Differentiation Techniques 1: Power, Constant Multiple, Sum and Difference Rules
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1 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules 97 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules Model : Fidig te Equatio of f '() from a Grap of f () a. f () = f( ) y b. y g () = c. y05 y y () = 8 6 y y86 Costruct Your Uderstadig Questios. Cosider Grap a i Model. a. Wat is te slope of te lie f( )? b. Is your aswer cosistet wit te grap of its derivative, f( ) 0?. Now cosider Grap b i Model. a. O Grap b, sketc te derivative of te fuctio g( ), sow tere. b. Determie te equatio of tis derivative: g( ). Now cosider Grap c i Model. d y p () = y a. O te aes (above, rigt) sketc te derivative of te fuctio sow i Grap c. To elp, te equatios of taget lies to te grap at 5,,, ad are give. b. From your grap, determie te equatio of tis derivative: ( ) c. f i Grap a is te derivative of te fuctio g i Grap b. Does te same relatiosip old for g ad i Grap b ad Grap c?
2 98 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules. (Ceck your work) f (o Grap a) is te derivative of g (o Grap b) ad g is te derivative of (o Grap c), so Kai guesses tat must be te derivative of p o Grap d. To ceck if tis is reasoable, e makes te followig table. Complete is table by fillig i empty boes wit positive, egative, or zero. Slope of lie taget to p i Grap d Value of te fuctio i Grap c positive positive 0 zero a. Kai cocludes tat o Grap c is a reasoable guess for te derivative of p o Grap d. Eplai ow te data i te table above supports tis coclusio. b. Te istructor asks Kai to eplai is reasoig. He begis is always positive ecept at = 0, were it is zero, ad te slopes of lies taget to f () = are positive ecept at = 0, were it is zero. Mark eac uderlied statemet TRUE or FALSE. c. (Ceck your work) Te istructor says: I like your tikig Kai, but lies taget to ay curve of te form k (were k is a positive costat) will be zero or positive. Is tis cosistet wit your aswer to part b? f () k y d. To figure out if Kai s guess is correct, te istructor suggests e use te defiitio of te derivative d f( ) lm i 0 f f( ) d to fid a epressio for. A partial solutio is sow o te et page, but take a d momet to derive a epressio for witout lookig aead.
3 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules 99 e. (Ceck your work) Is Kai s work sow at rigt cosistet wit your work o te previous page? d d Kai s Work o lim 0 d, as Kai was epectig? If ot, wat is d f. Is d lim 0 d lim 0 d ( ) lim 0 5. For fuctios of te form, we call te power of. Describe i words ow te power of caged i all te eamples so far i tis activity we goig from a fuctio f to its derivative f. 6. (Ceck your work) Look at te etries i te table i Model a (foud o te et page), ad decide if your aswer to te previous questio is cosistet. If ot, recosider te previous questio.
4 00 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules Model a: Derivatives of Power Fuctios For a Fuctio, f () ad its Derivative, f '() Equatio of f () Power of Equatio of f '() Power of ( ) f ( ) f Costruct Your Uderstadig Questios (to do i class) 7. Complete te first five rows of te table i Model a. Some etries are doe for you. 8. Based o te iformatio i te first five rows of Model a a. Write a rule describig ow to fid te derivative of a fuctio, were is a costat. b. Use tis rule to complete te saded (bottom si) rows of te table i Model a. c. (Ceck your work) Te derivative i te last row of Model a sould be. If tis does ot fit, go back ad recosider parts a ad b of tis questio.
5 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules 0 9. (Ceck your work ) Are your aswers to te previous two questios cosistet wit Summary Bo DT.? If ot, go back ad revise your aswers. Summary Bo DT.: Power Rule d were is a real umber 0. Eac fuctio i te table below ca be writte i te form. Complete te table. f( ) f () f () writte i te form f '() f( ) f( ). (Ceck your work) All but oe fuctio i te previous questio appears i te first colum of Model a. Are your aswers above cosistet wit your aswers i Model a? Model b: Derivatives of Power Fuctios For fuctios of te form c For a Fuctio, f () Equatio of f () Power of, we call te power of ad c te coefficiet of Coefficiet of ad its Derivative, f '() Equatio of f '() Power of. Coefficiet of f( ) ( ) ( ) f 0 f Note: Questios eplorig Model are foud o te et page.
6 0 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules Costruct Your Uderstadig Questios (to do i class). Complete te table i Model b. Some etries are doe for you.. For a fuctio c, write a epressio i terms of, c, ad for Model : Derivative Rules from Limit Laws Limit Law (Review from CA L) If te limit as approaces a of F( ) ad limit as approaces a of Geist, ( ) te lim cf( ) clim F( ) a a (Limit of a Costat Times a Fuctio Law) lim F ( ) G ( ) lim F ( ) lim G ( ) a a a (Limit of a Sum of Fuctios Law) lim F ( ) G ( ) lim F ( ) lim G ( ) a a a (Limit of a Differece of Fuctios Law) Costruct Your Uderstadig Questios. a. Wic Derivative Rule is proved at rigt? b. Wic Limit Law is used i te proof? Mark tis step. Correspodig Derivative Rule If f ad g are differetiable te d cf ( ) cf ( ) (Costat Multiple Rule) d f ( ) g ( ) (Sum Rule) d f ( ) g ( ) (Differece Rule) d cf ( ) c c lim Proof 0 f f lim c 0 ( ) ( ) f( ) f( ) clim cf '( ) 0 5. Complete te saded boes o te table i Model.
7 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules 0 Eted Your Uderstadig Questios (to do i or out of class) 6. Fid te derivative of eac fuctio. a. b. g( ) 6 c. d. 5 e. ( ) 9 f. f t () ( t) 7 g.. f () t t 5t i. ( ) g ( ) 7. Te rule at rigt is a combiatio of te Power Rule ad wat rule i Model? 8. (Ceck your work) Ceck tat tis combied rule is cosistet wit your aswer to Questio. Summary Bo DT.: Combied Rule d c c were c ad are real umbers 9. Fid te derivative of eac fuctio. a. b. c. ( ) 6 f 0. Cosider te fuctios i te previous questio: a. Costruct a eplaatio for wy 6 (i part c, above) is called te secod derivative of te fuctio (i part a, above). b. Wat is te tird derivative of? Eplai your reasoig. c. (Ceck your work) Are your aswers above cosistet wit te fact tat te fourt derivative of is zero?
8 0 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules Summary Bo DT.: Notatio for Secod, Tird, ad Higer Derivatives For a fuctio f ( ) (review) te first derivative is represeted as te secod derivative is represeted as f ( ) te tird derivative is represeted as te t ( derivative is represeted as f ) ( ) (were ) For eample, f () ( ) is used, ot f ( ).. Go back to Questio 9 ad, as sow i Summary Bo DT., add oe or more marks () to te f i parts b ad c to idicate eac fuctio s relatiosip wit te fuctio.. Fid f () ( ) for eac of te followig fuctios. Use te otatio i Summary Bo DT. to idicate te first troug tird derivatives tat you foud alog te way. a. 5 b. f 8 ( ) 5
9 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules 05 Notes
10 06 Differetiatio Teciques : Power, Costat Multiple, Sum ad Differece Rules Notes
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