Download Midterm 1

13-03-2012
NEM 352
MIDTERM I
(OVER 80)
1. (20 Points)
Consider the system
such as
a) (3 Points) Determine and , using 6 significant digits with chopping.
b) (3 Points) Use following iterative technique
Start with
, determine
c) (14 Points) Give an error bound for
. Determine the true error and compare with the error bound.
( Use
where is spectral radius of iteration matrix)
2. (5 Points)
A) The relative approximate error at the end of an iteration to find the root of an equation is 0.004%. What is
the least number of significant digits we can trust?
B) How many significant digits does the number 0.01970x103 have?
3. (10 Points) What is the expression for true error in calculating the derivative of Sin(2x) at x=π/4 in terms of h
by using the approximate expression:
4. (10 Points) The formula for normal strain in a longitudinal bar is given by
Where F is normal applied force, A is cross-sectional area of bar and E is the young’s modulus.
If
and
, what is the maximum error in the
measurement of strain.
5. (15 Points) Approximate the root of
on [0,1] with using the Bisection Method. Perform 6
iterations and find the absolute relative approximate errors at the end of each step.
6. (10 Points) A flat plate of mass falling freely in air with a velocity V is subject to a downward gravitational
force and an upward frictional drag force to air. The drag force
is given by the expression:
Terminal velocity is reached when the drag force equals to the gravitational force
Find the terminal velocity using the Bisection Method if m=1kg and
. Use an initial interval of
V=0 and 200 m/s. Iterate until the approximate error falls below 1% and calculate absolute relative
approximate error at the end of each iteration.
7. (10 Points) Image processing example
There is strong evidence that the first level of procedding what we see is done in the retina. It involves
detecting something called edges or positions of transitions from dark to bright of bright to dark points in
images. These points usually coincide with boundaries of objects to model the edges, derivatives of functions
such as:
need to be found.
Use central divided difference approximation of the first derivative of f(x) to calculate the functions derivative
at x=0.1 for a=0.24. use step size of Δx=0.05. Also, calculate the absolute relative true error.