Exponentials and Logarithms Review Part 2: Exponentials
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1 Eponentials and Logaritms Review Part : Eponentials Notice te difference etween te functions: g( ) and f ( ) In te function g( ), te variale is te ase and te eponent is a constant. Tis is called a power function. In te function f ( ), te variale is te eponent and te ase is a constant. Tis is called an eponential function. It is too ad tat ot of tese names refer to te eponent. It would e nice if te first one were called te ase function and te second one te eponential function ecause ten it would e clear were te variale was located. Tere are times in indsigt wen it would e convenient to cange names or notation, ut once te ave ecome estalised in te mainstream, tis is ver difficult to do. Meaning In tis section, we will e discussing eponential functions. We will start wat te notation means. Just as multiplication can e sortand for addition, for eample: , eponentials can e sortand for multiplication, for eample: n It is clear wat means if n is a natural numer greater tan, {,, }, ut wat does is it mean if te eponent is, 0 or negative? One wa to understand wat it sould mean is to look at wat appens eac time ou move down a row in te following sequence: Te eponent decreases on te left and ou lose one of te s on te rigt. But instead of tinking of it as losing a, tink of it as dividing te previous row.
2 Ten ou could add te net rows in tis pattern decreasing te eponents on te left and dividing te row on te rigt : 0 Rational Eponents Te set of real numers is te disjoint union of te rational and irrational numers. Te rational numers are te real numers tat can e written as q p were p and q are integers wit q 0. Wat sould / mean? Well, / sould fall etween 0 and (ecause 0 / ). ten n m nm Also, ecause / / / / sould satisf /. Te unique positive numer tat satisfies ot of tese conditions is : Because., ten and so te first ullet aove is satisfied. Because, ten te second ullet aove is satisfied. Hence, /. Likewise we ave / n n for all natural numers n and positive numers.
3 Wat sould / 7 mean? Well, / 7 sould fall etween 7 7 and 7 9 (ecause / ). n ( n) ( m) Also, ecause m ten / / / 7 sould satisf Te unique positive numer tat satisfies ot of tese conditions is 7 : Because 7. 9, ten 7 7 9, and so te first ullet aove is satisfied. Because 7 7, ten te second ullet aove is satisfied. Hence, / 7 7. Likewise, numer. m/ n n m wen m and n are integers, n 0, and is a positive Irrational Eponents It turns out tat wen rational numers are written in teir decimal form te eiter repeat or terminate. For eample: Repeat Terminate 7 8 Te irrational numers are te real numers tat do not repeat or terminate in decimal form. Because te are not usuall eas to represent, we sometimes give tem names. Some eamples of irrational numers are: Te last of tese as a pattern, ut it is a growing pattern, not a repeating pattern. Wat sould mean? Well sould satisf: or more precisel:
4 /0.. /0 or more precisel: etc. / /00 r Te ke is tat we know ow to define wen r is a rational numer. Hence, if we want to define wen is an irrational numer, we can find a sequence of rational numers tat approac (from te left or from te rigt) simpl using terminating decimal approimations of (as sown aove for te case of ). Net we will look at te laws of eponents. Properties of Eponents If a and are positive numers and and are an real numers, ten.... a a To see w propert is reasonale, consider te following eample: 7. To see w propert is reasonale, consider te following eample: To see w propert is reasonale, consider te following eample: 5 () (5) To see w propert is reasonale, consider te following eample:
5 5 Eample Simplif te following epressions. Assume tat and are positive. Solutions are at te end of te notes. a) 8 ) 5 5 c) d)
6 6 Eample Fill in te tale elow for f ( ) and g( ), plot tese points, and ten sketc teir graps on te same set of aes. Solutions are at te end of te notes. Give eact values, not decimal approimations f() g() a. Wat is te domain of f? g?. Do eiter of tese functions ave - or -intercepts? If so, wat are te? c. Do eiter of tese functions ave vertical asmptote(s)? Eplain. d. Do eiter of tese functions ave orizontal asmptote(s)? Eplain. e. If tere are orizontal asmptote(s), does eiter grap cross te orizontal asmptote(s)? Eplain. f. Wat is te range of f and g?
7 7 g. Sketc te graps of and on te same aes as g( ). Note: and ecause f ( ) and for are real numers.. How do te graps of and appear to e related? i. If te function f () is reflected aout te -ais, te new grap is given f ( ). If? f ( ), wat is its reflection aout te -ais? Is it te same as
8 8 Eample Grap sketcing and first. Tis sifts te grap of tree units rigt. Tis reflects te grap of aout te -ais. Tis sifts te grap of one unit up. Eample Assuming, matc te equation wit its grap. a.. c. d. e. f. A B C D E F
9 9 Solutions: a. D. F c. B d. A e. E f. C Eample Te graps of f ( ) and g( ) are sketced elow on te rigt along wit teir tangent lines at ( 0, ). We are going to approimate te slopes of teir tangent lines at te point ( 0, ). First we will do tis eealling te graps. Ten we will do tis approimating te slopes of secant lines wose slopes approac te slope of te tangent lines. Eealling: Estimate teir slopes using te given tangent lines. Rememer, ( 0, ) is on ot tangent lines. For f ( ) te tangent line looks like it goes troug (,.). Using tat approimation and te fact tat it also goes troug ( 0, ) gives us.. Slope For g( ) te tangent line looks like it goes troug (,.). Using tat approimation and te fact tat it also goes troug ( 0, ) gives us.. Slope.. 0 Now we will use te slopes of secant lines to approimate te slopes of te tangent lines. Rememer, tis is ow we came up wit te formula for te derivative. For f ( ) te slope of te secant line connecting (, f ( )) to ( 0, ) is given f ( ) 0.
10 0 As 0, te slope of te secant line approaces te slope of te tangent line. Te slope of te secant line corresponding to 0. is aout Te slope of te secant line corresponding to 0. 0 is aout Because te function is concave up and increasing, ot of tese are overestimates. However, te are prett good estimates ecause te slope of te tangent line is 0.69 rounded to decimal places. For g( ), te slope of te secant line connecting (, f ( )) to ( 0, ) is given f ( ). 0 In tis case, te slope of te secant line corresponding to 0. is aout.6 and te slope corresponding to 0. 0 is aout.07. Te slope of te tangent line is.0986 rounded to decimal places. Hence, te slope of te tangent line at ( 0, ) is less tan for tan for g ( ). f ( ) and more We see tat te slopes of te tangent lines at ( 0, ) increase as te ase increases (for functions of te form ( ) ). Matematicians wanted to find te value of for wic te slope was eactl. Clearl it was somewere etween and. It turns out to appen for equal to some irrational numer, so te gave it a name. (Just like te gave te irrational numer tat is te lengt of te circumference of a circle of diameter te name.) Does anone want to guess wat te named te numer etween and? e Tis means tat lim 0 e. ***
11 Te graps of f ( ), g ( ) and ( ) e are sown elow. We can use lim 0 e *** to find te derivative of f ( ) e. f ( ) lim 0 f ( ) f ( ) e lim 0 e e lim 0 e e e lim e 0 aove B *** e Tus, d d e e.
12 Solutions Eample Simplif te following epressions. Assume tat and are positive. a) 8 ) c) d) ecause0 and 0 / / or Eample Fill in te tale elow for f ( ) and g( ), plot tese points, and ten sketc teir graps on te same set of aes. Give eact values, not decimal approimations. f() g() - / /9 - / / 0 9 a. Wat is te domain of f? g? All real numers. Do eiter of tese functions ave - or -intercepts? If so, wat are te? Te ot ave -intercepts of ( 0, ) c. Do eiter of tese functions ave vertical asmptote(s)? Eplain. No, te are defined and continuous everwere. Tus, te grap does not grow, witout ound as ou approac an real numer. d. Do eiter of tese functions ave orizontal asmptote(s)? Eplain. Bot ave a HA of te -ais, 0. As -values ead towards, - values approac 0 from aove.
13 e. If tere are orizontal asmptote(s), does eiter grap cross te orizontal asmptote(s)? Eplain. No, te function is alwas positive; ence it will never touc te -ais. f. Wat is te range of f and g? ( 0, ) g. Sketc te graps of and on te same aes as ( ). Note: g numers. and ecause f ( ) and for are real. How do te graps of and It appears tat is te reflection of appear to e related? aout te -ais. i. If te function f () is reflected aout te -ais, te new grap is given f ( ). If f ( ), wat is its reflection aout te -ais? Is it te same as? f ( ). Yes, f ( ) Eample Grap sketcing and first. Tis sifts te grap of tree units rigt. Tis reflects te grap of aout te -ais. Tis sifts te grap of one unit up.
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![MATH Fall 08. y f(x) Review Problems for the Midterm Examination Covers [1.1, 4.3] in Stewart](/thumbs/87/95118706.jpg "MATH Fall 08. y f(x) Review Problems for the Midterm Examination Covers [1.1, 4.3] in Stewart")
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