numerical methods

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Unit III
Differentiation and Integration
Part A
1. What the errors in Trapezoidal and Simpson’s rule.
2. Write Simpson’s 3/8 rule assuming 3n intervals.
3. Evaluate
1
1
4 1 x
dx
using Gaussian quadrature with two points.
4. In Numerical integration what should be the number of intervals to apply
Trapezoidal, Simpson’s 1/3 and Simpson’s 3/8.
5. Evaluate
1
1
4
2
1 x
x dx
using Gaussian three point quadrature formula.
6. State two point Gaussian quadratue formula to evaluate
1
1
f (x)dx .
7. Using Newton backward difference write the formula for first and second order
derivatives at the end value x = x0 upto fourth order.
8. Write down the expression for
dx
dy
and
2
2
dx
d y
at x = x0 using Newtons forward
difference formula.
9. State Simpson’s 1/3 and Simpson’s 3/8 formula.
10. Using trapezoidal rule evaluate
0
sin xdx by dividing into six equal parts.
Part B
1. Using Newton’s backward difference formula construct an interpolating polynomial
of degree three and hence find f(-1/3) given f(-0.75) = - 0.07181250, f(-0.5) =
- 0.024750, f(-0.25) = 0.33493750, f(0) = 1.10100.
2. Evaluate
x y
dxdy
1
by Simpson’s 1/3 rule with x y = 0.5 where 0<x,y<1.
3. Evaluate I =
2
1
2
1 x y
dxdy
by using Trapezoidal rule, rule taking h= 0.5 and h=0.25.
Hence the value of the above integration by Romberg’s method.
4. From the following data find y’(6)
X : 0 2 3 4 7 9
Y: 4 26 58 112 466 922
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5. Evaluate
2
1
2
1
2 2 x y
dxdy
numerically with h= 0.2 along x-direction and k = 0.25 along y
direction.
6. Find the value of sec (31) from the following data
(degree) : 31 32 33 34
Tan : 0.6008 0.6249 0.6494 0.6745
7. Find the value of x for which f(x) is maxima in the range of x given the following
table, find also maximum value of f(x).
X: 9 10 11 12 13 14
Y : 1330 1340 1320 1250 1120 930
8. The following data gives the velocity of a particle for 20 seconds at an interval of
five seconds. Find initial acceleration using the data given below
Time(secs) : 0 5 10 15 20
Velocity(m/sec): 0 3 14 69 228
9. Evaluate
7
3
2 1 x
dx
using Gaussian quadrature with 3 points.
10. For a given data find
dx
dy
and
2
2
dx
d y
at x = 1.1
X : 1.0 1.1 1.2 1.3 1.4 1.5 1.6
Y: 7.989 8.403 8.781 9.129 9.451 9.750 10.031