Cologne University of Alied Sciences Winter-Semester 5-6 M: Mathematics Eam (8 minutes)- Problem Find: (a) the - coordinate of the centroid of the shaded area by hand integration. Detailed calculations required. Show all your work. (b) the integral y Setember, 5 quadratic arabola I y da (b) by hand integration (b) by numerical integration using GAUSS. Use the number of samle oints required for eact integration. (a) æ ö A ç + - d çè ø æ ö ç + - çè 9 ø 6+ - 6 æ ö da ç + - d çè ø S æ ö ç + - d çè ø æ ö ç + - çè 9 ø 8 9+ -6 9 9.75 da A 9.75 6.65 f f oben unten + + y da y dyd Sloe A + ( y ) d é êæ ö 8. ù d + - çè ø 7 êë úû æ 6 8 ö 8 ç + + + - çè 7 7 ø æ 5 7 8 ö 8 ç + + + - çè 5 89 8 ø» 5.657.. D:\THKoeln\MeineLehrveranstaltungen\MATHE\KlausurUndTests\Klausur \Mathe Klausur Sol.doc -Oct-5 /8
Cologne University of Alied Sciences Winter-Semester 5-6 Problem Aroimate the function y e - by a first, second, third and fourth degree Taylor olynomial at location. For - lot the function and the four Taylor aroimations into a single lot. The number of -values should be large enough to obtain smooth curves. f e f - e f e f - e f e - - - - IV - IV f() f () - f () f () - f () e e e e e () e - n n (-) ( )» + å - e e n! n () T - - e e ( -) e ( ) T - - + - e e e ( - + 5) e T - - + - - - e e e 6e (- + 6-5 + 6) 6e () T - - + - - - e e e 6e e ( -8 + -6 + 65) e + ( - ) For lotting it is easier to use the series eansion in () rather than the olynomial eansion in (). y 8 7 6 5 Function e - Linear TP Quadratic TP Cubic TP Quartic TP - - - D:\THKoeln\MeineLehrveranstaltungen\MATHE\KlausurUndTests\Klausur \Mathe Klausur Sol.doc -Oct-5 /8
Cologne University of Alied Sciences Winter-Semester 5-6 Problem Aroimate the function y cos by Lagrange interolation. (a) quadratic olynomials (samle oints,, ) (b) cubic olynomials (samle oints, /, /, ) (c) quartic olynomials (samle oints, /,,.5, ) (d) by iecewise cubic Lagrange olynomials (samle oints, /, /, for ) and four samle oints, /, 5 /, for. Plot five curves into a single figure. The number of -values should be large enough to obtain smooth curves. y.5 Quadratic Cubic Quartic Piecewise cubic Function y cos -.5-5 6 D:\THKoeln\MeineLehrveranstaltungen\MATHE\KlausurUndTests\Klausur \Mathe Klausur Sol.doc -Oct-5 /8
Cologne University of Alied Sciences Winter-Semester 5-6 Problem An investor invests 6. Si years later she invests an additional 8. (a) After years, the caital has grown to 5. Calculate the average rate of return in % over the eriod of years. (b) If the average rate of return is 5%, after how many years has her caital grown to 5. Use the Newton-Rahson rocedure on the formula æ i ö K å K + ç çè ø i discussed in class. t i K [ ] 5 (a) æ ö æ ö K 6 8 + + + ç è ø çè ø f 6 + + 8 + -5 9 f + + +.79.8%.5.5.5..5 i i 5 f ( i ) f ( i ) D i + i +.5 -.8.5.98.798.798 9.7.586 -.59.5.5..6787 -..7585.7585.8.55 -..5 5.5..58 -..85 6.85.. -..796 7.796.. -..796 (b) t t-6 f 6.5 + 8.5-5 t f 6 ln.5.5 + 8 ln.5.5 t-6 t 9.8 years Recall from school math: ln ln a e a lna e lna a a a Details of iteration not shown. D:\THKoeln\MeineLehrveranstaltungen\MATHE\KlausurUndTests\Klausur \Mathe Klausur Sol.doc -Oct-5 /8
Cologne University of Alied Sciences Winter-Semester 5-6 Problem 5 Solve + + y.96 7 + y + y 98.976 using the Newton Rahson method. Use 5, y 5 as your start oint and iterate until you reach convergence with reasonable accuracy. No hand calculations required. y + + -.96 y 7 + + - 98.976 Jacobian matri é y y ù é ù 6 + y y 7 ê + ú ê ú ë û J ë Course of iteration û y 7.79 6.657.58.9.5. 7.887 7.7..7.. del -.6 -. -. -.7 -.7 -. -.79 -..98.6.. 5..677.66.76.7.. 5..89.765.6.9.. (Numerical values of Jacobian matri J in each iteration not shown). The solution is., y.. D:\THKoeln\MeineLehrveranstaltungen\MATHE\KlausurUndTests\Klausur \Mathe Klausur Sol.doc -Oct-5 5/8
Cologne University of Alied Sciences Winter-Semester 5-6 Problem 6 Let % the robability that a certain tye of lightbulb will fail in a -hour test. Find the robability that a sign consisting of 5 such bulbs will burn hours with (a) no bulb failures. (b) not more than bulb failures. We use the Bernoulli distribution. æ5ö n 5! Pn ( failures in hours). ( -.). -. ç n è ø n! 5-n 5-n n 5-n (a) 5 P (no failures).98.786 7.9% (b) æ5ö P( failure). -. ç çè ø 5!..98! 5!.7857 5.7857 checks (a) 5 æ5ö P( failure). -. ç çè ø 5!..98!! 5..756.6 æ5ö P( failures). -. ç çè ø 5!..98!! 5..769. P(not more than failures).786 +.6 +..997 99.7% D:\THKoeln\MeineLehrveranstaltungen\MATHE\KlausurUndTests\Klausur \Mathe Klausur Sol.doc -Oct-5 6/8
Cologne University of Alied Sciences Winter-Semester 5-6 Problem 7 Find mean and standard deviation of the robability distribution given by its robability density function f cos Solve the roblem by (a) hand calculation, show integration stes. (b) numerical integration using the SIMPSON rule (5 segments, i.e. 5 samle oints) (a) f cos m fd cosd sin - sin d + cos -».578 a ; b i/; N 5; [,w,del] simsonsamleandweights(a,b,n); y cos(); y.*cos(); y.^.* cos(); int sum(y.* w * del/) %should be int sum(y.* w * del/) %mean int sum(y.* w * del/) %second moment var int - int^ %variance std sqrt(var) %standard deviation function [,w,del] simsonsamleandweights(a,b,n) %INPUT: %a: lower limit %b: uer limit %N: number of samle oints E fd s s cosd sin - sin d + cos - cos d + -sin - E -m æ ö -- ç - çè ø -- + - -».6 -».76 %OUTPUT: %: samle oints %w: weights %del: sacing del (b-a) / (N-); [a:del:b]; w zeros(,n); w([ N]) ; w([::n-]) ; w([::n-]) ; D:\THKoeln\MeineLehrveranstaltungen\MATHE\KlausurUndTests\Klausur \Mathe Klausur Sol.doc -Oct-5 7/8
Cologne University of Alied Sciences Winter-Semester 5-6 Problem 8 Let the random variable X be the roduct of three dice throws. (a) Of the 6 ossible combinations, how many different roducts are ossible? (b) Find PX, i.e. what is the robability that the roduct is equal to? (c) Find PX, i.e. what is the robability that the roduct is smaller or equal to? (d) What are the values of the roducts that have the largest robability of occurring, i.e. what values of X have the largest robability? What is that robability? (e) Plot the PDF of X. No hand calculations required. (a) The number of different roducts is. (b) The robability that the roduct is equal to is zero. (c) Of the 6 ossible roducts, 99 are smaller than or equal to. The robability is thus P / 6.9. (d) Of the 6 ossible roducts, the roducts X and X show u 5 times. The robability is thus P 5 / 6.69 (e) Product Frequency Cum. Frequ. Probability of occurrence.6.9 7.9 6.78 5 6.9 6 9 5.7 8 7. 9 5.9 6.78 5 56.69 5 6 6.78 6 6 68.78 8 9 77.7 9 86.7 5.69 5.9 7 5.6 7.556.9 6.556 6 8.78 5.9 8 9 5.7 5 5.9 5 56.9 6 68.556 6 69.6 7 9 78.7 75 8.9 8 8.9 9 6 9.78 96 9.9 96.9 8 99.9 6 5.78 5 6.6 9.9 5.9 8 5.9 6 6.6 å 6. Probability Frequency/6 5 bars Frequeny 5 no bar @ X tallest bar at X and X with height F 5 5 5 X D:\THKoeln\MeineLehrveranstaltungen\MATHE\KlausurUndTests\Klausur \Mathe Klausur Sol.doc -Oct-5 8/8