4.904 / 217.6796
5.035 / 173.9268
6.113 / 864.7335
6.745 / 498.8405
9.020 / 751.6802
8.712 / 679.904
11.088 / 363.2905
11.494 / 809.8741
11.985 / 625.5712
13.385 / 565.7193
15.445 / 848.9764
15.612 / 681.9436
15.759 / 1026.122
17.570 / 1248.545
18.522 / 1276.586

4. Sort the following table from lowest to highest

568
261
9
422
213
427
675
124
988
614
69
775
866
991
151
950

5. Plot the following series of data

0 / 0
5 / 0.919551
10 / 1.613341
15 / 1.931852
20 / 1.850833
25 / 1.4757
30 / 1
35 / 0.633194
40 / 0.524005
45 / 0.707107
50 / 1.092396
55 / 1.500589
60 / 1.732051
65 / 1.643683
70 / 1.205737
75 / 0.517638
80 / -0.22324
85 / -0.78888

6. Integrate the previous set of data from question 5.

7. Find the instantaneous slope at each point in the previous set of data from question 5.

II. Equation Solving

8. Find the intersection of the following two lines

y = 10x -20

y = 32x +5

9. Find the intersection of the following two curves

X^2+y^2=100

Y = 5x^2 – 15 x +5

10. Determine if the following lines in a 3-dimensinoal space intersect, and if so at what values of x,y and z.

z = x+y+1

z = 2 x+10 y

z = -x - 5y-1

III. Differential Equations

Find the genera; solutions (i.e. w/o initial conditions) to the following differential equations

11. Y’’ = c1

12. y’’ + y’ – y =0

13. y’’ + x y’ – y = 0

IV. Calculus

14. Integrate f(x) = x^2 + sin(x)

15. Differentiate f(x) = sin(x)cos(x)

16. Integrate f(x) = sin(x)*e^(-x)

V. Matrix Operations

B=4354-2155-6-1

17. Find the inverse of B

18. Find the determinant of B

4 / 3 / 5 / 6 / 10
4 / -2 / 15 / 20 / 10
C = / 5 / -6 / -1 / 12 / -4
12 / 123 / -45 / -2 / -45
12 / 65 / 78 / 90 / -101

19. Find the determinant of C

20. Find the inverse of C.

IV. Minimization

21. Find the minimum of f(x) if

f (x)= x^2 – x

22. Find the saddle point of f(x,y) if

f(x) = x^2 –xy – y^2

23. Find the local extremum (minimum or maximum value) of the function f(x1,x2,x3) if {x1,x2,x3} are all between {0, 10} and

f(x1,x2,x3) = (x1-1)^2 +(x2+1)^2 +x3^2 +10 (x1-x3)^2 +10 (x2-x3)^2+ 10 (x1-x2)^2

Programming

24. Write a short program that sums 1/x where x is every third multiple of 7 (i.e. 1/21+1/42+1/63+…,) for x between 1 and 2100.