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CE 190 Final Exam
Closed Book, Closed Notes
Three pages of notes allowed (8.5” x 11” sheet of paper, front and back)
Tuesday, December 13th (9:45 am – 12 noon)
10 problems, 17 total points
Please do all of your work on a separate sheet of paper
Final answers should be circled next to your work
To receive full (or partial) credit, you must show your work
Note: each problem has different weighting
For problems 1–2, use the following set of data:
x:
f(x):
4
8
5
30
9
10
1a.
What order polynomial should be used to fit the data? (1 point)
1b.
We will evaluate the interpolating polynomial at x = 6. How should the data be
sorted? (1 point)
2.
Using a Newton Divided-Difference Interpolating Polynomial, find the y-value
when x is 6. (2 points)
For extra credit (1 point): Find the Lagrange Interpolating Polynomial. (It should yield
the same results at x = 6)
3.
The following equation can be used to calculate the second derivative. (this is the
backward, low-accuracy version)
f '' ( x i ) 
f ( x i )  2 f ( x i 1 )  f ( x i  2 )
h2
Using two Taylor series approximations, derive this equation. (2 points)
Hint: Define h clearly. Generate the first three terms of f(xi-1) and f(xi-2).
Multiply the first equation by 2 and subtract.
4.
Explain two ways in which you can reduce the error of a derivative
approximation. (1 point)
For problems 5–6, use the following data:
x:
f(x):
1
5
2
2
3
1
4.5
3.25
5.
Find the area under the curve. (i.e. integrate f(x) from x = 1 to x = 4.5) Use
approximations that will minimize the amount of error. (2 points)
6.
The above data points are based on f(x) = x2 – 6x + 10. Calculate the relative
error between the true answer and the above approximation. (1 point)
For extra credit (1 point): Calculate an approximation for this integral using the original
data and trapezoids only. (the relative error here should be worse)
For problems 7–8, the following differential equation is to be solved numerically:
dy
 (1  2 x) y
dx
y(0) = 1
1

It can be shown that the solution to this equation is y ( x)   x( x  1)  1
2

7.
2
Use Heun’s method with a step size of 0.5 to find y(0.5) and y(1). Use y(0) = 1 as
the starting point. Show your results clearly. Indicate the following results:
a.
b.
c.
d.
e.
f.
x
dy/dx (using the previous values of x and y)
dy/dx (using the current value of x and the predictor equation)
y (using Heun’s method)
y (true value)
Relative error
Results should be accurate to 4 decimal places.
Total value of this problem: 3 points
8.
Explain why this method is more accurate that Euler’s method. (1 point)
9.
One of the ways to find the x-intercept of a function numerically is to use the
midpoint rule. Explain how you use this technique. (1 point)
10.
Systems of equations can be solved numerically using elimination methods
(typically matrices) or iterative methods (which are approximations). Give one
example of each technique. Briefly describe how each method is implemented.
(2 points)