Cubic Functions: Local Analysis

Transcription

1 Cubic function cubing coefficient Capter 13 Cubic Functions: Local Analysis Input-Output Pairs, 378 Normalized Input-Output Rule, 380 Local I-O Rule Near, 382 Local Grap Near, 384 Types of Local Graps Near, 385 Local Features Near, 386 Addition Formula For Cubes, 388 Local I-O Rule Near 0, 392 Local Coefficients Near 0, 394 Derivative Functions, 401 Local Grap Near 0, 403 Local Features Near 0, 406. Cubic functions are functions specified by a global input-output rule of te form CUBIC a,b,c,d CUBIC a,b,c,d () = a 3 + b 2 + c + d were CUBIC a,b,c,d is te name of te cubic function and were a, b, c and d stand for te four bounded numbers tat are needed to specify te cubic function CUBIC a,b,c,d. 1. In order to facilitate our investigation of cubic functions, it is necessary to begin by introducing some language special to cubic functions. Given te cubic function CUBIC a,b,c,d, tat is te function specified by te global input-output rule CUBIC a,b,c,d CUBIC a,b,c,d () = a 3 + b 2 + c + d Te given signed number a is called te cubing coefficient of te cubic function CUBIC a,b,c,d and a 3, tat is te cubing coefficient multiplied 373

2 374 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS cubing term squaring coefficient squaring term linear coefficient linear term constant coefficient constant term by tree copies of te input, is called te cubing term of te cubic function CUBIC a,b,c,d. Te given signed number b is called te squaring coefficient of te cubic function CUBIC a,b,c,d and b 2, tat is te squaring coefficient multiplied by two copies of te input, is called te squaring term of te cubic function CUBIC a,b,c,d Te given signed number c is called te linear coefficient of te cubic function CUBIC a,b,c,d and c 1, tat is te linear coefficient multiplied by one copy of te input, is called te linear term of te cubic function CUBIC a,b,c,d Te given signed number d is called te constant coefficient of te cubic function CUBIC a,b,c,d and d 0, te constant coefficient multiplied by no copy of te input, is called te constant term of te cubic function CUBIC a,b,c,d. In oter words, te constant term is te same as te constant coefficient and wat words we will use will depend on wat viewpoint we will be taking. EXAMPLE 1. Te function T AT A 3,+5, 2,+4 specified by te input-output rule T AT A 3,+5 2,+4 T AT A 3,+5, 2,+4 () = is te cubic function wose cubing term is ( 3) 3, tat is te cubing coefficient 3 multiplied by tree copies of te input squaring term is (+5) 2, tat is te squaring coefficient +5 multiplied by two copies of te input linear tern is ( 2), tat is te linear coefficient 2 multiplied by one copy of te input constant term is +4, tat is te constant coefficient +4 multiplied by no copy of te input. In oter words, te constant term is te same as te constant coefficient. 2. Unless tere is a need for te above way of writing tings, as tere will be, for instance, wen we ave to deal wit more tan one function at a time, we will write only watever information is needed. a. Instead of writing te complete global input-output rule as above, CUBIC a,b,c,d CUBIC a,b,c,d () = a 3 + b 2 + c + d we may let te coefficients go witout saying since tey appear in te output and we will often write te simplified global input-output rule CUBIC CUBIC() = a 3 + b 2 + c + d

3 375 since tere is no loss of information in tat te coefficients, a, b, c, are still given. EXAMPLE 2. Instead of writing T AT A 3,+5, 2,+4 T AT A 3,+5, 2,+4 () = we will often write just T AT A T AT A() = because tere is no loss of information. b. Actually, since te name of te function, CUBIC, occurs in two CUBIC places, above te arrow,, and in te name of te output, CU BIC(), te simplified global input-output rule could be simplified even furter and written into eiter one of two even simpler ultra-simplified global input-output rules. Indeed: Some people write te name of te function, CUBIC, only once, in te name of te output, CUBIC() and do not write te name of te CUBIC function, CUBIC, on top of te arrow,. EXAMPLE 3. Instead of writing some people just write T AT A T AT A() = T AT A() = because tere still is no loss of information. Oter people prefer to do te opposite and write te name of te function, CUBIC, only on top of te arrow, but not in te CUBIC output. EXAMPLE 4. Instead of writing oter people prefer to write T AT A() = T AT A because tere still is no loss of information. In tis tet, toug, we will use te simplified global input-output rule mentioned above, tat is we will not write te coefficients in te name of te functions but we will write te name of te function bot above te arrow and in te name of te output as:

4 376 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS full name it cannot urt, it is not muc additional writing and it migt prevent mistakes c. On te oter and, instead of specifying a cubic function by its complete global input-output rule CUBIC a,b,c,d CUBIC a,b,c,d () = a 3 + b 2 + c + d or even by its simplified global input-output rule CUBIC CUBIC() = a 3 + b 2 + c + d we may use just its full name, tat is its name including te coefficients. CUBIC a,b,c,d because, again, tere is no loss of information in tat bot te name of te function and te coefficients, a, b, c, d, are still given. EXAMPLE 5. Instead of saying: Given te cubic function T AT A 3,+5, 2,+4, tat is T AT A 3,+5, 2,+4 te function specified by te global input-output rule T AT A 3,+5, 2,+4 () = or even instead of saying Given te cubic function T AT A T AT A 3,+5, 2,+4 specified by te global input-output rule we can just as well say, witout loss of information, Given te cubic function T AT A 3,+5, 2,+4 because tere is no loss of information. 3. Tere are tree ways to look at cubic functions and wic view we will take will depend on wic is te most convenient for us in te matematical situation we are in. a. We can look at te cubic function CUBIC a,b,c,d as a combination of te first four non-negative-eponent power functions: UNIT UNIT () = 0 IDENT IT Y IDENT IT Y () = +1 SQUARING SQUARING() = +2 CUBING CUBING() = +3 Tis is te way we will look at polynomial functions in general.

5 377 b. We can look at te cubic function CUBIC a,b,c,d as te result of adding to te quadratic function quadratic part QUADRAT IC b,c,d QUADRAT IC b,c,d () = b 2 + c + d to be called te quadratic part of te cubic function CUBIC a,b,c,d,, an a-dilation of te cubing function tat is te function CUBING CUBING() = +3 a CUBING a CUBING() = a +3 We will use tis view wen investigating and discussing local graps. c. We can look at te cubic function CUBIC a,b,c,d as te result of adding to an a-dilation of te cubic function tat is to te function CUBING CUBING() = +3 a CUBING a CUBING() = a +2 te squaring function, QUADRAT IC b,c,d, tat is a function specified by te global input-output rule QUADRAT IC b,c,d QUADRAT IC b,c,d () = b 2 + c + d EXAMPLE 6. Given te cubic function input-output rule T AT A 3,+5, 2,+4 specified by te global T AT A 3,+5, 2,+4 T AT A 3,+5, 2,+4 () = ( 3) +3 + (+5) +2 + ( 2) +1 + (+4) 0 = te function Quadratic Part of T AT A 3,+5, 2,+4 is te squaring function specified by te global input-output rule Affine Part of T AT A 3,+5, 2,+4 = (+5) +2 + ( 2) +1 + (+4) 0 = We will use tis view wen investigating and discussing te essential bounded grap. POINTWISE ANALYSIS

6 378 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS 13.1 Input-Output Pairs As wit any function specified by a global input-output rule, in order to get te output for a given input we must: i. Read and write wat te global input-output rule says, ii. Replace in te input-output rule by te given input, iii. Identify te resulting specifying prase. 1. In te case of a cubic function CUBIC a,b,c,d, tat is of a function specified by te global input-output rule CUBIC CUBIC() = a 3 + b 2 + c + d and given an input 0, in order to get te output, we proceed as follows. i. We read and write wat te input-output rule says: Te input-output rule reads: Te output of CUBIC a,b,c,d is obtained by multiplying a by 3 copies of te input, adding b multiplied by 2 copies of te input, adding c multiplied by 1 copy of te input and adding d. We write, or at least tink CUBIC CUBIC() = a + b + c + d ii. We indicate tat is about to be replaced by te given input 0 0 CUBIC CU BIC() 0 = a 3 + b 2 + c + d 0 wic gives us te following specifying-prase = a b c + d iii. Wen we ave specific values for a, b, c, d and for 0, we can ten identify te specifying-prase a b2 0 + c 0 + d to get te specific value of te output CUBIC( 0 ). EXAMPLE 7. Given te function T HORA specified by te global input-output rule T HORA T HORA() = and given te input 3, in order to get te input-output pair, we proceed as follows.

7 13.1. INPUT-OUTPUT PAIRS 379 i. We read and write wat te input-output rule says: Te input-output rule reads: Te output of T HORA is obtained by multiplying by 2 copies of te input, adding multiplied by 1 copy of te input, and adding We write T HORA T HORA() = ii. We indicate tat is about to be replaced by te given input 3 3 T HORA T HORA() wic gives us te following specifying-prase iii. We identify te specifying-prase 3 = = ( 42.17)( 3) 3 + ( 32.67)( 3) 2 + ( 52.39)( 3) + (+71.07) = = In practice we usually will not write all of tis since tere is a lot of redundant information. i. On te input side of te arrow tat represents te function, instead of writing 0 we will write just te result of replacing by 0, tat is 0 ii. On te output side of te arrow tat represents te function, instead of writing te full CUBIC a,b,c,d () 0 we will similarly write just te result of replacing by 0, tat is CUBIC a,b,c,d ( 0 ) iii. But, on te rule side will we keep te indication tat is to be replaced by 0 = a 3 + b 2 + c + c 0 because tat is were te action will be.

8 380 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS So, altogeter, given an input 0, we will write 0 CUBIC CUBIC( 0 ) = a 3 + b 2 + c + d 0 EXAMPLE 8. or 3 T HORA T HORA( 3) = = ( 42.17)( 3) 3 + ( 32.67)( 3) 2 + ( 52.39)( 3) + (+71.07) = = Altogeter, and depending on te circumstances, we can ten write: 0 CUBIC a b c 0 + d CUBIC( 0 ) = a b c 0 + d or ( 0, a b2 0 + c 0 + d) is an input-output pair for te function CUBIC. EXAMPLE 9. Altogeter, and depending on te circumstances, we can ten write: 3 T HORA or or. T HORA( 3) = ( 3, ) is an input-output pair for te function T HORA 13.2 Normalized Input-Output Rule Because, in te case of given cubic functions, we will almost always be able to carry out our investigations all te way to te quantitative stage, qualitative investigations migt seem as if tey will not make our life tat muc simpler. Still, tey will allow us make general statements about cubic functions and tat will be well wort te time.

9 13.2. NORMALIZED INPUT-OUTPUT RULE In te case of cubic functions, te feature of te input-output rule tat will be important to us will be: te sign of te cubic coefficient In fact, toug, tere is anoter feature of te input-output rule tat will turn out to be also very important to sort out cubic functions and tat is te sign of te discriminant of a function tigtly associated wit te given function but we will let tat discriminant come up naturally in te course of our investigations, tat is, in te net capter. In oter words, at tis point, we will carry out our general investigations witout concerning ourselves wit te size of te cubic coefficient a and wit eiter te sign or te size of b, c and d. Only te sign of a will matter. 2. From te qualitative viewpoint, tere will terefore be two types of cubic functions: Sign cubic coefficient TYPE + P EXAMPLE 10. Te function M Y LE specified by te global input-output rule MY LE MY LE() = is a cubic function wose input-output rule as te following feature Te cubic coefficient is Negative, So, te cubic function MY LE is of type N. 3. Given an cubic function, we will normalize it by just normalizing te cubic coefficient as follows: If te cubic coefficient is: Positive +1 Negative 1 N we will normalize it to: EXAMPLE 11. Te function LOLA specified by te global input-output rule P N normalize will be normalized to LOLA LOLA() = LOLA LOLA () = wic, owever, we will usually write =

10 382 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS 4. So, te two types of cubic functions ave te following normalized input-output rules: NOTE. of TYPE Normalized input-output rule P P P () = b 2 + c + d N N N() = 3 + b 2 + c + d It is wort keeping in mind tat te in front of 3 is better read as opposite LOCAL ANALYSIS NEAR INFINITY 13.3 Local Input-Output Rule Near Infinity As wit all functions, past, present and future, our first move in te investigation of cubic functions will usually be to get te local grap near. As was already te case wit affine functions and quadratic functions and will be te case wit all functions, given any cubic function CUBIC, it will be convenient to begin by specifying a new function wose local grap near will turn out to be te local grap near of CUBIC. 1. Given a cubic function CUBIC a,b,c,d tat is a function specified by te global input-ouput rule CUBIC a,b,c,d CUBIC a,b,c,d () = a 3 + b 2 + c + d te global input-output rule says: i. multiply te cubic coefficient a by tree copies of te input, ii. add te quadratic term, tat is te quadratic coefficient b multiplied by two copies of te input, iii. add te linear term, tat is te linear coefficient c multiplied by one copy of te input, iv. add te constant term d, tat is te constant coefficient d multiplied by no copy of te input. We will now be using: Te fact tat a bounded number multiplied by tree copies of a large-insize number gives a result tat is large-in-size, Te fact tat a bounded number multiplied by two copies of a large-insize number gives a result tat is large-in-size but not as large-in-size as

11 13.3. LOCAL I-O RULE NEAR 383 te preceding one so tat, regardless of its sign, adding tis number to te preceding one will not cange te size of te result wic will still be large-in-size, Te fact tat a bounded number multiplied by one copy of a large-in-size number gives a result tat is large-in-size but not as large-in-size as te preceding one so tat, regardless of its sign, adding tis number to te preceding one will not cange te size of te result wic will still be large-in-size, Te fact tat, regardless of its sign, adding a bounded number to a largein-size number will not cange te size of te result wic will still be large-in-size. So, wen te input is large-in-size, after we ave multiplied te cubic coefficient a by tree copies of te input, adding te quadratice term b 2, te linear term c and te constant term d, regardless of teir sign, will not cange te size of te output. In oter words, in te case of cubic functions, multiplying te cubing coefficient by tree copies of te input results in te principal term near of te output, tat is te cubic term a 3 makes up most of te output wen te input is near and te quadratic part of te output, b 2 + c + d, can be safely ignored wen te input is near. 2. So, given a cubic function CUBIC a,b,c,d, te new function tat we will introduce, P RINCIP AL T ERM of CUBIC a,b,c,d will be te function wose output is te cubing term of CUBIC a,b,c,d, tat is te function specified by te global input-output rule P P L. T ERM. of CUBIC a,b,c,d P P L. T ERM. of CUBIC a,b,c,d () = a 3 3. However, even for large inputs, te output of CUBIC is not quite equal to te output of P P L T ERM of CUBIC because, even for large inputs, wile te quadratic part of CUBIC is too small to matter ere, it is not 0. EXAMPLE is not equal to 1 3 because = 0.99 wile = 1 and so we can only write 1 3 = [...] But, in matematics, we do want to write equalities if only because tey are easier to work wit. So, in order to be able to write an equality, we will use again [...] as a sortand for someting too small to matter ere.

12 384 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS We will ten be able to write, completely trutfully, CUBIC a,b,c,d near CUBIC a,b,c,d () near = a 3 + [...] wic we will of course call te approimate local input-output rule near of te cubic function CUBIC a,b,c,d EXAMPLE 13. Given te cubic function MOAN , , tat is te function specified by te global input-ouput rule MOAN MOAN() = te function P P L. T ERM. of MOAN is te function specified by te global inputoutput rule P P L. T ERM. of MOAN P P L. T ERM. of MOAN() = and we can write trutfully near MOAN MOAN() near = [...] In oter words, we ave: THEOREM 1 (Approimation Near ). Near te output of a cubic function is approimately te same as te output of its principal term Local Grap Near Infinity Since we already investigated te local grap of te cubing function, te Approimation Near Teorem ten gives us a procedure by reduction, tat is a procedure tat reduces te current problem to a problem we ave already been able to solve, for finding te local grap near of affine functions: i. Get te approimate local input-output rule near ii. Normalize it, iii. Draw its local grap near.. EXAMPLE 14. Given te cubic function ALMA 38.38, 21.36, 45.78,+53.20, find its local grap near. ALBA 21.36, 45.78, is specified by te global input-output rule: ALMA ALMA() =

13 ) ( ) TYPES OF LOCAL GRAPHS NEAR 385 i. Te approimate local input-output rule near is: near ALMA ALMA() near = [...] ii. We normalize ALMA to ALMA near ALMA ALMA () near = +3 + [...] iii. Te local grap near of ALMA is: Output Ruler + Screen ) + Input Ruler 13.5 Types of Local Graps Near Infinity Since te local grap near of a cubic function is approimately te grap of its principal term near, we get tat te local grap near of a cubic function is approimately eiter te function CUBING or te function OP P -CUBING CUBING CUBING() = + 3 OP P -CUBING OP P -CUBING() = 3 Tey are sown in te table below as seen from two viewpoints: i. As seen from not too far, tat is we see te screen and only te part of te local grap near tat is near te transition, tat is for inputs tat are large but not tat large so tat we can still see te slope. ii. As seen from faraway. Indeed, in order to see really large inputs, we need to be faraway and te parts of te local grap near tat we see are still straigt and vertical.

14 386 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS standard notation Input-output rule From not too far From faraway Output Ruler + 0 Screen CUBING CUBING() = Input Ruler Output Ruler + 0 Screen OP P -CUBING OP P -CUBING() = Input Ruler 13.6 Local Features Near Infinity As we saw in Capter 3, Section 7, we can read te local features near off te local grap near. 1. In order to state Local Feature Teorems as simply as possible, it will be convenient to use te standard notation in wic we tink of and as being positive, tat is as + and as being negative, tat is as 2. Ten, based on te local graps in te previous section, we ave: THEOREM 2 (Local Features Near ). Given a cubic function CUBIC a,b,c,d, te local features near are: Heigt-sign CUBIC near = (Sign a, Sign a, ) Slope-sign CUBIC near = (Sign a, Sign a, ) Concavity-sign CUBIC near = (Sign a, Sign a, ) EXAMPLE 15. Given te cubic function MAON 38.82,+34.54, 40.38, 94.21, tat is te function specified by te global input-output rule MAON MAON() = we find its local features near as follows. i. Te local features near according to te Local Features Teorem are:

15 ) ( ) ( ) LOCAL FEATURES NEAR 387 Heigt-sign near = (, +) Slope-sign near = (, ) Concavity-sign near = (, +) wic, translated back into our language, gives Heigt-sign near = (, +) Slope-sign near = (, ) Concavity-sign near = (, ) ii. Sould we want to ceck tat te Local Features Teorem gave us te correct information, we would get te normalized approimate local input-output rule near near MAON MAON () near = 3 + [...] and ten te local grap near Output Ruler + Screen ) from wic we get te following local features Output Ruler Input Ruler Screen ) wic are te same as tose tat were given by te Local Features Teorem keeping in mind tat te local grap near is viewed ere by Mercator rater tan by Magellan. ) + Input Ruler LOCAL ANALYSIS NEAR A FINITE INPUT

16 388 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS addition formula for cubes initial cube 13.7 Addition Formula For Cubes In order to get te local input-output rule of a given cubic function near a given finite input 0, we need first to get an addition formula to give us ( 0 + ) 3 in terms of 0 and and wic we will call te addition formula for cubes. We prefer to establis it ere, aead of time, rater tan in te midst of te developing te local input-output rule. Tere are two approaces. 1. Te computational approac to te addition formula for cubes is to multiply tree copies of ( 0 + ): a. We begin by multiplying two copies of ( 0 + ): b. We multiply , te result of te multiplication of two copies of ( 0 + ), by a tird copy of ( 0 + ): Te grapic approac to te addition formula for cubes is to go back to te real world and to te definition of double multiplication in terms of te volume of a rectangular bo so tat ( 0 + ) 3 is te volume of a 0 + by 0 + by 0 + cube: ( 0 +) 3 Wat we will do is to start wit an 0 by 0 by 0 cube, te initial cube,

17 13.7. ADDITION FORMULA FOR CUBES 389 and ten see ow enlarging te sides of te cube square by enlarges te volume of te cube. For te sake of clarity, we will enlarge te initial cube one step at a time: i. Te tree sides of te initial cube are equal to 0 and te volume of te initital cube is terefore 3 0 : 0 slab grove rod 0 0 ii. We now enlarge te initial cube wit tree ( 0 + ) by ( 0 + ) by slabs: We glue te tree slabs to te initial cube wic, owever, leaves tree groves: iii. We fill te tree groves wit tree 0 by by rods:

18 390 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS indentation We glue te tree rods in te tree groves wic leaves an indentation in te corner: iv. We finis te enlargement of te initial cube by adding one by by cube to fill te indentation in te corner: v. Altogeter, te volume of te enlarged cube, wose side is 0 +,

19 13.7. ADDITION FORMULA FOR CUBES is terefore: Contrary to wat one migt tink at first, te grapic approac is greatly preferable because it as tree major advantages over te computational approac: i. Te terms in te sum automatically come in order of diminising sizes. Indeed, since 0 is finite and is small, all tree dimensions of te initial cube are finite, say in te ones, so 3 0 is also finite or in te ones, two dimensions of te slabs are finite, say in te ones, but te tird dimension is small, say in te tents, so te volume of te slabs is small: since 0 is in te ones and is in te tents, ten will also be in te tents, one dimension of te rods is finite, say in te ones, but te oter two dimensions are small, say in te tents, so te volume of te rods is smaller: since 0 is in te ones and is in te tents, ten will be in te unfredts, all tree dimensions of te little cube are small so tat te volume of te little cube is even smaller tan te volume of te slabs: if is in te tents, ten 3 will be in te tousandts. ii. If all we need is only a particular one of te terms, and tis will very often be te case, we can get it straigt from te picture witout aving to go troug te wole multiplication. iii. Later on, wen we sall need formulas for ( 0 + ) 4, ( 0 + ) 5, etc, not only will te lengt of te computational approac get very rapidly out of and but, as we sall see, since we will never need more tan te first two or tree terms of te result, te computational approac will also become

20 392 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS more and more inefficient. On te oter and, even toug we will not be able to draw pictures as we ave been able to do so far, one can etend te patterns we ave found so far in te grapic approac and te grapic approac will tus survive. In any case, we ave THEOREM 3 (Addition Formula For Cubes). ( 0 + ) 3 = Local Input-Output Rule Near A Finite Input Wen we investigated power functions near 0, we replaced by small in te global input-output rule and computed te size of te output using te Multiplication Teorem. Here, we will proceed as follows i. We will replace by 0 + (were is small) in te global inputoutput rule ii. We will use te Addition Formula For Squares and te Addition Formula for Cubes iii. We will use te Multiplication Teorem to compute wit In oter words, even toug we will want to be able to investigate cubic functions near any finite input 0, tis will involve only one etra step, namely te use of addition formulas. 1. More precisely, given te cubic function CUBIC a,b,c,d, tat is a function specified by te global input-output rule CUBIC CUBIC() = a 3 + b 2 + c + d and given a finite input 0, we find te local input-output rule of CUBIC near 0 just as in te case of quadratic functions: i. We localize te function CUBIC at te given input 0, tat is we count inputs from te given imput 0 instead of from te origin of te input ruler. In oter words, we use te location of in relation to te given input 0, tat is we replace in te global input-output rule by 0 +, (were is small). 0 + CUBIC a,b,c,d CUBIC a,b,c,d () 0 + = a3 + b 2 + c + d = a[ 0 + ] 3 + b[ 0 + ] 2 + c[ 0 + ] + d 0 +

21 13.8. LOCAL I-O RULE NEAR X ii. using te Addition Formula Teorem, we get [ = a ] [ + b ] [ ] + c d = a a a a 3 + b b 0 + b 2 + c 0 + c + d iii. Collecting like terms, = [ a b c 0 + d ] + [ 3a b 0 + c ] + [ 3a 0 + b ] 2 + [ a ] 3 2. Since 0 is given for te duration of te local investigation and since it is terefore tat is te actual input, we will often want to tink in terms of te local function CUBIC (0 ) tat is te function wic returns for te same output tat te global function CUBIC would return for = 0 +. Te computation we just did gives us te local input-output rule tat specifies te local function CUBIC (0 ): THEOREM 4 (Local Input-Output Rule). Given te cubic function CUBIC specified by te global input output rule CUBIC CUBIC() = a 3 + b 2 + c + d te local function CUBIC (0 ) is specified by te local input-output rule CUBIC ( 0 ) CUBIC 0 () = [ a3 0 + b c 0 + d ] + [ 3a b 0 + c ] + [ 3a 0 + b ] 2 + [ a ] 3 EXAMPLE 16. Given te cubic function BEN BEN() = , and given an input, say 0 = 4, we get te local input-output rule near 0 = 4 by localizing at 4 and ten just do te computations as follows: 4 + BEN BEN( 4 + ) = = 2 [ 4 + ] [ 4 + ] 2 5 [ 4 + ] + 7 and, using te Addition Formula Teorem wit 0 = 4, [ = 2 ( 4) 3 + 3( 4) 2 + 3( 4) 2 + 3] [ + 3 ( 4) 2 + 2( 4) + 2] 5 [ = ] [ ] [ ] = [ ]

22 394 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS local constant coefficient local linear coefficient local quadratic coefficient local cubic coefficient and, collecting like terms, = [ + 203] + [ 125] + [ + 27] 2 + [ 2] 3 So, te localization of BEN wen is 4 is te function specified by te local inputouput rule: ZEN ( 4) ZEN 4 () = Local Coefficients Near A Finite Input Wen looking only for one of te local features, instead of computing te wole local input-ouput rule, we will only compute te single term of te local input-ouput rule tat controls te local feature. At first, getting just tis one single term rater tan te wold local input-output rule will look more difficult because it cannot be done in a umdrum manner but tat will not last. More precisely, given a function CU BIC specified by te global inputoutput rule: CUBIC CUBIC() = a 3 + b 2 + c + d and wose local input-output rule near 0 is terefore CUBIC( 0 ) CUBIC( 0 ) 0 + = [ a b c 0 + d ] + [ 3a b 0 + c ] + [ 3a 0 + b ] 2 + [ a ] 3 we will say tat [ a b2 0 + c 0 + d ] is te local constant coefficient, [ 3a b 0 + c ] is te local linear coefficient, [ 3a 0 + b ] is te local quadratic coefficient. [ a ] is te local cubic coefficient. and we will now investigate ow to get just a single one of te local coefficients witout getting te wole local input-output rule. At first, getting just tis one single term rater tan te wold local input-output rule will

23 13.9. LOCAL COEFFICIENTS NEAR X look more difficult tan getting te wole local input-output rule but, wit a little bit of practice, writing less and less eac time, tis will soon get easy. We begin by writing 0 + CUBIC a,b,c,d CUBIC a,b,c,d () 0 + = a3 + b 2 + c + d = a[ 0 + ] 3 + b[ 0 + ] 2 + c[ 0 + ] + d A first approac is to work from te addition formulas but its drawback will be readily apparent in tat it requires tat te addition formulas be readily available, one way or te oter. a. To find te local constant coefficient, we proceed as follows: i. Since te Addition Formula for Cubes is: ( 0 + ) 3 = te contribution of a[ 0 + ] 3 to te local constant term will be a ii. Since te Addition Formula for Squares is: ( 0 + ) 2 = te contribution of b[ 0 + ] 2 to te local constant term will be b iii. Te contribution of c[ 0 + ] to te constant term will be c 0 0 iv. Te contribution of d to te local constant term will be d 0 Altogeter, te local constant term adds up to: a b c d 0 = a b c d 0 and terefore te local constant coefficient is: a b c 0 + d = [ a b c 0 + d ] 0 b. To find te local linear coefficient, we proceed as follows i. Since te Addition Formula for Cubes is: ( 0 + ) 3 = te contribution of a[ 0 + ] 3 to te local linear term will be a ii. Since te Addition Formula for Squares is: ( 0 + ) 2 = te contribution of b[ 0 + ] 2 to te local linear term will be b 2 0

24 396 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS local cubing coefficient iii. Te contribution of c[ 0 + ] to te local linear term will be c iv. Te contribution of d to te local linear term will be noting. Altogeter, te linear term adds up to: a b c = 3a b 0 + c and terefore te local linear coefficient is: 3a b 0 + c = [ 3a b 0 + c ] c. To find te local squaring coefficient, we proceed as follows: i. Since te Addition Formula for Cubes is: ( 0 + ) 3 = te contribution of a[ 0 + ] 3 to te local squaring term will be a ii. Since te Addition Formula for Squares is: ( 0 + ) 2 = te contribution of b[ 0 + ] 2 to te local squaring term will be b 2 iii. Te contribution of c[ 0 + ] to te local squaring term will be noting iv. Te contribution of d to te local squaring term will be noting. Altogeter, te squaring term adds up to: a b 2 = 3a b 2 and terefore te local squaring coefficient is: 3a 0 + b = [ 3a 0 + b ] 2 d. To find te local cubing coefficient we proceed as follows: i. Since te Addition Formula for Cubes is: ( 0 + ) 3 = te contribution of a[ 0 + ] 3 to te local cubing term will be a 3 ii. Since te Addition Formula for Squares is: ( 0 + ) 2 = te contribution of b[ 0 + ] 2 to te local cubing term will be noting

25 13.9. LOCAL COEFFICIENTS NEAR X iii. Te contribution of c[ 0 + ] to te local cubing term will be noting iv. Te contribution of d to te local cubing term will be noting. Altogeter, te cubing term adds up to: a 3 = a 3 = [ a ] 3 and terefore te local cubing coefficient is: a 2. Keeping in mind te pictures tat are at te basis of te addition formulas, namely te pictures of te enlarging of a square and of a cube, allows us to compute te local coefficients a lot more efficiently by dispensing wit writing te wole addition formulas and, instead, by focussing on te appropriate term by visualizing te corresponding piece(s) used to enlarge te base square or base cube. a. To find te local constant coefficient, we proceed as follows: i. We start wit te 0 term of an enlarged cube wic is te measure of te original cube tat is te volume of : tat is 3 0 wic we must multiply by te coefficient a ii. To wic we must add te 0 term of an enlarged square wic is te measure of te original square tat is te area of:

26 398 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS tat is 2 0 wic we must multiply by te coefficient v iii. To wic we must add te 0 term of an enlarged segment wic is te measure of te original segment tat is te lengt of: 0 0 tat is 1 0 wic we must multiply by te coefficient c iv. To wic, finally, we must add wic we must multiply by te coefficient d Altogeter, te 0 terms add up to: a b c d 0 = a b c d 0 = [ a b c 0 + d ] 0 and terefore te local constant coefficient is: a b c 0 + d b. To find te local linear coefficient, we proceed as follows i. We start wit te 1 term of an enlarged cube wic is te measure of te tree add-on slabs tat is wic we must multiply by te coefficient a ii. To wic we must add te 1 term of an enlarged square wic is te measure of te two add-on rectangles

27 13.9. LOCAL COEFFICIENTS NEAR X tat is wic we must multiply by te coefficient b iii. To wic we must add te 1 tern of an enlarged segment wic is te measure of te add-on tat is 0 0 wic we must multiply by te coefficient c Altogeter, te 1 terms add up to: a b c = 3a b 0 + c and terefore te local linear coefficient is: 3a b 0 + c = [ 3a b 0 + c ] c. To find te local squaring coefficient, we proceed as follows i. We start wit te 2 term of te enlarged cube wic is te measure of te net tree add-ons tat is wic we must multiply by te coefficient a

28 400 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS ii. To wic we must add te 2 term of te enlarged square wic is te measure of te add-on square tat is 2 wic we must multiply by te coefficient b Altogeter, te 2 terms add up to: a b 2 = 3a b 2 and terefore te local squaring coefficient is: 3a 0 + b = [ 3a 0 + b ] 2 EXAMPLE 17. Given te cubic function BEN BEN() = , and given te input 4, find te local squaring term. We start as usual: 4 + BEN BEN( 4 + ) = = 2 [ 4 + ] [ 4 + ] 2 5 [ 4 + ] + 7 now, instead of using te full Addition Formula Teorems, we just pick te 2 terms in eac addition formula: [ ] [ = 2 ( + 3( 4) ] [ ] 5 + were 3( 4) 2 are te tree rods in te enlarged cube and + 2 is te little square in te enlarged square. So, and, collecting like terms, [ ] [ = ] [ ] 5 + = = [ ] + [ ] + [ + 27] 2 + [ ] 3 So, te quadratic term in te local input-output rule of BEN wen is near 4 is: +27 2

29 DERIVATIVE FUNCTIONS 401 d. To find te local cubing coefficient, we proceed as follows i. We start wit te 3 term of te enlarged cube wic is te measure of te little cube add-on To wic tere is noting to add. Altogeter, te cubing term adds up to: a 3 = a 3 = [ a ] 3 and terefore te local cubing coefficient is: a Derivative Functions We saw just above tat te local constant coefficient for a cubic function CUBIC(a, b, c, d) is: a b c 0 + d It so appens, of course, tat a b2 0 + c 0 + d is te output returned by CUBIC for te input 0 : 0 CUBIC CUBIC() 0 = a 3 + b 2 + c + d 0 = a b c 0 + d In order for te oter local coefficients also to be outputs of functions, we will automatically associate wit CU BIC several functions wose purpose will indeed be, given an input 0, to output te local coefficients. We will call tese functions te Derivative functions of te function CU BIC. More precisely, since te local input-output rule of CUBIC near 0 is: CUBIC( 0 ) CUBIC( 0 ) 0 + we will say tat: = [ a b c 0 + d ] + [ 3a b 0 + c ] + [ 3a 0 + b ] 2 + [ a ] 3

30 402 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS 1 st Derivative of CUBIC CUBIC 2 nd Derivative CUBIC a,b,c,d CUBIC weird multiplier 3 rd Derivative CUBIC a,b,c,d CUBIC 0 t Derivative recursive Te 1 st Derivative of CUBIC is te function, wic we will call CUBIC, tat outputs te local linear coefficient, tat is te coefficient of 1 in te local input-output rule of CUBIC near 0. In oter words, te function 1 st Derivative CUBIC a,b,c,d is specified by te global input-output rule 1 st Derivative CUBIC 1 st Derivative CUBIC() = 3a 2 + 2b + c 2 nd Derivative CUBIC a,b,c,d is te function, wic we will call CUBIC, tat outputs twice te local squaring coefficient, tat is twice te coefficient of 2 in te local input-output rule of CUBIC near 0. In oter words, te function 2 nd Derivative CUBIC a,b,c,d is specified by te global input-output rule 2 nd Derivative CUBIC 2 nd Derivative CUBIC() = 2 (3a + b) NOTE. We will eplain below te reason for te weird multiplier twice. 3 rd Derivative CUBIC a,b,c,d is te function, wic we will call CUBIC, tat outputs trice twice te local cubing coefficient, tat is trice twice te coefficient of 3 in te local input-output rule of CUBIC near 0. In oter words, te function 3 rd Derivative CUBIC a,b,c,d is specified by te global input-output rule 3 rd Derivative CUBIC 3 rd Derivative CUBIC() = 3 2 a In fact, we will tink of te original function CUBIC as being its own 0 t Derivative because it completes te pattern inasmuc as it outputs te coefficient of 0 0 t Derivative CUBIC 0 t Derivative CUBIC() = a 3 + b 2 + c + d NOTE. Later on, Derivative functions will appear as te central concept in Differential Calculus and te reason for te weird multipliers, twice and trice twice is because tere it will make te computation of te derivative recursive: 2 nd Derivative of CUBIC = 1 st Derivative of 1 st Derivative of CUBIC 3 rd Derivative of CUBIC = 1 st Derivative of 1 st Derivative of 1 st Derivative of CUBIC Etc

31 LOCAL GRAPH NEAR X wic will make it easier to compute derivative functions since tere it involves a single procedure tat needs only be repeated. In tis tet, toug, te main use of te derivative functions will be to facilitate writing a number of similar teorems and so te weird multipliers, twice and trice twice will in fact complicate our life but, fortunately, not too muc. vanis Local Grap Near A Finite Input Te local grap of a given cubic function CUBIC a,b,c,d near a given input 0 is te grap of te local function CUBIC (0 ), tat is te grap of te function specified by te local input-output rule of CUBIC a,b,c,d near 0. We will proceed essentially in te same manner as we did wit quadratic functions ecept for one difference: ere we will usually not ave to all te way up to te cubic term. More precisely, we will usually use only te quadratic part and we will not add-on te cubic term. Te reason is quite simple: te cubic term is so small compared to te quadratic part tat it cannot cange te local grap qualitatively. Te eception is only wen, for one reason or te oter, te squaring term of te local input-output rule appens to vanis, tat is te squaring coefficient turns out to be 0. In tat case, as little as te cubic term contributes to concavity, since tat is all tere is tat accounts for concavity, we must include te cubic term in order not to lose concavity as it would terefore get us a local grap tat would be qualitatively different. Observe tat we need to use te cubic term only wen it is te squaring term tat vanises. Wen it is te constant term or te linear term tat vanises, tat does not cange te local qualitatively because, as well as contributing to concavity, te squaring term also contributes bot to te eigt and to te slope even toug te contributions are small. 1. Wen te quadratic part is complete, in order to construct te grap of CUBIC 0, i. We will get te local grap of te constant term of CUBIC 0, ii. We get will te local grap of te linear term of CUBIC 0, iii. We will construct te local grap of te affine part of CUBIC 0 by adding-on te local grap of te linear term to te local grap of te constant term iv. We will get te local grap of te squaring term of CUBIC 0 v. We will construct te local grap of CUBIC 0 by adding-on te local grap of te squaring term to te local grap of te affine part.

32 404 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS EXAMPLE 18. Given a cubic function SHIP and given tat te local input-output rule of SHIP near te input 4 is SHIP ( 4) SHIP ( 4) () = construct te local grap of SHIP near 4, tat is te grap of SHIP ( 4). i. We get te local grap of te constant term of SHIP ( 4) : +75 tat is Output Ruler Screen ( ) 4 ii. We get te local grap of te linear term of SHIP ( 4) : Input Ruler 29 tat is Output Ruler Screen 0 ( iii. We get te affine part of SHIP ( 4) by adding-on te linear term to te constant term: 4 ) Input Ruler tat is

33 LOCAL GRAPH NEAR X Output Ruler Screen ( ) 4 Input Ruler iv. We get te local grap of te squaring term of SHIP ( 4) : +3 2 tat is Output Ruler Screen 0 ( v. We get te full local grap of SHIP ( 4) by adding-on te squaring term to te affine part: 4 ) Input Ruler tat is Output Ruler Screen ( ) 4 vi. And since te grap as concavity, we are done! 2. In te case wen te squaring term appens to vanis, we will of course i. get te local grap of te constant term of CUBIC 0, ii. get te local grap of te linear term of CUBIC 0, Input Ruler

34 406 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS iii. construct te local grap of te affine part of CUBIC 0 by adding-on te local grap of te linear term to te local grap of te constant term iv. get te local grap of te cubing term of CUBIC 0 v. construct te local grap of CUBIC 0 by adding-on te local grap of te cubing term to te local grap of te affine part. 3. In te case wen it is te constant term and/or te linear term tat vanises we will of course i. get te local grap of watever term tere is, if any, ii. get te local grap of te squaring term of CUBIC 0 iii. construct te local grap of CUBIC 0 by adding-on te local grap of te squaring term to te local grap of watever term, if any, as not vanised Local Features Near A Finite Input Once we ave te local grap of te function CUBIC near 0, we can read te local features off te local grap as discussed in Capter 3. Tis is indeed a good way to proceed initially because it is visual. However, te local grap acts only as an intermediary between te local input-output rule and te local features and it is really te coefficients in te local input-output rule tat control te local features so tat, if all we want is only one particular local feature, it is very inefficient to ave to compute te wole local input-ouput rule to get te local grap. Indeed, Wy compute coefficients tat control features we don t need? As we investigate more functions, computing te wole local input-output rule is rapidly turning out to be more and more labor-intensive, if only because te addition formulas get to be longer and longer and more and more complicated but for oter reasons as well, as we will see wit rational functions. More precisely, given te cubic function CUBIC a,b,c,d, tat is a function specified by te global input-output rule CUBIC CUBIC() = a 3 + b 2 + c + d wat controls te local features of CUBIC near 0 is te quadratic part of te local input-output rule near 0 CUBIC( 0 ) CUBIC( 0 ) 0 + = [ a b c 0 + d ] + [ 3a b 0 + c ] + [ 3a 0 + b ] 2 + [...]

35 LOCAL FEATURES NEAR X and terefore it is te coefficients of te local quadratic part, tat is te tree lowest eponent terms tat eac control a particular local feature: Since te local coefficients are te output of te derivative functions of CUBIC (up to te weird multipliers but since we will only be interested in te sign of te derivative functions, te weird multipliers will not matter), we will state te resulting teorems in terms of te Derivative functions of CUBIC so as to display te overall pattern. 1. Te local constant term [ a3 0 + b c 0 + d ] gives most of te local eigt. Tis is because, as long as remains small, tat is as long as remains near 0, te oter tree terms of te local input-ouput rule, namely te linear term, te squaring term and te cubing term contribute very little eigt to te total local eigt and certainly not enoug to cange te sign of te total local eigt. Te local constant coefficient is terefore wat determines te eigt-sign near any finite input 0 wic will terefore be determined by te sign of CUBIC( 0 ) tat is by te sign of 0 t Derivative CUBIC( 0 ): THEOREM 5 (Heigt-sign Near 0 ). For any quadratic function CUBIC a,b,c : W en CUBIC( 0 ) = +, Heigt-sign CUBIC near 0 = (+, +) W en CUBIC( 0 ) =, Heigt-sign CUBIC near 0 = (, ) W en CUBIC( 0 ) = 0, Heigt-sign CUBIC near 0 is given by te sign of 1 st Derivative CUBIC( 0 ) In oter words: W en 0 t Derivative CUBIC( 0 ) = +, Heigt-sign CUBIC near 0 = (+, +) W en 0 t Derivative CUBIC( 0 ) =, Heigt-sign CUBIC near 0 = (, ) W en 0 t Derivative CUBIC( 0 ) = 0, Heigt-sign CUBIC near 0 is given by te sign of 1 st Derivative CUBIC( 0 ) 2. Te local linear term [ 3a b 0 + c ]

36 408 CHAPTER 13. CUBIC FUNCTIONS: LOCAL ANALYSIS is wat controls te local slope. Tis is because te constant term as no slope and bot te squaring term and te cubing term contribute very little to te total slope as long as remains small and certainly not enoug to cange te sign of te total slope. Te local linear coefficient is terefore wat determines te slope-sign near any finite input 0 wic will be determined by te sign of 1 st Derivative CUBIC( 0 ): THEOREM 6 (Slope-sign Near 0 ). For any cubic function CUBIC a,b,c,d : W en 1 st Derivative CUBIC( 0 ) = +, Slope-sign CUBIC near 0 = (, ) Wen 1 st Derivative CUBIC( 0 ) =, Slope-sign CUBIC near 0 = (, ) Wen 1 st Derivative CUBIC( 0 ) = 0, Slope-sign CUBIC near 0 is given by Using te standard notation, tat is tinking of and as being positive, tat is as + and as being negative, tat is as te sign of 2 nd Derivative CUBIC( 0 ) allows for a muc nicer statement of te teorem, one tat brings out te similarity of te Slope-sign Near 0 Teorem wit te Heigt-sign Near 0 Teorem: THEOREM 6 (Slope-sign Near 0 )For any cubic function CUBIC a,b,c,d : Wen 1 st Derivative CUBIC( 0 ) = +, Slope-sign CUBIC near 0 = (+, +) Wen 1 st Derivative CUBIC( 0 ) =, Slope-sign CUBIC near 0 = (, ) Wen 1 st Derivative CUBIC( 0 ) = 0, Slope-sign CUBIC near 0 is given by 3. Te local squaring term [ 3a 0 + b ] 2 te sign of 2 nd Derivative CUBIC( 0 ) is wat controls te local concavity. Tis is because bot te constant term and te linear term ave no concavity and te cubic term contributes very little to te total concavity as long as remains small and certainly not enoug to cange te sign of te total concavity.

37 LOCAL FEATURES NEAR X Te local squaring coefficient is terefore wat determines te concavity-sign near any finite input 0 wic will be determined by te sign of 2 nd Derivative CUBIC( 0 ): THEOREM 7 (Concavity-sign Near 0 ). For any cubic function CUBIC a,b,c,d : W en 2 nd Derivative CUBIC( 0 ) = +, Concavity-sign CUBIC near 0 = (, ) Wen 2 nd Derivative CUBIC( 0 ) =, Concavity-sign CUBIC near 0 = (, ) Wen 2 nd Derivative CUBIC( 0 ) = 0, Concavity-sign CUBIC near 0 is given by Using te standard notation, tat is tinking of and as being positive, tat is as + and as being negative, tat is as te sign of 3 rd Derivative CUBIC( 0 ) allows for a very striking statement of te Concavity-sign Near 0 Teorem wic brings out te similarity wit te Heigt-sign Near 0 Teorem and te Slope-sign Near 0 Teorem: THEOREM 6 (Concavity-sign Near 0 )For any cubic function CUBIC a,b,c,d : Wen 2 nd Derivative CUBIC( 0 ) = +, Concavity-sign CUBIC near 0 = (+, +) Wen 2 nd Derivative CUBIC( 0 ) =, Concavity-sign CUBIC near 0 = (, ) Wen 2 nd Derivative CUBIC( 0 ) = 0, Concavity-sign CUBIC near 0 is given by. te sign of 3 rd Derivative CUBIC( 0 )