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# Class Note for MATH 250A at UA

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This 30 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15

MATH 250a Fall Semester 2007 Section 2 J M Cushing Thursday September 6 httprnathariz0naeducushing250ahtrnl Rational Functions f x l px qx are polynomials Basic properties 1 Defined continuous differentiable at all points Where the denominator qx does not equal 0 x2 l l l x l l xzj l xzj l x2 Rational Functions f x l px qx are polynomials Basic properties 1 Defined continuous differentiable at all points Where the denominator qx does not equal 0 Sometimes we can remove a discontinuity 1 1 is discontinuous atxl and l x Forx l 1x M 1 l x2 Wl x l x Rational Functions f x l px qx are polynomials Basic properties 1 Defined continuous differentiable at all points Where the denominator qx does not equal 0 Sometimes we can remove a discontinuity 1 1 is discontinuous atxl and l x For Xi li 1x M 1 l x2 Wl x l x Defined and continuous at x l and equals 12 there Rational Functions f x l px qx are polynomials Basic properties 1 Defined continuous differentiable at all points Where the denominator qx does not equal 0 Sometimes we can remove a discontinuity l x l x2 f x 1 E for x l discontinuity at x 1 cannot be removed Why not for all x 7 i1 now continuous at x l Rational Functions f x l px qx are polynomials Basic properties 2 At nonremovable discontinuities x has a vertical asymptote l gt 0 for x lt l y l x lt0 forxgtl 1 2 gt 0gtincreasing clx l x 2 dx2 l x3 dzy l concave up for x lt l gt concave down for x gt 1 Rational Functions f x l px qx are polynomials Basic properties 2 At nonremovable discontinuities x has a vertical asymptote 1 V I I I I I O 1 0391 w Rational Functions f x l px qx are polynomials Basic properties 2 At nonremovable discontinuities x has a vertical asymptote V 1x 1 for Xiil l x2 l x y A 0391 1 O I I I I I O 1 0391 w Rational Functions f x Z px qx are polynomials Basic properties 3 Has horizontal asymptote if and only if deg q ideg p l i x2 l x y Rational Functions f x Z px qx are polynomials Basic properties 3 Has horizontal asymptote if and olnly if deg q ideg p l x2 Z x Inn y 2 11m 2 11m x gtoo x gtoo 1 x x gtoo 1 2 00 X Rational Functions f x Z px qx are polynomials Basic properties 3 Has horizontal asymptote if and oply if deg q ideg p 1 x2 In y hr In x 2 00 no hor1zontal x 00 x ool X x OO 1 asymptotes x 1 l x2 Inn y 11m 2 11m x oo x gt oo x gt oo 1 x x gt oo 1 Rational Functions f x Z px qx are polynomials Basic properties 3 Has horizontal asymptote if and only if deg q ideg p l i x2 l x3 y Rational Functions f x Z px qx are polynomials Basic properties 3 Has horizontal asymptote if and only if deg q ideg p I E 2 3 limylim1x 39 x 960 x gtoo x gtoo 1 x x gtoo 1 1 Rational Functions f x Z px qx are polynomials Basic properties 3 Has horizontal asymptote if and only if deg q ideg p l l x2 0 11m 2 11m O y x gtoo y x gtoo 1 x3 IS an horizontal asymptote l x2 x1113 y x1113 1 x3 0 in both directions Rational Functions f x Z px qx are polynomials Basic properties 3 Has horizontal asymptote if and only if deg q ideg p lx l x y Rational Functions f x Z px qx are polynomials Basic properties 3 Has horizontal asymptote if and only if deg q ideg p 1 1 l 1 11m y 11m 36 11m x 2 2 1 x gtoo x gtoolx x gtool 1 X Rational Functions f x Z px qx are polynomials Basic properties 3 Has horizontal asymptote if and only if deg q ideg p 1 1 1 1 x 1 y 1 H1210 y H1210 1 x is an horizontal asymptote l x 11m y 2 11m l in both directions x gt OO x gt oo Rational Functions f x Z px qx are polynomials Basic properties 3 Has horizontal asymptote if and only if deg q ideg p y Rational Functions Enzymesubstrate reactions aS b S substrate concentration Vs reaction rate Rational Functions Enzymesubstrate reactions aS V S lt D bS art5 ab 2gt0 increasing dS bS d2VS 46 2 3 lt 0 concave down dS b 5 lim VS lim am S gtoo S gtoo b S a horizontal asymptote Rational Functions Enzymesubstrate reactions VS 5 bS Rational Functions Enzymesubstrate reactions VS V S Rational Functions Enzymesubstrate reactions 5 VS V Rational Functions Enzymesubstrate reactions S S V VS 1V 2 max 5 1 Vmax max b S 2 S b Rational Functions Enzymesubstrate reactions VS V S m m 1 VmaX V E VmaX VmaX 1 max 1V b S 2 2 max often denoted by Km S 2 b 0 SJ An Application S V substrate ethanol M lL Ab b 39 0 W mm mm enzyme alcohol dehydrogenase ADH 00070 00600 product acetaldehyde 00150 0 l 100 00310 01600 00680 02100 V 01000 02300 3 39 o o 0 02000 02800 539 o 020 0 03000 02900 015 0 04000 02800 010 o 005 390 000 u u l 00 01 02 03 04 S K Dendinskas et al J Chemical Education 827 1068 2005 An Application s V V i MolL Absorbancemin m K m S 00070 00600 00150 01100 00310 01600 00680 02100 V 01000 02300 3 39 o o 0 02000 02800 539 0 03000 02900 392 0 04000 02800 quotquot5quot o 010 o 005 o 000 u u l 00 01 02 03 04 S K Dendinskas et 31 J Chemical Education 827 1068 2005 An Application s V V m i MolL Absorbancemin Km S 00070 00600 Eyeball estimates 00150 01100 Vmax 028 00310 01600 Km R 002 00680 02100 01000 02300 0393039 02000 02800 539 03000 02900 392 015 04000 02800 010 005 000 00 01 02 03 04 S K Dendinskas et al J Chemical Education 827 1068 2005 An Application S V V 028 5 MolL Absorbancemin 002 S 00070 00600 Eyeball estimates 00150 0 1 100 00310 01600 00680 02100 01000 02300 03930 02000 02800 5 03000 02900 392 015 04000 02800 010 005 000 I u u 00 01 02 03 04 S K Dendinskas et al J Chemical Education 827 1068 2005 An Application S V MolL Absorbancemin 00070 00600 00150 01100 00310 01600 00680 02100 01000 02300 02000 02800 03000 02900 04000 02800 K Dendinskas et al J S 002 S Eyeball estimates V2028 Fit not so good We can do better 030 025 020 015 010 005 000 I u u 00 01 02 03 04 S Chemical Education 827 1068 2005

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