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.

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.

The bacteria concentration in a reservoir varies as

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

where t 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)

The time versus velocity data of a particle is given in the table below. Use Lagrange’s
interpolation formula to find the distance moved by a particle and its acceleration at
the end of 3 seconds.
t: 0, 1, 2, 5
v: 2, 3, 12, 147