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Nepal Engineering College

Changunarayan, Bhaktapur

Subject: Numerical Methods

Subject Code: MTH 317.3 Tutorial No.: 3

Title: Numerical Differentiation and Integration Date: Dec,20, 2006

1. Find dy/dx at x = 0.7 and 1.3 from the following table.

|x |0.7 |0.8 |0.9 |1.0 |1.1 |1.2 |1.3 |

|y |0.644218 |0.717356 |0.783327 |0.841471 |0.891207 |0.932039 |0.963558 |

2. Find dy/dx and d2x/dx2 of y = x1/3 from the following data at x = 51. Compare the results with the exact values.

|x |50 |51 |52 |53 |54 |55 |56 |

|y |3.684 |3.7084 |3.7325 |3.7563 |3.7798 |3.8030 |3.8259 |

3. Given the following pairs of values of x and y.

|x |1 |2 |4 |8 |10 |

|y |0 |1 |5 |21 |25 |

Determine numerically the first and second derivatives at x = 4.

4. Calculate the area bounded by the curve y = x² + 4, and the lines y = -1, x = 1 and x = 4 by Trapezoidal rule, taking number of subintervals as 6.

5. Calculate the area bounded by the curves y = x² - 4 and y = x² + 4, and the lines x = 1 and x = 4 by Newton Cotes 2 ordinates formula, using more than 4 subintervals.

6. Evaluate [pic]using Simpson's 1/3 rule and initial interval of 1. Go on halving the interval and find the integral with relative error less than 0.01.

7. Evaluate I = [pic]correct up to 3-decimal places with h = 1/6 and 1/12 by Simpson's 1/3 and Simpson's 3/8 rules. Discuss on the results.

8. Evaluate [pic]using Simpson’s 1/3rd rule with h = 0.2.

9. Calculate by Simpson’s 1/3rd rule the value of [pic]with at least 5 subintervals.

10. Use Romberg's method to compute [pic]correct to 4 decimal places.

11. Using the Gauss Legendre three ordinate quadrature formula, evaluate [pic].

12. Find the value of the integral [pic]using Gauss Legendre two-point and three-point integration rules. The nodes and weights are,

n nodes weights

1 ( 0.5773502692 1.0000000000

2 0.000000000000 0.8888888889

( 0.7745966692 0.5555555556

13. The values of stress corresponding to different values of strain, as obtained in a plane strain compression test, are given below.

|Strain (e) |0.25 |0.30 |0.35 |0.40 |0.45 |

|Stress (σ) |24.1 |25.5 |26.6 |27.3 |27.9 |

Find the strain energy, which is the area under the stress-strain curve. [Hint: strain energy = [pic]σ de]

14. Find your own interpolatory quadrature formula for integration using the nodes {-2, -1, 1, 2} in the range

[-1,1]. Also use this formula to evaluate [pic]

Submit by July 31, 2007

Nepal Engineering College

Changunarayan, Bhaktapur

Subject: Numerical Methods Practice Problems in Chapter 3 (not for submission)

Subject Code: MTH 317.3 (for real students who want to learn more than minimum required for exam)

Title: Numerical Differentiation and Integration

1. Find dy/dx and d²y/dx² at (a) x = 1.0 and (b) x = 1.5, 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

. 2. A slider in a machine moves along a fixed straight rod. Its distance x cm. along the rod is given below for various values of the time t seconds. Find the velocity of the slider and its acceleration when t = 0.1 second.

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 f'(20) from the following data:

x 3 5 11 27 34

f(x) -13 23 899 17315 35606

. 4. Given sin 0( = 0.000, sin 10( = 0.1736, sin 20( = 0.3420, sin 30( = 0.5000, sin 40( = 0.6428,

. (a) find the value of sin 32(

(b) find the numerical value of dy/dx at x = 12 for y = sin x

(c) find the numerical value of d²y/dx² at x = 30 for y = sin x.

. 5. The population of Duwakot VDC is shown below in the following table. Estimate the population in the years 1966 and 1993. Also find the rate of growth of population in 1991.

Year 1951 1961 1971 1981 1991

Population 1996 3965 5881 7721 9991

6. A rod is rotating in a plane. The following table gives the angle t (radians) through which the rod has turned for various values of the time t seconds. Calculate the angular velocity and the angular acceleration of the rod, when t = 0.6 second.

. 7. Evaluate [pic]by using (i) Trapeziodal rule, (ii) Simpson's 1/3 rule, and (iii) Simpson's 3/8 rule. Compare the results with its actual value.

. 8. The velocity v (km/min) of Safa Tempo which starts from rest is given at fixed intervals of time t (min) as follows. Estimate approximately the distance covered in 17 minutes.

t 2 4 6 8 10 12 14 16 18 20

v 10 18 25 29 32 20 11 5 2 0

. 9. A solid of revolution is formed by rotating about the x-axis, the area between the x-axis, the lines x = 0, x = 1 and a curve through the points with the following co-ordinates

x 0.00 0.25 0.50 0.75 1.00

y 1.0000 0.9896 0.9589 0.9089 0.8415

Estimate the volume of the solid formed using Simpson's rule.

. 10. Integrate numerically [pic]

. 11. A curve is drawn to pass through the points given by the following table.

x 1 1.5 2 2.5 3 3.5 4

y 2 2.4 2.7 2.8 3 2.6 2.1

Estimate the area bounded by the curve, and the lines y = 1, x = 1, x = 4.

. 12. A river is 80 ft. wide. The depth d in ft. at a distance x ft. from one bank is given by the following table.

x 0 10 20 30 40 50 60 70 80

y 0 4 7 9 12 15 14 8 3

Find approximately the area of the cross-section and the mean depth of the river.

.

. 13. A curve is given by the table

x 0 1 2 3 4 5 6

y 0 2 2.5 2.3 2 1.7 1.5

The x-coordinate of the C.G. of the area bounded by the curve, the end ordinates and the x-axis is given by where A is the area. Find [pic]using Simpson's 1/3 rule.

14. Estimate the length of the arc of the curve 3y = x3 from (0,0) to (1,3), using Simpson's 1/3 rule taking 8 sub intervals.

15. A body is in the form of a solid of revolution. The diameter D in cms of its sections at

. distances x cm. from one end are given below. Estimate the volume of the solid.

x 0 2.5 5.0 7.5 10.0 12.5 15.0

D 5 5.5 6.0 6.75 6.25 5.5 4.0

. 16. The following table gives the velocity v of a particle at time t. Find acceleration at t = 12.

t 0 2 4 6 8 10 12

v 4 6 16 34 60 94 136

17. A rocket is launched from the ground. Its acceleration f (cm/sec²) is registered during the first 80 seconds and is given in the table below. Using Simpson's 1/3 rule, find the velocity of the rocket at t = 80 seconds.

t 0 12 20 30 40 50 60 70 80

f 30 31.63 33.34 35.47 37.75 40.33 43.25 46.69 50.67

18. A reservoir discharging water through sluices at a depth of h below the water surface has a

surface area A for various values of h as given below.

h(ft) 10 11 12 13 14

A(ft²) 950 1070 1200 1350 1530

If t denotes time in minutes, the rate of fall of the surface is given by dh/dt = -48 (h/A. Estimate the time taken for the water level to fall from 14 to 10 ft. above the sluices.

19. Use Romberg's method to compute [pic]correct to 4 decimal places.

20. Using trapezoidal rule evaluate [pic], taking four sub-intervals.

21. Using Simpson's rule evaluate the integral [pic]

[pic]

22. Evaluate the following integrals using

(i)) 4 ordinate Gauss-Legendre quadrature

23. Evaluate the following integrals using the 2, 3, 4- ordinate Gauss-Laguerre quadrature.

[pic]

[pic]

[pic]

24. Use Gauss-Hermite and Gauss-Chebysev

quadrature to approximate the integrals with n = 2, 3, 4

Nepal Engineering College

Changunarayan, Bhaktapur

Subject: Numerical Methods

Q. 1 Evaluate the following integrals using:

[pic]

i) Newton-Cotes (closed)

4 ordinates & 5 ordinates

ii) Simpson's 1/3 Rule

iii) 3 ordinate Gauss-Legendre

[pic]

Q. 2 Evaluate the following integrals using

(i) 5 ordinate Newton-Cotes formula

(ii) 4 ordinate Gauss-Legendre quadrature

Q. 3 Evaluate the following integrals using the 2, 3, 4- ordinate Gauss-Laguerre quadrature.

[pic]

[pic]

[pic]

Q. 4 Use Gauss-Hermite and Gauss-Chebysev

quadrature to approximate the integrals with

n = 2, 3, 4

Q. 5 Find dy/dx at x = 1 from the following table by constructing a central difference table.

|x |0.7 |0.8 |0.9 |1.0 |1.1 |1.2 |1.3 |

|y |0.644218 |0.717356 |0.783327 |0.841471 |0.891207 |0.932039 |0.963558 |

(0.54030)

Q. 6 Given the following pairs of values of x and y.

| |x |1 |2 |4 |8 |10 |

| |y |0 |1 |5 |21 |25 |

Determine numerically the first and second derivatives at x = 4. (2.883, 0.861)

Old questions from Chapter 3 in Final Examination (2053 BS and 2054 BS, Tribhuvan University)

1 a) Write a program to find [pic]using Simpson's 1/3 rule and initial interval

of 1. Go on halving the interval and find the integral with relative error less than 0.001.

b) Write a flowchart to calculate the area under the curve by Trapezoidal rule.

2 a) Find dy/dx and d2x/dx2 of y = x1/3 from the following data.

| |X |50 |51 |52 |53 |54 |55 |56 |

| |Y |3.684 |3.7084 |3.7325 |3.7563 |3.7790 |3.8030 |3.8259 |

(at x=50, 0.02455, -0.0003)

b) Evaluate I = [pic]correct up to 3-decimal places with h = 0.25 and 0.125 by Simpson's 1/3 rule. Discuss on the results.

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[pic]

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