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# Class Note for MATH 250A at UA

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Date Created: 02/06/15

MATH 250a Fall Semester 2007 Section 2 J M Cushing Thursday August 30 httpmathariz0naeducushing250ahtml GRAPHS OF EXPONENTIALS Sketch the graph of Px ex Notes 611 l a 2 ex gt 0 Implles graph IS increasing 6121 x 2 dxz e gt 0 Implles graph 15 concave up 3 Px gt O for all x graph lies above the horizontal xaXis 4 PO e0 l graph passes through point 01 5 lim Px oo lim Px0 x gtoo x gt OO GRAPHS OF EXPONENTIALS Sketch the graph of Px ex P xaXis is called a horizontal asymptote GRAPHS OF EXPONENTIALS Sketch the graph of Px 6 36 Re ect graph of e x through the vertical axis GRAPHS OF EXPONENTIALS Sketch the graph of Px 6 36 GRAPHS OF EXPONENTIALS Sketch the graph of Px 62 dP 1 d 2xe x2 lt 0 forx gt 0 gt implies graph is decreasing forx gt 0 x d2P concave down for O lt x lt xE 2 dxz 2lt2x2 1gtex2 2 concave up for xE 2 lt x 3 Px gt O PO 1 lim Px O xH 00 P 10 GRAPHS OF EXPONENTIALS Sketch the graph of Px 62 dP l d 2xe x2 lt 0 forx gt 0 gt implies graphi decreasing forx gt 0 x d2P concave down for O x lt 22 dx2 2 2lt2x2 lgtex2 concave up for xE lt x 3 Pxgt0 POl r1111 Px0 P P x Px means graph is symmetric 10 with respect to the vertical axis Such a function is called even GRAPHS OF EXPONENTIALS Sketch the graph of Px 62 dP l d 2xe x2 lt 0 forx gt 0 gt implies graph is decreasing forx gt 0 x d2P concave down for O lt x lt xE 2 dx2 2 2lt2x2 lgtex2 concave up for xE 2 lt x 3 Px gt O PO 1 lim Px O xH 00 P x Px means graph is symmetric with respect to the vertical axis Such a function is called even GRAPHS OF LOGARITHMIC FUNCTIONS Sketch the graph of Px 1nx Inverse of Px ex yaXis is a vertical asymptote GRAPHS OF LOGARITHMIC FUNCTIONS Sketch the graph of Px Znx Notes 611 1 gt 0 for x gt 0 implies graph 1s increasing x d2 1 2 lt 0 implies graph is concave down dxz x APPLICATIONS OF EXPONENTIAL FUNCTIONS l Exponential Decay How long does it take a decreasing exponential function Pt that starts at P0 at time t O to decrease by 50 How does this halving time th depend on P0 How does th depend on the exponential rate of decay APPLICATIONS OF EXPONENTIAL FUNCTIONS 1 Exponential Decay Pt Poe rt O lt r decay rate 1 For what value 0ft 2 th does P0 rth1 Solve 1306 EPO for th First cancel 1 1 1 6 Wk 2 gt th ln 2 r 2 Halving time th 1n 2 APPLICATIONS OF EXPONENTIAL FUNCTIONS l Exponential Decay How long does it take a decreasing exponential function Pt that starts at P0 at time t O to decrease by 50 1 ANSWER th ln2 I Where r is the exponential decay rate How does this halving time th depend on P0 ANSWER it doesn t depend on P0 at all How does th depend on the exponential rate of decay ANSWER it is inversely proportional to r APPLICATIONS OF EXPONENTIAL FUNCTIONS l Exponential Decay How long does it take a decreasing exponential function Pt that starts at P0 at time t O to decrease by 50 1 ANSWER th ln2 I Where r is the exponential decay rate th is called the half life of P APPLICATIONS OF EXPONENTIAL FUNCTIONS l Exponential Decay EXAMPLE Suppose that the amount P of a drug of a medicinal drug present in a patient s bloodstream decreases by p per hour a Show that P is a decreasing exponential function and nd its halflife b For the antibiotic ampicillis p 40 What is the halflife Draw a graph of P as a function of time APPLICATIONS OF EXPONENTIAL FUNCTIONS l Exponential Decay EXAMPLE Suppose that the amount P of a drug of a medicinal drug present in a patient s bloodstream decreases by p per hour a Show that P is a decreasing exponential function and nd its halflife L L LZ PlPOl 100 P2Pll 100POl L L3 P3P2l 100POl APPLICATIONS OF EXPONENTIAL FUNCTIONS l Exponential Decay EXAMPLE Suppose that the amount P of a drug of a medicinal drug present in a patient s bloodstream decreases by p per hour a Show that P is a decreasing exponential function and nd its halflife P1 PO1 Observe the pattern and induct p t 2 P t P O l P2POl O 100 This is an exponential function L 3 P3POl 100 APPLICATIONS OF EXPONENTIAL FUNCTIONS l Exponential Decay EXAMPLE Suppose that the amount P of a drug of a medicinal drug present in a patient s bloodstream decreases by p per hour a Show that P is a decreasing exponential function and nd its halflife Pt P0at where a 1 1 lt1 lOO Pt P0elnat Pt POe where r lna gt O APPLICATIONS OF EXPONENTIAL FUNCTIONS l Exponential Decay EXAMPLE Suppose that the amount P of a drug of a medicinal drug present in a patient s bloodstream decreases by p per hour a Show that P is a decreasing exponential function and nd its halflife Pt P0at where a 1 1 lt1 lOO Pt P0elnat Pt POe where r lna gt 0 ln 2 2 2 Half life th r lna lnl plOO APPLICATIONS OF EXPONENTIAL FUNCTIONS 1 Exponential Decay EXAMPLE Suppose that the amount P of a drug of a medicinal drug present in a patient s bloodstream decreases by p per hour b For the antibiotic ampicillis p 40 What is the halflife Draw a graph of P as a function of time APPLICATIONS OF EXPONENTIAL FUNCTIONS 1 Exponential Decay EXAMPLE Suppose that the amount P of a drug of a medicinal drug present in a patient s bloodstream decreases by p per hour b For the antibiotic ampicillis p 40 What is the halflife Draw a graph of P as a function of time 2 Half life th m 137 hour 1n1 40100 Pt P0erl Where r s 051 APPLICATIONS OF EXPONENTIAL FUNCTIONS 1 Exponential Decay EXAMPLE Suppose that the amount P of a drug of a medicinal drug present in a patient s bloodstream decreases by p per hour b For the antibiotic ampicillis p 40 What is the halflife Draw a graph of P as a function of time P PU P0e7051t P0 0 1 2 3 4 5 6 7 8 9 APPLICATIONS OF EXPONENTIAL FUNCTIONS 1 Exponential Decay EXAMPLE Suppose that the amount P of a drug of a medicinal drug present in a patient s bloodstream decreases by p per hour b For the antibiotic ampicillis p 40 What is the halflife Draw a graph of P as a function of time P PU P0e7051t P0 P02 0 1 2 3 4 5 6 7 8 9 hm137 APPLICATIONS OF EXPONENTIAL FUNCTIONS 1 Exponential Decay EXAMPLE Suppose that the amount P of a drug of a medicinal drug present in a patient s bloodstream decreases by p per hour b For the antibiotic ampicillis p 40 What is the halflife Draw a graph of P as a function of time P PU P0e7051t P0 P02 P04 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE World Population billions Population numbers P Year World Population billions 1900 1655 1910 1750 1920 1860 1930 2070 1940 2300 1950 2519 1960 3024 1970 3697 1980 4442 1990 5280 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 t APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population Assume exponential Pt P0 6 billions Problem nd P0 and r 1900 1 65 5 Interpolation 1910 1750 1920 1860 Force exponential to pass through 1930 2070 two data p01nts 1940 2300 1950 2519 1960 3024 1970 3697 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population Assume exponential Pt P0 6 billions Problem nd P0 and r 1900 1 65 5 Interpolation 1910 1750 1920 1860 For example interpolate the first 1930 2070 two data pomts as follows 1940 2300 P1900 1655 1655 Poe1900r 1950 2519 P19101750 1750 Poemm 1960 3024 1970 3697 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE 39 rt Year World Population Assume exponential Pt P0 e billions Problem nd P0 and r 1900 13965 5 Interpolation 1910 1750 1920 1860 For exarinple interpolatelthe first 1930 2070 two ata p01nts as o ows 1940 2300 1655 19069005 1750 Poe1910r 1950 2519 1960 3024 2 equations in 2 unknowns 1970 3697 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population Assume exponential Pt P0 6 billions Problem nd P0 and r 1900 1 65 5 Interpolation 1910 1750 1920 1860 For example interpolate the first 1930 2070 two data pomts as follows 1940 2300 1655 190690 1750 Poe1910r 1950 2519 1900r 1960 3024 PO 1396556 1970 3697 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population Assume exponential Pt P0 6 billions Problem nd P0 and r 1900 1655 Interpolation 1910 1750 1920 1860 For example interpolate the first 1930 2070 two data pomts as follows 1940 2300 1655 19069001 1750 Poe1910r 1950 2519 490m 1960 3024 PO 1396556 1970 3697 1750 1655elor 1980 4442 1990 5280 1 equation 1n 1 unknown APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth Year World Population billions 1900 1655 1910 1750 1920 1860 1930 2070 1940 2300 1950 2519 1960 3024 1970 3697 1980 4442 1990 5280 EXAMPLE Assume exponential Pt P0 6 Problem nd P0 and r Interpolation For example interpolate the first two data points as follows 1655 190690 1750 Poe1910r Po 1655e 1900r 7 1750 1655e10r 1 1750 1 ln 10 1655 5581gtlt103 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth Year World Population billions 1900 1655 1910 1750 1920 1860 1930 2070 1940 2300 1950 2519 1960 3024 1970 3697 1980 4442 1990 5280 EXAMPLE Assume exponential Pt P0 6 Problem nd P0 and r Interpolation For example interpolate the first two data points as follows 1655 190690 1750 Poe1910r PO 1655e 1900r m 4104 gtlt10 5 1 1750 1 ln 1655 5581gtlt103 10 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population Assume exponential Pt P0 6 billions Problem nd P0 and r 1900 1 65 5 Interpolation 1910 1750 1920 1860 For example interpolate the first 1930 2070 two data pomts 1940 2300 P0 m 4104 gtlt10 5 1950 25 rm558lgtlt10 3 1960 3024 1970 3697 PU m 4104 gtlt105e5 581gtlt103 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population billions 1900 1655 1910 1750 1920 1860 1930 2070 1940 2300 1950 2519 1960 3024 1970 3697 1980 4442 1990 5280 Population numbers P World Population billions Interpolated points 0 1 I I I I I I I I I I 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 t 43 PO m 4104gtlt105 6558lgtlt10 1 Poor t APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth w Year World Population Other interpolations are possible billions 1900 1655 For exampl we could use the same 1910 1750 algebraie procedure to interpolate 1920 1860 the first amp last data pOlnts 1930 2070 1940 2300 1950 2519 1960 3024 1970 3697 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population Other interpolations are possible billions 1900 1655 For example we could use the same 1910 1750 algebraie procedure to interpolate 1920 1860 the f1rstamp last data p01nts 1930 2070 Turns out 1940 2300 PU 3821X10 11gte1289gtlt102t 1950 2519 1960 3024 1970 3697 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE World Population billions Population numbers P Interpolated points PU m 3821x10 Year World Population billions 1900 1655 1910 1750 1920 1860 1930 2070 1940 2300 1950 2519 1960 3024 1970 3697 1980 4442 1990 5280 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 t e1289gtlt10 2t Still not a good t APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population What is the best exponential t billions Can we use the 1900 1655 Method of Least Squares 1910 1750 Recall 1920 1860 d d 2 1930 2070 67701 lt ala pre lCllOI l 2 1940 2300 2 PM Ptdam 1950 2519 P P Nd t 2 1960 3024 lt M 06 i 1970 3697 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population What is the best exponential t billions Can we use the 1900 1655 Method of Least Squares 1910 1750 Recall 1920 1860 h d h 1930 2070 met 0 minimizes t e sum total of all errors 1940 2300 rt 2 d 1950 2519 E 2pdm poe ata 1960 3024 1970 3697 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population What is the best exponential t billions Can we use the 1900 1655 Method of Least Squares 1910 1750 Recall 1920 1860 h d h 1930 2070 met 0 minimizes t e sum total of all errors 1940 2300 rt 2 d 1950 2519 E 2pdm poe ata 1960 3024 1970 3697 1 P d 1980 4442 non 1near 1n 0 an r 1990 5280 Nonlinear Least Squares APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population Nonlinear Least Squares Method b39ll39 1 Ions le cult to apply 1900 1655 1910 1750 Numerical dif culties often arise 1920 1860 Instead a common procedure is to 1930 2070 search for a lmear relatlonshlp 1940 2300 between transformed variables 1950 2519 1960 3024 and then use L1near Least Squares 1970 3697 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population P P e billions 0 I 1quot 1900 1655 IHP 21110158 gt 1910 1750 In Z N lnpo 1920 1860 1930 2070 This is a linear relationship between 1940 2300 y lnP and t 1950 2519 y m 1960 3024 1970 3697 r slope ln P0 1ntercept 1980 4442 1990 5280 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population We can linearly regress on In scale y lnP and t 1900 0504 1910 0560 y rt lnP0 1920 0621 Turns out use computer or calculator 1930 0728 1940 0833 lnpo 24604 1950 0924 m1315x1o 2 1960 1107 1970 1308 y lt1315gtlt102t 24604 1980 1491 1990 1664 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth EXAMPLE Year World Population World Population billions 1n scale 2 1900 0504 1910 0560 3 1920 0621 f 139 1930 0728 3 1940 0833 1950 0924 1910 190 1930 1930 191 1920 1920 1930 193980 193990 1960 1107 t 1970 1308 y 1315gtlt102t 24604 1980 1491 1990 1664 APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth Year World Population billions 1900 1655 1910 1750 1920 1860 1930 2070 1940 2300 1950 2519 1960 3024 1970 3697 1980 4442 1990 5280 w Linear regression on a log scale lnP0 x 24604 r 1315gtlt10 2 y lt1315gtlt102gtt 24604 In terms of the original variables P0 2063 gtlt1011 rmaisxio 2 PO 2 Poe l m 330 gtlt1011e13930gtlt102l APPLICATIONS OF EXPONENTIAL FUNCTIONS 2 Exponential Growth Year World Population billions 1900 1655 1910 1750 1920 1860 1930 2070 1940 2300 1950 2519 1960 3024 1970 3697 1980 4442 1990 5280 EXAMPLE World Population billions Population numbers P N I 1 I I I I I I I I I I 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 t PO 2 Poe l m 330 gtlt1011e13930gtlt102l

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