MAE2360 FINAL EXAM
1pm-2pm, August 10, 2023
Last Name
First Name
Last 4 ID #s
  1. Solve the following equations:
    x + a y + a2 z
    =
    a,
    a x + a2 y + z
    =
    a2,
    a2 x+y + a z
    =
    1
  2. How can you use Euler's method to solve y"(t)−y(t)=0, y(0)=0,  y′(0)=1 ? Describe your approach.
  3. Obtain y(t) at t = 1.4 for
    dy
    dt
    		 = (y − t) y ,     y(1) = 1,
    using Euler's method with h (step size) = 0.2.
  4. Approximate f′(1.0) from the following table using the formula:
    f′(x) ≈ 3 f(x) − 4 f(x − h) + f(x−2h)
    2 h
    x 0.2 0.4 0.6 0.8 1.0
    f(x) 2 369 13
  5. In a sound file (*.wav), vocals are low frequency signals and music is a high frequency signal. How can you separate vocals from music to make a karaoke file ?
  6. Numerically approximate
    ⌠
    ⌡     		1

    0 
    		 1
    1+x
    		 dx
    by hand using Simpson's rule with h = 0.5 (step size). You need only two partitions.
  7. Find the cubic root of 5 using the Newton-Raphson method:
    xn+1 = xn − f(xn)
    f′(xn)
    		 .
    Compute x3 starting with x1 = 1.
  8. What value does the following MATLAB/Octave program output ? 
    s=0;
for i=2:4
 for j=3:2:5
  s=s+i-j;
 end
end
fprintf('%f\n', s);
  9. What value does the following FORTRAN program output ? 
          FUNCTION F(X)
      REAL F, X
      F=X**2-1.0
      RETURN
      END

      REAL SUM, F
      INTEGER I
      SUM = 0.0
      DO 2015 I=1,5,2
        SUM=SUM+F(FLOAT(I))
        IF (SUM.GT.4.0) THEN 
          GOTO 2023
        ENDIF
 2015 CONTINUE

 2023 CONTINUE
      
      WRITE(*,*) SUM
      STOP
      END

    10. (a) TRUE or FALSE: The integration of sin([1/(x)]) is −cos([1/(x)]).
      (b) TRUE or FALSE: CGI programming uses remote computer's resources.
      (c) TRUE or FALSE: Both C and Matlab require a semicolon (;) at the end of each statement while Fortran does not.
      (d) When does the Newton-Raphson method fail ?
      (e) Name one of the standard methods to solve a system of 9 simultaneous linear equations ?
(1,0)250
Do not write below:
1 2 3 4 5 6 7 8 9 10
Soluion
1. x = 0, y=1, z=0
2. Let y1 ≡ y, y2=y1′, then, we have
d y1
dt
=
y2
d y2
dt
=
y1,
with y1(0)=0 and y2(0)=1.
3.
y(1)=1, y(1.2)=y(1)+h f(1, 1)=1, y(1.4)=y(1.2)+h f(1.2, 1)=0.96.
4.
(3×13−4×9+6)/(2×0.2) = 22.5.
5. Use a high pass filter in FFT to filter out lower frequency signals.
6.

f(0)+4 f(0.5)+f(1.0)
6
 = 0.69444.
7.

f(x)=x3−5, x2=1−f(1)/f′(1)=7/3=2.3333, x3=7/3−f(7/3)/f′(7/3) = 821/441=1.86168.
8.-6.
9. 8
10.F T F When f'(a)=0. Gauss-Seidel.

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On 18 Apr 2024, 23:23.