SRM University 2007 B.Tech Food Process Engineering NUMERICAL DIFFERENTIATION AND INTEGRATION - Question Paper

UNIT – III
NUMERICAL DIFFERENTIATION AND INTEGRATION
PART – A
1. Write the formula for and using forward difference operator and at using backward difference operator.
2. Why Simpson’s rule is called a closed formula.
3. State Trapezoidal rule to evaluate
4. State Simpson’s rule to evaluate
5. State Simpson’s rule to evaluate
6. Distinguish ranging from Trapezoidal rule and Simpson’s rule
7. What is the geometrical meaning of Trapezoidal rule ?
8. Evaluate using simpson’s 1/3 rule provided that .
9. Why trapezoidal rule is so called.
10. When does Simpson’s rule provide exact outcome
11. Evaluate using trapezoidal rule by taking six intervals.
12. Say actual or False. Trapezoidal rule is the lowest right formula.
13. Say actual or False. For using Simpson’s rule the interval of integration must be divided into an even number of subintervals of width h.
14. Error in the Trapezoidal rule is of order ____________
15. Error in simpson’s rule is of order _____________
16. Which 1 is more reliable, simpson’s rule or Trapezoidal rule?
17. For the subsequent data, obtain the area bounded by y=f(x), x-axis and x=7.47 to 7.52 using Trapazodial rule.
X : 7.47 7.48 7.49 7.50 7.51 7.52
F(x) : 1.93 1.95 1.98 2.01 2.03 2.06
18. What is the fault in Trapezoidal rule of numerical Integration?
19. What is the fault in simpson’s rule of numerical Integration?
20. What is the fault in quadrature formulae?
21. Write the Numerical double Integration formula for Trapezoidal rule?
22. Write the Numerical double Integration formula for simpson’s rule?
23. Use Trapezoidal rule to evaluate considering 5 sub-intervals.
24. Evaluate by simpson’s rule.
25. Find y1(5) from the subsequent Table
X : 0 1 2 3 4 5
Y : 4 8 15 7 6 2
26. Find the 1st derivative at x =1 for the subsequent table
X : 1 2 4 8 10
Y : 0 1 5 21 27
27. Find y1(0.4) for the subsequent Table
X : 0.1 0.2 0.3 0.4
Y(x) : 1.10517 1.22140 1.34986 1.49182
PART-B
1. Given that
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
obtain and at x=1.0 and x=1.6
2. A slider in a machine moves along a fixed straight rod. Its distance x.cm along the rod is provided beneath for different values of the time t seconds. obtain the velocity of the slider and its acceleration when t = 0 and 0.6 seconds.
T : 0 0.1 0.2 0.3 0.4 0.5 0.6
X : 30.13 31.62 32.87 33.64 33.95 33.81 33.24
3. Find the 1st and 2nd derivatives of f(x) at x = 1.5 if
X : 1.5 2.0 2.5 3.0 3.5 4.0
F(x) : 3.375 7.000 13.625 24.000 38.875 59.000
4. The velocity v(km/min) of a moped which begins from rest, is provided at fixed intervals of time `t’(min) as follows:
T: 2 4 6 8 10 12 14 16 18 20
V 10 18 25 29 32 20 11 5 2 0
Estimate approximately the distance covered in 20 minutes.
5. Evaluate by using (i) Trapezoidal rule and (ii) simpson rule. Verify your outcomes by true Integration.
6. By dividing the range into 10 equal partts, evaluate by Trapezoidal and simpson’s rule. Verify your ans with true Integration.
7. A curve persses through the points (1,2) (1.5, 2.4) (2.0, 2.7), (2.5, 2.8), (3,3), (3.5, 2.6) and (4.0, 2.1). find the area bounded by the curve, the x-axis and x=1 and x=4. Also obtain the quantity of solid of revolution got by revolving this area about the x-axis.
8. Using Trapezoidal rule evaluate
I = , taking 4 sub-intervals.
10. Apply simpson’s rule to evaluate the Integral
I =
11. Using Trapezoidal and Simpson’s rules, evaluate dxdy, taking 2 sub-intervals.
12. Apply Trapezoidal rule to evaluate , taking 2 sub-intervals.
13. Evaluate , using simpson’s rule.
14. Given the subsequent table of values of x and y

x 1.0 1.05 1.10 1.15 1.20 1.25 1.30
y 1.0000 1.0247 1.0488 1.0723 1.0954 1.1180 1.401
obtain dy/dx at x =1, 1.3 compute the angular velocity and the angular acceleration of the rod when t=0.6sec.

15. For the provided data
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
Find f?(x) and f?(x) at x=1.0
16. Evaluate numerically and hence obtain the value of ? taking h = 0.2
b. Evaluate the above integral using Simpson’s rule taking h = ¼.
17. The population of a certain town is shown in the subsequent table
Year 1931 1941 1951 1961 1971
Population in thousands 40.6 60.8 79.9 103.6 132.7
Find the rate of growth of the population in the year 1971

18. Evaluate by using (i) Trapezoidal rule (ii) Simpson’s rule iii) Simpson’s rule.
19. Evaluate dx by dividing the range of integration into four equal parts using (i) Trapezoidal rue (ii) Simpson’s rule.
20. Compute the value of the definite integral using (i) Trapezoidal rule (ii) Simpson’s rule.
b. Find the value of from dx using Simpson’s rule with h = 0.25.
21. When a train is moving at 30 meters /sec steam is shut off and brakes are applied. The speed of the train (V) in m/sec after t seconds is givn by
t 0 5 10 15 20 25 30 35 40
v 30 24 19.5 16 13.6 11.7 10 8.5 7.0
Using simpson’s 1/3 rule to determine the distance moved by train in 40 seconds.
22. The velocity v of a particle at distances from a point on its path is provided by the table.
t 0 10 20 30 40 50 60
v 47 58 64 65 61 52 38
Estimate the time taken to travel 60 feet by using Simpson’s 1/3 rule compare the outcome with Simpson’s 3/8 rule.
23. A rive is feet wide. The depth ‘d’ in feet at a distance x feet from 1 bank is provided by the subsequent table.
X 0 10 20 30 40 50 60 70 80
D 0 4 7 9 12 15 14 8 3

Find approximately the area of cross part of the river using Simpson’s 1/3 rule.

24. Evaluate the integral I= using trapezoidal rule with h=k=0.5

25. . Using Simpson’s rule evaluate I=

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Earning:   Approval pending.