Notice

Due to the festival of Rakhi, both the assignments (1 and 2) will be checked in the week starting from 6/8/12 during respective tutorials. No assignment is due next week (starting 30/7/12).

2nd Assignment of Numerical Methods in Engineering (ME 309)

Beant College of Engineering & Technology Gurdaspur

5^th Semester Mech (Section A&B)

2^nd  Assignment of  Numerical Methods in Engineering (ME 309) (10 Marks)

Due date: In the week starting 06/08/12 during respective Tutorials.

Q1.      Use bisection method to find square root of 30, correct up to 4 decimal places.

Q2.      Using bisection method, find the real root of the equation f(x) = 3x – (1 + sin x)^1/2 = 0, correct up to 3 decimal places.

Q3.      Find the real root of the equation f(x) = xe^x – 3 = 0, using Regula Falsi (false position) method, correct up to 3 decimal places.

Q4.      Find the real root of the equation f(x) = x² – log_e x – 3 = 0, using Regula Falsi (false position) method, correct up to 3 decimal places.

Q5.      Find the approximate root of the equation f(x) = e^-x – sin x  = 0, using Newton-Raphson  method, correct up to 4 decimal places. Start at x₀ = 0.6.

Q6.      Find the real root of the equation f(x) = 3x – cos x – 1 = 0, using Newton-Raphson method, correct up to 4 decimal places.

Q7.      The root of the equation f(x) = sin x – 5x – 2 = 0, lies near 0.5. This equation can be written in two possible ways to find its root by iterative method. Which of two possible ways will not yield result and which one will yield result and hence find the root of the equation correct up to four decimal places?

Q8.      Find the root of the equation f(x) = xe^x = 0, correct up to three decimal places using secant method.

Q9.      Find the root of the equation f(x) = 5x – cos  x – 3  =  0, correct up to three decimal places using Aitken’s Δ² method.

1st Assignment of Numerical Methods in Engineering (ME 309)

Beant College of Engineering & Technology Gurdaspur

5^th Semester Mech (Section A&B)

1^st  Assignment of  Numerical Methods in Engineering (ME 309) (10 Marks)

Due date: In the week starting 30/07/12 during respective Tutorials.

Q1.       Provide the number of significant digits in each of the following numbers:

(a)  0.0000055 g  _____   (c) 1.6402 g  _____   (e) 16402 g ______

(b)  3.40 x 10³ mL  ______  (d)  1.020 L  _____   (f) 1020 L _______

Q2.      Perform the operation and report the answer with the correct number of significant digits.

(a)    (10.3) x (0.01345) =  ___________________ 

(b)   (10.3) + (0.01345) = ______________________

(c) [(10.3) + (0.01345)] ÷ [(10.3) x (0.01345)]  =____________________________

Q3.      If r = 2×h×(h⁴ - 3), find the percentage error in r at h = 2, if the percentage error in h is 5.   
    

Q4.      If R = 4x³y²/z⁴ and relative errors in x, y, z are 0.03, 0.01, 0.02 respectively at x = 2, y = 1, z = 3. Calculate the absolute, relative and percentage error in evaluating R.

Significant Digits

Rules for Significant Digits

A.  Read from the left and start counting significant digits when you encounter the first non-zero digit

1. All non zero numbers are significant (meaning they count as significant digits)

617 has three significant digits

123456 has six significant digits

2. Zeros located between non-zero digits are significant

2006 has four significant digits

102 has three significant digits

40000000000000002 has 17 significant digits!

3.  Trailing zeros (those at the end) are significant only if the number contains a decimal point; otherwise they are insignificant (they don’t count)

5.240 has four significant digits

170000. has six significant digits

130000 has two significant digits – unless you’re given additional information in the problem

4. Zeros to left of the first nonzero digit are insignificant (they don’t count); they are only placeholders!

0.000456 has three significant digits

0.052 has two significant digits

0.000000000000000000000000000000000012 also has two significant digits!

B.  Rules for addition/subtraction problems

  

Your calculated value cannot be more precise than the least precise quantity used in the calculation. The least precise quantity has the fewest digits to the right of the decimal point. Your calculated value will have the same number of digits to the right of the decimal point as that of the least precise quantity.

In practice, find the quantity with the fewest digits to the right of the decimal point. In the example below, this would be 11.1 (this is the least precise quantity).

7.915 + 6.74 + 17.3 = 31.755, However, In this case, your final answer is limited to one significant fig to the right of the decimal or 31.8 (rounded up).

C.  Rules for multiplication/division problems

  

The number of significant digits in the final calculated value will be the same as that of the quantity with the fewest number of significant digits used in the calculation.

In practice, find the quantity with the fewest number of significant digits. In the example below, the quantity with the fewest number of significant digits is 17.3 (three significant digits). Your final answer is therefore limited to three significant digits.

 (17.3 x 13.235) ¸ 1.732 = 132.197171. However, in this case, since your final answer it limited to three significant digits, the answer is 132.

D.  Rules for combined addition/subtraction and multiplication/division problems

  

First apply the rules for addition/subtraction (determine the number of significant digits for that step), then apply the rules for multiplication/division.