Quantitative Methods Answers

In recording measured values, a digit is significant if and only if it affects the -------------- of the measurement

A. absolute error
B. relative error
C. inherent error
D. truncation error

8 The zeroeth divided difference of the function f, with respect to
x i
, denoted by
f[x i ]
is the same as



f[x i ]=f(x i )



f[x 0 ]=f(x i )



f[x]=f(x i )



f[x]=f(xi)





9 The quantity
L 0 (x)
of the Lagrange’s interpolating polynomial P(x) is equal to



(x−x 1 )(x−x 2 )(x−x 3 )(x 0 −x 1 )(x 0 −x 2 )(x 0 −x 3 )



(x−x 1 )(x1−x 2 )(x 0 −x 3 )(x 0 −x 1 )(x 0 −x 2 )(x 0 −x 3 )



(x−x 1 )(x−x 2 )(x−x 3 )(x−x 1 )(x−x 2 )(x−x 3 )



(x−x 1 )(x−x 0 )(x−x 3 )(x 0 −x 4 )(x 0 −x 2 )(x 0 −x 3 )





10 A square matrix is called ………….. if all the elements above the main diagonal vanish.

upper triangular
triangular

lower triangular

rectangular

3 If f(1) = 1, f(3) = 19, f(4) = 49 and f(5) = 101, find the Lagrange’s interpolation polynomial of f(x).

a. P(x)=x 3 −x 2 +1
b. P(x)=x 3 –3x 2 −5x–4
c. P(x)=x 3 –3x 2 +5x–6
d. P(x)=2x 2 –3x+5x–6
4 The first divided difference of f with respect to
x i
and
x i+1
denoted by
f[x i ,x i+1 ]
is defined as
a. f[x i ,x i+1 ]=f[x i+1 ]−f[x i ]x i+1 −x i
b. f[x i ,x i+1 ]=x i+1 −x i f[x i+1 ]−f[x i ]
c. f[x i ,x i+1 ]=f[x i ]−f[x i+1 ]x i+1 −x i
d. f[x i ,x i+1 ]=f[x i ]−f[x i+1 ]f[x i+1 ]−x i

if f(1) = 1, f(3) =19, f(4) = 49 and f(5) = 101, find the lagrange's interpolation polynomial of f(x)

Let f(n) = 560*n^3 +3*n+107 and g(n) = 3*n^3 +5000*n^2. Which of the following is true?
a) f(n) is O(g(n)), but g(n) is not O(f(n))
b) g(n) is O(f(n)), but f(n) is not O(g(n))
c) f(n) is not O(g(n)) and g(n) is not O(f(n))
d) f(n) is O(g(n)) and g(n) is O(f(n))

How many elementary operations are used in the following algorithm?
Step 1 Set a=1, b=1 c=2, and k=1.
Step 2 while k<n
(a) Replace c with a+b
(b) Replace a with b
(c) Replace b with c
(d) Replace k with k+1
endwhile
Step 3 Print b.

an inland revenue services (IRS) regional office in the United States has plotted a large sample of tax refund amounts and found that they form a bell-shaped distribution symmetrical about the central line. The average refund amount was found to be close to $750 with a standard deviation of $125. Most refund values are fairly close to the central line,suggesting normal distribution.
The regional office is especially interested in the largerefunds. A refund larger than $1000 is considered large, and the office would
like to know what percentage of current funds based on the normal distribution
are in excess of this amount.
A second question involves the office’s concern with newwithholding guidelines for taxpayers. It has been estimated that the average
refund amount will rise by $120 after these guidelines go into effect, and the
IRS office wonders how the percentage of large refunds will be affected.

1. Calculatedthe percentage of refunds expected to exceed $1000 under the current
withholding guidelines

2. Calculate the percentage increase in therefund exceeding $1000 if the average refund increases by $120. Assume that the
degree of variability in refund amounts remain unchanged when the average
refund increases.

3. What wouldbe the effect on the percentage of refunds over $1000 if the average refund
amount actually drops by $70

4. Whatchange in the current average refund over $1000 will produce a 5% increase in
the current percentage of refunds over $1000. (Assume no change in the degree of
variability in refunds amounts.)

The bacteria concentration in a reservoir varies as

c= e^t - (t^3/6 ) (e^0.3t) - t^2/2 - t

where is the time in seconds. Use the Newton-Raphson method to estimate the
time required for the bacteria concentration to reach 1 (correct up to 2 decimal
places)

Create two multi-dimensional arrays:
- a 10 x 10 x 10 numerical array (3-D) and
- a 5 x 5 x 5 x 5 numerical array (4-D)

where each value in each array corresponds to the multiplication of its indices.