1

Tutorial 2

Lecture NotesDr. Rakhmad Arief SiregarUniversiti Malaysia Perlis

Applied Numerical Method for Engineers

2

Quiz (60 Minutes)

What do you know about mathematical model in solving engineering problem? (10 marks)

Use zero through third order Taylor series expansions to predict f(2.5) for f(x) = ln x using a base point at x = 1. Compute the true percent relative error for each approximation. (15 marks)

Determine the real root of f(x)= 5x3-5x2+6x-2 using bisection method. Employ initial guesses of xl = 0 and xu = 1. iterate until the estimated error a falls below a level of s = 15% (20 marks)

3

Quiz (60 Minutes)

What do you know about mathematical model in solving engineering problem? (10 marks)

Use zero through third order Taylor series expansions to predict f(2.5) for f(x) = ln x using a base point at x = 1. Compute the true percent relative error for each approximation. (15 marks)

Determine the real root of f(x)= 5x3-5x2+6x-2 using bisection method. Employ initial guesses of xl = 0 and xu = 1. iterate until the estimated error a falls below a level of s = 15% (20 marks)

4

Solution 2

True value:f(2.5) = ln(2.5) = 0.916291...

5

Solution 1

True value:f(2.5) = ln(2.5) = 0.916291...

6

Solution 1

The process seems to be diverging suggesting that a smaller step would be required for convergence

7

Quiz (60 Minutes)

What do you know about mathematical model in solving engineering problem? (10 marks)

Use zero through third order Taylor series expansions to predict f(2.5) for f(x) = ln x using a base point at x = 1. Compute the true percent relative error for each approximation. (15 marks)

Determine the real root of f(x)= 5x3-5x2+6x-2 using bisection method. Employ initial guesses of xl = 0 and xu = 1. iterate until the estimated error a falls below a level of s = 15% (20 marks)

8

Solution 3

Graphically

-8

-4

0

4

8

-1 0 1

9

Solution 3

First iteration 563-583

10

Solution 3

First iteration The process can be repeated until the approximate error falls

below 10%. As summarized below, this occurs after 5 iterations yielding a root estimate of 0.40625.

11

Problems 5.1

Solution A plot indicates that a single real root occurs at

about x = 0.58

-8

-4

0

4

8

-5 0 5 10

12

Solution 5.1

Using quadratic formula

First iteration:

40512.6

40512.1

)5.0(2

)5.4)(5.0(4)5.2(5.2 2

x

13

Solution 5.1

Second iteration:

14

Solution 5.1

Third iteration:

15

Solution 5.4

Solve for the reactions:

16

Solution 5.4

A plot of these equations can be generated

-500

0

500

1000

0 4 8 12

0<x<3 3<x<6 6<x<10 10<x<12