Intrductin Lecture 1 Bimathematics Bimatematika prgram. I. Függvény fgalma, A függvények tulajdnságai. Elemi és összetett függvények. Határérték. Függvények határértéke, flytnsság. Számsrzatk és végtelen srk. Határérték. II. A differenciahányads és határértéke. Elemi függvények deriváltja. Differenciálási szabályk. Magasabbrendű deriváltak. Függvények hatványsrba fejtése. Taylr sr. Az e x, sin x és cs x srfejtése. Függvények szélső értéke, alkalmazásk. III. Határzatlan integrál. Alapintegrálk. Integrálási szabályk. Parciális integrálás. Integrálás helyettesítéssel. Összetett függvények integrálása. Határztt integrál. Newtn-Leibniz frmula. IV. A differenciálegyenletek, differenciálegyenletek típusai. Az y' = k y típusú differenciálegyenlet megldása. Elsőrendű differenciálegyenletek megldása. Alkalmazásk: kémiai reakciók,enzimkinetika. Magasabbrendű differenciálegyenletek. 1
M easure the m easurable and try t render m easurable what is nt yet G alile G alilei (1564-1642) Simple pharmaclgical prblem. Drug in cell envirnment RBC k 2 k 1 c i utflux influx Pharmackinetic mdels: drug absrptin, drug distributin, ib ti drug metablism Pharmacdynamic mdels: drug reactin, drug interactins Initial cnditins: t = 0, c ut = c 0, c in = 0 ut 0 in c 0 = c ut + c i Δc i= k1 c ut Δt - k 2 c i Δt 2
Example: time curse fr ttal systemic absrptin f a drug at subcutaneus dept at time t = 0 Absrptin f drug, a.u. 0.300 0.250 0.200 0.150 F = 1 { k e t /( k e / k a ) k e e e t } k e 1 k a k e =0.04381s -1 k a = 0.02638 s -1 0.100 0.050 0.000 Time, a.u. 0.00 20.00 40.00 60.00 80.00 100.00 120.00 Influenza Epidemic in a Bys Barding Schl 1978 βsi γi SIR mdel ds = rsi dt di = rsi ai dt dr = ai dt Threshld: rsi ai >> 0 r S > a / r Br. J. Med. 280. 1978 3
Membrane transprt envirnment: drug Passive transprt: cell c in k 2 k1 c ut utflux influx mass transprt acrss a membrane Δc in =k 1 c ut Δt - k 2 c in Δt Δc in /Δt = k 1 (c tt - c in ) - k 2 c in Cmpartment mdel: c in k 1 k 2 c ut dc in /dt = k 1 c tt - (k 1 + k 2 ) c in cell envirnment Initial cnditins: t = 0, c in = 0 and c ut = c tt c tt = c in + c ut Cncentratin, a.u. C, a.u. c ut (t) c ut = k 2 c tt /(k 1 + k 2 ) c tt C equ c in (t) c in = k 1 c tt /(k 1 + k 2 ) 0 1 2 3 T, óra c in (0) = 0 and c ut (0) = c tt 4
Average value f cncentratin Cncentratin, a.u. Cav (t2 t1) = T 2.0 c c av t 2 = c() t dt t1 c1 c2 c1 lg( c ) av = 1 2 2 1.5 1.0 0.5 c t c = c(t) 0 50 100 150 200 250 t 1 t 2 Time, sec Symbls and Definitins A set is a cllectin f bjects, called elements: A = {a 1,a 2,, a n, } Useful sets: N : {1,2,3, } Z : {, -2., -1, 0, 1,2, } Q the set f ratinal numbers, R the set f real numbers, C the set f cmplex numbers, (e.g. x 2 + 1 = 0 equ. has slutin in set C) U the set f all elements, (universal set, usually cntext-dependent) If x is an element f R, we write: x R, e.g. 2 R; 2 Q The sets satisfy: N Z Q R C, where (is cntained in), e.g. A B. 5
USEFUL SETS FOR NUMBERS Q R C C: R: Q: Z: N: Cmplex Real Ratinal Integer Natural Z N Definitins (i) The intersectin f A and B is dented by A B. (ii) The unin f A and B is written as A B. element in cmmn, then If sets A and B have n A B = where is the empty set. (iii) If the sets A and B such that A B = and A B = U, then A is cmplement f B, writing B = \ Α r A = \ B (\ excluded) B = Ā 6
Venn diagrams A A U B B Ũ A A U B B Symbls and Ntatins Belngs t Tends t Is cntained in [a, b] Clsed interval Intersectin f: A B Functin mapping A int B Unin x a x tends t a frm \ Excluded x <,> a values less/greater than a The empty set δx, Δx Small changes in x lim x a Limit as xtendst a expx = ex Value f expnential functin at x lge = lnx Value f lgarithm f x t base e f 1 Inverz functin f f ( f g) The cmpsitin f the functins f and g 7
Interval ntatin It is pssible t describe sets using interval ntatin. If a and b are real numbers such that a < b, we define the pen interval (a,b) as the set f all numbers between but nt including a and b, the set f all x fr which a < x < b. (a,b) : = {x a < x < b} The pints a and b are the endpints f the interval. (a, ) : = { x x > a } (-, ):={x x is a real number } The clsed interval is the set f values f x which satisfy the inequality a x b [a,b] : = {x a x b } Interval representatin pen interval (a, b) a b clsed interval a b psitive [a, b] O E P 0 1 unit length 8
Definitin f functin Crrespndence A Dmain f(x) B Range Functin: A functin is a crrespndence between a first set, called dmain, and a secnd set, called range, such that each member f the dmain crrespnds t exactly ne member f the range. Relatin: f : A B f -1 (x) x f(x) (x є D f ) A relatin is a crrespndence between a first set, called dmain, and a secnd set, called range, such that each member f the dmain crrespnds t at least ne member f the range. f(x) Y 60 f(x) = x 3 f(x) = x2 40 20 vertical-line test 0-20 -40 x < 0 x > 0 x -60-15 -10-5 0 5 10 15 9
Cmpsed functin Dmain 2 A x Dmain 1 φ B φ (x) (Range) f C f[φ(x)] Range (f φ)(x) = f(φ(x)) Inverse Inverz functin f(x) Dmain D f A B Range R R f Range f(x) f -1 (x) Dmain R -f = D f D = R -f f f -1 (f(x) = x 10
Table feature f a functin Temperature dependence f a resistance Temperature, CC Resistance, Ω 19.1 76.30 25.0 77.80 30.1 79.75 36.2 80.50 40.7 82.47 Analytical frmulas f functins y = sinx y y = x+ 4 4 x ( ) ( ) 3x5 tg 2x+ 3 = lg 1+ x 11
Death Rate / 100000 Males 1200 1100 y = 93.28x 2-1336 x + 5460.8 1000 900 800 700 600 Hurs f Sleep 5 6 7 8 9 Data frm Dr. Harald J. Mrwitz (Yale University) 12