Lab 2 Answer Sheet

Lab Activity #2 Answer Sheet

Each problem is worth 1/4 point.

Part 1 Matlab constants

1. For this question first type the following expression at the Matlab prompt:

>> string = [ 'y = ' ,    _______num2str(number)_okay also accept ___num2str(1.2345)_________________________ , ' * cos(x)' ];

number           ____________double array______________________________________

string               ____________char array______________________________________

Part 2 Matlab operators

To answer the following questions type the following commands at the Matlab prompt:

>> a = 2 : 3 : 17;

>> b = 1:6;

2. Fill in the blanks with the response Matlab gives when you type the following at the Matlab prompt. If the result is an error then write ERROR.
    >> a
    a =
                ____ 2     5     8    11    14    17 ______

3. >> length(a)
    ans =

                ____6_____

4. >> size(a) , size(b)
ans =

_____1 6

ans =

_____1 6_____

5. >> 3 a
ans =

_____ERROR_______

6. >> 3 * a
ans =

6 15 24 33 42 51 ______

7. >> 3 .* a
ans =

6 15 24 33 42 51

8. >> a * b
ans =

_________ERROR____________

9. >> a .* b

ans =

2 10 24 44 70 102

10. >> a .^ 2
ans =

4 25 64 121 196 289

11. >> 2 .^ b
ans =

2 4 8 16 32 64

12. >> (a .* b) ./ b
ans =

2 5 8 11 14 17

13. >> (a .* b) ./ a
ans =

1 2 3 4 5 6

Part 3. Vector carving (subscripting)

Type the following expression at the Matlab prompt then either answer the question or fill in the blanks with the response Matlab returns. If there is an error write ERROR.

>> x = linspace(0,3*pi,100);

>> y = [ 1  5  9  13  17 21]  ;

14. How many elements does the vector x have?

_____100___________________

15. >> y(3:end)
ans =

9 13 17 21

16. >> y(1:end-1)
ans =

1 5 9 13 17

17. >> x( [1 2])
ans =
______ 0 0.0952 (approx.)

18. >> x(1:2)
ans =
______ 0 0.0952__________(approx.)

19. Fill in the blank below to carve out the last two values of y (in reverse order) and prepend zero to these two values. The correct answer would display
      z =
             0    21 17

   >> z =     [ 0 , _____y(end:-1:end-1) OR y([6,5])___OR y(end), y(end-1)_____ ]
   (The correct answer must use the variable y and subscriping on y)

20. Fill in the blanks below to flip the values of y. The correct answer would display
flipy =
21 17 13 9 5 1

>> flipy = y( end : ___-1___ : ___1____ )

Part 4 Matlab functions

21. Complete the following Matlab expression to simulate the roll of one, six-sided die. Note the die values are 1,2,3,4,5,6 and are integers.
(Hint: use the function rand with proper scaling)

ceil(  _____6_____   * rand(1,1) )

22. Write a Matlab expression to display a row vector with 2 random integers in the range from 2 to 5 inclusive.

>> random = ceil( ______4________* rand(1,2) + _____1________ )

Part 6 Symbolics in Matlab

To answer the following questions, first type,
>> syms x

23. Write the response Matlab returns after typing the following,
     >> factor(x^3 - 1)

    _______(x-1)*(x^2 + x+1)____________________

24. Write the response Matlab returns after typing the following,
       >> simple(sin(x)^2+cos(x)^2)

(Just write the answer, don't write the intermediate results)     ans = ______________1______________________

25. Fill in the blank below to simplify the mathematical function cos(2x) / (cos²(x)-sin²(x)). (The domain is a subset of the Reals).

     >> simple(__cos(2*x) / (cos(x)^2-sin(x)^2)___________)

Part 7 Discretization in Matlab

26. For the function y = 1/x choose the sample of x values as [ -2 -1 1 2] type the following command at the Matlab prompt
>> x = [-2 -1 1 2];

        Fill in the blank below to compute the values for y, (you should type this at the Matlab prompt for use in Part 8)

             >> y = ____1./x OR x.^-1 but NOT 1/x_____________

That's it, we have now discretized y = 1/x . How good is our discretization? We find out in Part 8.

27. y = 1.234 * sin(x) Fill in the blanks below to discretize this function. Choose the sample of x values as 50 evenly spaced values from 0 to 2*pi. We will use x2 and y2 so that we can keep these separate from the x and y in problem #26.

        >> x2 = _____linspace(0, 2*pi, 50)__________________________

        >> y2 = _______1.234 * sin(x2)________________________

Part 8 Plotting

28. Type the following comand to plot the discretized parabola we created in Part 7.
>> plot(x,y,'g') ('g' for green)

Was the discretization in #26 good? No, since 1/x is a hyperbola and is undefined for x = 0. Fix the problem by plotting y = 1/xusing 101 points from -2 to 2. The plot should be in red, no lines connecting the points and each point should be marked with a *(star) not a . (period).

>> x = ___linspace(-2, 2, 101)_________________

>> y = _____1./x OR x.^-1________

>> plot( ____x________ , _________y___________, _____'r*'_________)

>> grid on

29. We will now plot the discretized function from problem #27.
>> plot(x2,y2)

Now put two plots on one figure window, type,
>> hold on
>> plot( x2,y2)
>> grid on
Does plot(x2,y2) give the same results as plot(y2,x2) ______NO__________(YES/NO)?

Part 9 Differentiation

Since every real number in Matlab number takes up eight bytes of memory and there is eight bits per byte, this means that every real number takes up 64 bits. A bit is a 0 or a 1 so there are two possibilities for each bit and thus 2⁶⁴ different possible real numbers can be represented in Matlab. That's not infinite, and since there are an infinite number of real numbers (actually an infinite number between any two real numbers), Matlab has lots of holes or gaps in its representation of the real numers. This problem is called roundoff error. If you take CS/Math 257 you will learn a lot more about this. But let's do one example.

Approximate the derivative of cos(x) at x = pi/2. From Calculus we know the formula for the slope of the derivative
d/dx(cos(x)) = -sin(x) so plug in x = pi/2 and we get -1.0 exact. Now from the instructions let's use the Centered Difference formula to approximate the derivative, however we will use various values of h, type,
>> format long e
>> h = 10 .^ -(2:14)
>> derivative = (cos(pi/2 + h/2) - cos(pi/2 - h/2))./h
>> error = abs(derivative + sin(pi/2))
>> loglog(h,error) % loglog is like plot except it plots the log of both h and error

30. Which value of h gives us the best approximation to the derivative, that is, which value of h gives us the smallest error?

h = __________1.0e-5_______________________

Because of roundoff error arbitrarily small values for h won't necessarily give us better values for the derivative or in any numerical calculation( like the definite integral). However, in CS101 we will be in the "denial stage" and will try to ignore roundoff error. If you can't do this then you are welcome to take a course on Numerical Analysis like CS/Math 257.

Using the formulas derived in Part 9 of the instructions,

>> midpoints = (x(1:end-1) + x(2:end))/2 ; 

>> yprimes = diff(y) ./ diff(x) ;

fill in the blanks below in order to compute and plot the derivative of y = sin(x) on the interval [0, 2*pi] using 10 equally spaced points.

>> x2 = linspace(0,2*pi,10);
>> y2 = sin(x2);

31.
>> midpoints = __(x2(1:end-1) + x2(2:end))/2_____________ ;

32.
>> yprimes = _____diff(y2) ./ diff(x2)_______ ;

>> plot(midpoints, yprimes , 'g')
>> hold on
>> plot(midpoints , cos(midpoints), 'r') (we know that the derivative of sin(x) is cos(x) so let's compare symbolice vs. discrete)

Part 10 Logical values and operators

33. Type the following assignment statement at the Matlab prompt,

>> v = [ -1 0 2 0 3 0 4 0 5 0 6]

Fill in the blank in the Matlab expression below that uses subscripting to extract all non-zero values from v and assign these to the variable w.

>> w = v ( ____________v ~=0_______________________________)

w =

-1 2 3 4 5 6

34. Fill in the blank in the Matlab expression below that uses subscripting to extract all zero values from v and assign these to the variable w.

>> w = v ( ____________v == 0_____________________________)

w =

0 0 0 0 0

35. Suppose that we are given the discretized values of a function y = f(x) as below. We want to find all the roots of the function f . That is, we want to find all the values r such that 0 = f(r).

Type the following assignment statements at the Matlab prompt,
>> y = [ 1 0 5 0 -3 0 4 0 5 0 ] ;
>> x = [ .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 ] ;

Fill in the blank in the Matlab expression below that uses subscripting to find all the roots of the discretized function. Yes the answer is [ .2 .4    .6    .8    1.0] but your answer should be a Matlab command that gives you the correct answer no matter what x and y are, assuming of course they have the same number of elements.

r =    ______________x( y == 0)___________________________________ ;

r =
          0.2000 0.4000    0.6000    0.8000    1.0000

36. For the x and y values from problem #35 find the x values that correspond to the smallest value in y. The answer is .5 since -3 is the smallest value for y. Write Matlab code to produce this answer. Your answer should work for any values for x and y. Use subscripting and the Matlab function named min.

result = _________x( y ==min(y))_____________________________________ ;

result =
0.5000

Congratulations, you are done!