FACULTY OF ENGINEERING/EUROPEAN UNIVERSITY OF LEFKE MATH 224 (MATH 208/302/305) ENGINEERING MATHEMATICS (NUMERICAL METHODS) SPRING FINAL EXAM

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1 Page 1 / 5 FACULTY OF ENGINEERING/EUROPEAN UNIVERSITY OF LEFKE MATH 224 (MATH 208/302/305) ENGINEERING MATHEMATICS (NUMERICAL METHODS) SPRING FINAL EXAM Date: Instructor: Prof. Dr. Hüseyin Oğuz Time: 09:00-11:00 Room#: AS Student Registration No: Student Name-Surname: Important Note: Your own scientific calculator is only allowable gadget to use during the exam with the prohibition of its exchange. Please choose 4 questions out of 5 by indicating in this sheet. 1. (25 p) The electrical resistance, (), of a thermistor varies with temperature according to: ()=100(1+ ) where is in Ω, = ,,= ,, is the temperature in. Find the temperature corresponding to a resistance of 200 Ω by using Newton-Raphson method with = =.%. 2. (25 p) An engineer supervises the production of the three types of components. Three kinds of material-metal, plastic, and rubber-are required for production. The amounts needed to produce each component are given in the table below: Component Metal Plastic Rubber If totals of 3.89, 0.095, and kg of metal, plastic, and rubber, respectively, are available each day, how many components can be produced per day. Use LU Factorization method. 3. (25 p) The following data gives the approximate population of the world for selected years from 1850 until Year Population(billions) Assume that the population growth can be modeled with an exponential function = where is the year and is the population in billions. Write the equation in a linear form, and use linear leastsquares regression to determine the constants and for which the function best fits the data. Use the model equation to estimate the population in the year (25 p) Evaluate the following integral: (1. ) a) Analytically, b) By using composite Simpson s 1/3 Rule with =. c) Calculate absolute relative true error as percentage (,%) 5. (25 p) a) Determine the value of ( 16 9), expressing the result in polar and rectangular forms. b) Determine the moduli and arguments of the complex roots (4 3) / Recall: = ( ) =; =; =;= ( ) = ; ( = ) ( ) ( )+4,,( )+2,, +( ) 3

2 MATH 302 NUMERICAL METHODS SPRING FINAL EXAM SOLUTIONS Date: Instructor: Prof. Dr. Hüseyin Oğuz 1. The solution can be formulated as: ()=100(1+ ) ()=0 ()=100( ) 200=0 ()=( ) 100=0 ()= = ( ) ( ) Applying Newton-Raphson formula given above gives with the initial guess of =250 Iteration 1: Iteration 2: = ( ) (250) (250) 100 ( ) = (250) =250 ( ) = = =6.15%>0.05% Page 2 / 5 = ( ) ( ) ( ) 100 ( ) = ( ) = = =0.016%<0.05% = h 3 2. a) Let =.The system of linear algebraic equations modeling the production run can be set up by using table data given as follows : = = =282 b) The system can be written in matrix form as follows: = The solution of the above three simultaneous linear equations by using LU Factorization method will give the value of,, 3890 = ;= Forward Elimination of Unknowns: Since there are three equations, there will be two steps of forward elimination of unknowns.

3 Page 3 / 5 First step: Divide Row 1 by 15 and then multiply it by 0.30 ( =. =0.02) and subtract the result from Row 2: (0.30)= Divide Row 1 by 15 and then multiply it by 1 ( = =0.0667) and subtract the results from Row 3: (1)= = Second step: We now divide Row 2 by (0.06) and then multiply by (0.0667) ( =.. =1.1117) and subtract the results from Row 3: Second step (Forward substitution): Third step (Back substitution): The solution vector is: (0.0667)= = = 1 0= = = =3890 =95 (0.02)(3890)=17.2 = (3890) (17.2)= = = = = = 17.2 (0.17)(77.086) 0.06 = = (68.256) 19(77.086) 15 =84.334

4 84 =68 77 Page 4 / 5 3. Employing exponential model requires transformation (linearization) of the model to use leastsquare method as follows: = ln=ln()+ = + Apply Least-Square method for the transformed form of exponential model to find regression coefficients as follows: = ( = ) Table. Summations of data to calculate coefficients of the linearized model: x x x x x x10 6 By substituting the corresponding values at the bottom of the table to the least square equations given above, we have: = 5( ) (9680)(5.1044) 5( ) (9680) =0.0104= = = =ln() = =. = Exponential model to estimate the population for the year 1970: = = () = a) Analytically: Apply u-substitution for the second integral as follows: = ,= (1. )=. = =10+ b) Applying Composite Simpson s 1/3 Rule: () =5.421 ( ) ( )+4,,( )+2,, +( ) 3

5 = =0,= = =10,=4 (),h= =7.5, =10 = (10 0) (0)+4(2.5)+(7.5)+2(5)+(10) 12 Page 5 / 5 =2.5, =2.5, =5, (0)=1.() =0;(2.5)=1.(.) =0.3680; (5)=1.() =0.6006; (7.5)=1.(.) =0.7476; (10)=1.() = (10 0) (0.6006) = c) (,%) (%)= 100= =0.019 % <0.05% h 3 5. a) =( 16) +( 9) tan = (3 ) 16 Applying De Moivre s theorem: ( 16 9) = = 337 6( )= By adding 360 (counterclockwise) two times, equivalent angle will be (2 nd Quadrant angle) or equivalently = (2 ). The final result will be given in polar form as follows: For rectangular form: ( 16 9) = ( ) =(cos+sin) = cos( )+sin( )=10 ( ) b) Determine the moduli and arguments of the complex roots: (4 3) / =() / = 1 1 () /= ( ) The roots are symmetrically displaced from one another = =120 apart round an Argand diagram as follows: (4 3)=(4) +( 3) tan 3 4 Applying De Moivre s theorem: = (4h ) (4 3) = =(5) ( )= (1 1 ), ( )= (2 2 ) ( )= (3 3 ) 3 rd root can be given equivalently as follows: =0.342 ( )= (3 3 )