TEACHER NOTES MATH NSPIRED
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1 Naural Logarihm Mah Objecives Sudens will undersand he definiion of he naural logarihm funcion in erms of a definie inegral. Sudens will be able o use his definiion o relae he value of he naural logarihm funcion o he area under a curve. Sudens will be able o use he Fundamenal Theorem of Calculus o compue he derivaive of he naural logarihm funcion. Sudens will be able o use he derivaive logarihm o deermine properies of is graph. æ ö ç of he naural è ø Sudens will look for and make use of srucure. (CCSS Mahemaical Pracice) Sudens will reason absracly and quaniaively. (CCSS Mahemaical Pracice) Vocabulary naural logarihm funcion area under a curve increases wihou bound derivaive Fundamenal Theorem of Calculus Abou he Lesson This lesson involves consrucing he graph of he naural logarihm funcion from is definiion. As a resul, sudens will: Conjecure abou he value of he naural logarihm funcion as increases wihou bound. Conjecure abou he value of he naural logarihm funcion as approaches 0 from he righ. Use he derivaive o discover properies of he graph of he naural logarihm funcion. TI-Nspire Navigaor Sysem Send ou he Naural_Logarihm.ns file. Monior suden progress using Class Capure. Use Live Presener o spoligh suden answers. Aciviy Maerials Tech Tips: This aciviy includes screen capures aken from he TI- Nspire CX handheld. I is also Nspire family of producs including TI-Nspire sofware and TI-Nspire App. Sligh variaions o hese direcions may be required if using oher echnologies besides he handheld. Wach for addiional Tech Tips hroughou he aciviy for he specific echnology you are using. Access free uorials a hp://educaion.i.com/calculao appropriae for use wih he TI- rs/pd/us/online- Learning/Tuorials Lesson Maerials: Suden Aciviy Naural_Logarihm_Suden.pdf Naural_Logarihm_Suden.doc TI-Nspire documen Naural_Logarihm.ns Compaible TI Technologies: TI-Nspire CX Handhelds, TI-Nspire Apps for ipad, TI-Nspire Sofware 204 Teas Insrumens Incorporaed educaion.i.com
2 Naural Logarihm Discussion Poins and Possible Answers Tech Tip: If sudens eperience difficuly dragging a poin, make sure hey have no seleced more han one poin. Press d o release poins. Check o make sure ha hey have moved he cursor (arrow) unil i becomes a hand ( ) geing ready o grab he poin. Also, be sure ha he word poin appears. Then selec / o grab he poin and close he hand ({). When finished moving he poin, selec d o release he poin Tech Tip: To change he value of, sudens can ouch heir finger o he poin and hen drag i along he -ais. TI-Nspire Navigaor Opporuniy: Class Capure and/or Live Presener See Noe a he end of his lesson. Move o page.2.. As sudens grab and drag poin o he righ along he horizonal ais, he compued area of he shaded region is equivalen o ln, he value of he naural logarihm funcion. Sudens may also use he up/down arrows in he op-righ porion of he page o change he value of. a. Complee he following able. Answer: ln b. Eplain wha happens o he value of ln as increases. Answer: As increases, he value of ln also increases (a a slower rae). c. Eplain your answer in par b geomerically. Answer: As increases, he area under he graph of y =, above he -ais, and beween he verical lines a and is increasing. 204 Teas Insrumens Incorporaed 2 educaion.i.com
3 Naural Logarihm 2. Drag poin o he lef of (bu greaer han 0), or use he up/down arrows o change he value of. a. Complee he following able. Answer: ln b. Eplain wha happens o he value of ln as decreases (ges closer o 0). Answer: As he value of ges closer o 0, he value of ln decreases. c. Eplain your answer in par b geomerically. Answer: As ges closer o 0, he area under he graph of y, above he -ais, and beween he verical lines a and is increasing. However, since he lower bound on he inegral is, and <, he value of he definie inegral is negaive. Use he following propery o illusrae his: d d. Move o page A par of he graph of y = ln is displayed. Have sudens grab poin or use he up/down arrows o change he value and move i along he horizonal ais o he righ o consruc he remaining par of he graph of y = ln. The values of he naural logarihm funcion are displayed on he righ screen. a. Eplain wha happens o he graph of y = ln as increases wihou bound (as ). Answer: As increases wihou bound, he graph of y = ln also increases, a a slower rae. Some sudens may say ha he graph appears o be leveling off. b. Eplain wha happens o he graph of y = ln as approaches 0 from he righ (as 0 + ). Answer: As 0 +, he graph of y = ln decreases quickly. I appears o be approaching. 204 Teas Insrumens Incorporaed 3 educaion.i.com
4 Naural Logarihm c. Eplain why = 0 is no in he domain of he funcion y = ln. Answer: The naural logarihm is defined in erms of a definie inegral. In order o evaluae he definie inegral d d, mus be defined on, he inegrand, he closed inerval [, ]. The funcion is no defined for = 0. Therefore, we canno evaluae he definie inegral, and he naural logarihm funcion is no defined for = 0. d. The funcion f ( ) is defined for < 0. For eample, f( 2) =. Eplain why he 2 definiion of he naural logarihm funcion canno be eended o include negaive numbers. b Answer: If he funcion f is coninuous, hen he definie inegral f ( ) d eiss. If he a funcion f has only a finie number of jump disconinuiies, ha is, f is piecewise coninuous, hen he definie inegral also eiss. However, consider he inegral d. The inegrand, 2, has an infinie disconinuiy a 0. As 0+, +. And as 0,. For any value of < 0, he inerval [, ] will include 0. Therefore, his definiion of he naural logarihm funcion canno be eended o negaive numbers. e. Use he Fundamenal Theorem of Calculus o find he derivaive of f() = ln. Deermine he inervals on which he graph of y = f() is increasing and he inervals on which i is decreasing. Find he absolue ereme values for f. Deermine he inervals on which he graph of y = ln is concave up and he inervals on which i is concave down. Answer: Using he Fundamenal Theorem of Calculus, d f ( ) d d. Since f () = > 0 for > 0, he graph of he naural logarihm funcion is always increasing. Since he graph of y = f() is always increasing, here is no absolue maimum value. Since he domain is > 0, here is no absolue minimum value. d f ( ) 2 d Since f () < 0 for > 0, he graph of he naural logarihm funcion is always concave down. 204 Teas Insrumens Incorporaed 4 educaion.i.com
5 Naural Logarihm Wrap Up Upon compleion of he discussion, he eacher should ensure ha sudens undersand: The relaionship beween he area under he curve and he definiion of he naural logarihm funcion. The properies of he graph of he naural logarihm funcion. A he end of his aciviy, sudens discover ha he graph of he naural logarihm funcion is always increasing and ha here is no absolue maimum or minimum value. However, we do no know wheher here is an upper bound or lower bound for he funcion. Some relaed opics for discussion include he area of an unbounded region and comparing he area under he graph of y, above he -ais, greaer han =, wih he harmonic series. TI-Nspire Navigaor Noe Quesion, Class Capure and/or Live Presener As sudens begin his aciviy, use Class Capure o ensure ha each suden is able o grab and drag he appropriae poin. 204 Teas Insrumens Incorporaed 5 educaion.i.com
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