EDEXCEL NATIONAL CERTIFICATE UNIT 41 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 1

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1 EDEXEL NATIONAL ERTIFIATE UNIT FURTHER MATHEMATIS FOR TEHNIIANS OUTOME ALGEBRAI TEHNIQUES TUTORIAL SIMPLIFYING/EXPANDING EQUATIONS ONTENTS Apply algebaic techniques Algebaic manipulation:, manipulation of engineeing fomulae, gaphical solution of simultaneous, quadatic and thid ode polynomial equations, intoduction to mati aithmetic, matices, deteminant of a mati, solution of two simultaneous equations using mati algeba; intoduction to seies eg: sigma notation, simple aithmetic/geometic seies, summation of seies, factoial notation fo combinations and pemutations, the binomial coefficient and binomial theoem omple numbes: intoduction to comple numbe fom, addition, subtaction, multiplication and addition of comple numbe in atesian fom, vecto epesentation of comple numbes, modulus and agument, pola epesentation of comple numbes, agand diagams, otating vecto, pola to atesian fom and vice vesa Logic cicuits: Boolean algeba zeo and unit ules, complement elations; commutative, distibutive, associative laws and elementay Boolean epessions fo logic cicuits Applications: solids, bodies in motion, vectos, mechanical enegy, heat, fluids, electical enegy, electic cicuits and logic cicuits It is assumed that the student has completed the module MATHEMATIS FOR TEHNIIANS. The answes to all the eecises ae given at the end. D.J.Dunn

2 SUMMATION This wod is used to mean the esult of adding (summing) a numbe of tems and the symbol Σ (capital sigma) is used. onside the two diagams. The left diagam is divided into many equal squaes of aea δa. The total aea may be epessed as A ΣδA meaning add all the small aeas togethe. The ight diagam is made up of stips all of width δ and heights y, y... y n The y values ae called odinates. The aea is the sum of all the stips so we can wite A Σ(y δ) We can define how many stips (n in this case) to sum as shown and the common facto δ may be taken outside so we should wite: A δ n y If we wanted to epess the aea between any two odinates, say y and y then we would wite: A δ y This idea in vaious foms can be applied to many poblems and helps simplify them. FATORIALS A factoial numbe is indicated with! so factoial is witten! And so on. A factoial numbe means simply all the integes (whole numbes) multiplied togethe so fo eample:! Thee is pobably a button on you calculato (e.g.!) that will let you do this quickly ty pactising. Note that the factoial of negative numbes ae an infinite seies fo eample: -! (-)(-)(-)(-)... -! (-)(-)(-)(-7)... THE MEANING OF n This is an epession that comes up in outcome and late in this tutoial. It is a factoial epession with the following meaning. n n(n -)(n - )(n - )...(n -{ })! The top line is the fist factos of n and the bottom line is factoial WORKED EXAMPLE No. Evaluate n and. The top line will be the fist factos of 0 D.J.Dunn

3 WORKED EXAMPLE No. Evaluate n and 0 SELF ASSESSMENT EXERISE No.. Evaluate the following:! 7! 0! (Answes, 00 and 8800!). Evaluate the following! 8!! 8!! (Answes 880 and 7). Evaluate the following. a a a (Answes, and ) In the last poblem you might notice that If we eamined this close we would see that in any seies fo a given value of n stating at n 0 and ending at n n the values ae symmetical aound the cente value stating and ending with. onside the case n (Take this as tue but it is had to see how to evaluate it) 0 0 It follows that 0 In geneal n n n- D.J.Dunn

4 ARITHMETIAL/GEOMETRI SERIES One of the most widely used eamples of an aithmetical seies is the BINOMIAL EXPANSION. BINOMIAL THEOREM This theoem povides us with a method fo evaluating y ( + ) n onside the case of n y ( + ) (+) (+) (+) (+) Multiplying out the backets yields y This is a seies with each tem containing to a powe in ascending ode fom 0 to n. Since o and we nomally wite as simply the equation is moe commonly witten as: y In geneal we can say y ( + ) n a o 0 + a + a + a + a a n n If we had a method fo finding a 0, a, a...a n the seies would be useful. Without poof, it can be shown that any a say a n Hence y ( + ) n n 0 + n + n + n + n n n n Noting that n 0 n n this becomes y ( + ) n + n + n + n + n n n n n The binomial epansion can be epessed as ( + ) ( ) 0 WORKED EXAMPLE No. Epand ( + ) with the binomial theoem. (+ ) n ( ) 0 we know that ( + ) as solved ealie 0 ( + ) heck this out by putting in any value of say (+) 79 ( + ) + + ()() + () + (0) + () + () D.J.Dunn

5 WORKED EXAMPLE No. Using the binomial theoem epand y (z + a) n Fist we must eaange the epession into a fom that can be epanded. n n n a (z a) z + + let a/z n n ( ) n (z + a) z + z Epanding we get y z n ( + ) n z n [ + n + n + n + n n ] y z n ( + ) n z n [ + n (a/z) + n (a/z) + n (a/z) + n (a/z) (a/z) n ] y z n ( + ) n z n [ + n z - a + n z - a + n z - a + n z - a z -n a n ] y z n ( + ) n z n + n z n- a + n z n- a + n z n- a + n z n- a a n ] WORKED EXAMPLE No. Using the binomial theoem epand ( ) answe is n - (+ ) (+ ) ( )( ) + + ()() + - y + and show that if is vey small then the +... ( )( )( ) + ()()() +... (+ ) The esult is an infinite seies and it is only useful fo evaluation when is small such that highe powes ae negligible. heck if 0.0 then y WORKED EXAMPLE No. Using the binomial theoem epand y and evaluate when y ( + ) putting n - we can epand. + n n n y ( + ) ( )( ) ( )( )( ) y ()() ()()() y y Note fo small values of y is quite accuately given by - D.J.Dunn

6 SELF ASSESSMENT EXERISE No.. Epand y ( + ) using the binomial theoem.. Epand y ( p) - to fou tems using the binomial theoem.. Epand y ( q) - to fou tems using the binomial theoem.. Epand y and show that fo small numbes y +. Epand ( + ) y and show that fo small numbes y + 9 This wok is continued in outcome tutoial. D.J.Dunn