5.1 The derivative or the gradient of a curve. Definition and finding the gradient from first principles

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1 Capter 5: Dierentiation In tis capter, we will study: 51 e derivative or te gradient o a curve Deinition and inding te gradient ro irst principles 5 Forulas or derivatives 5 e equation o te tangent line and te equation o te noral line 54 Increasing and decreasing unctions Stationary points 55 Maxiu and iniu points Metods to ind axiu and iniu points 56 e cain rule 57 e second derivative 58 Applications o dierentiation 51 e derivative or te gradient o a curve Deinition and inding te gradient ro irst principles Consider a generic unction ( x ) We want to calculate te equation o te tangent line to ( x )) at soe generic point on te grap P a, ( a ), suc as in te ollowing igure: G (to te grap o Figure 1: A generic tangent line to First, ow do we deine te tangent line to G at P a, ( a ) G at Pa, ( a )? is is deined as te line wic intersect te curve only at P (i we look close enoug to P), and wic ollows te direction o te curve near P In order to ind te equation o tis tangent line, we need to ind te slope o tis line: know two points on te tangent line, we are orced to consider: Since we do not (1) ( a ) ( a) as 0 or, in sort: () ( a ) ( a) li 0 Geoetrically tis represents calculating te slope o te tangent line as te liit value o te slope o secant lines, wit te point Qa, ( a ) approacing te point P a, ( a ) igure:, as we can see in te ollowing 1

2 Figure : Calculating as li 0 ( a ) ( a) Note ow te secant lines approac te tangent line as 0 Exaple: Find te equation o te tangent line to te grap o ( x) x at te point P1,1 : Using (), we ave : 1 (1) 1 1 li li li li Note tat we ad to sipliy one power o (in eac nuerator and denoinator) to be able to calculate tis liit is is typical in all liits o type (tereore in all calculations o In our case, since te graps below ), ten te equation o te tangent line is: y 1 x 1 y x 1 is is conired by Figure : angent line to ( x) x at te point P1,1 Note: e slope o te tangent line to te curve at a, and it is denoted by: () G at, ( ) P a a is called te derivative o ( x ) at a, or te gradient o dy ( a ) ( a) '( a) ( a) ( a) li dx 0 It represents te steepness o te grap o ( x ) at a (a steep grap will ave a ig value o te slope o te tangent line, and tereore a ig derivative, eiter positive or negative) More generally, we can calculate '( x ) at any point x using te orula: (4) ( x ) ( x) '( x) li 0

3 Forula (4) gives te slope o te tangent line at any point Px, ( x ) on G 5 Forulas or derivatives Using te orula (4) above, we can deduce orulas to calculate derivatives o polynoials, rational and irrational unctions easily We tereore deduce: ( x ) '( x ) n x nx n 1 c 0 x 1 x x ( x) g( x) '( x) g '( x) ( x) g( x) '( x) g '( x) k ( x ) k '( x ) able 51: Forulas or derivatives n n 1 Note tat te orula: (1) Exaple: Calculate te ollowing derivatives: 1 x x 6 ' ' x ' ' x x ' nx olds even or n Z, nqand even or nr Q 7 x ' 4 x x' and do probles 4,, 7 and 8 ro Exercise set 5B

4 5 e equation o te tangent line and te equation o te noral line : Since '( a) represent te slope o te tangent line to line to G at Pa, ( a) is (1) y ( a) '( a) x a G at Pa, ( a ), ten te equation o te tangent We deine te noral line to troug Pa, ( a ) G at, ( ) P a a as te line perpendicular to te tangent line, and wic goes Since te noral is perpendicular to te tangent, we ave tat: () 1 1 n ( a), and tereore te equation o te noral line to ( a) '( a) G at Pa, ( a) is: 1 '( a) () y ( a ) x a Exaple: Fro Exercise Set 5C: probles 6, 1 and ro Exercise Set 5D: probles 9, 19 and 0 54 Increasing and decreasing unctions Stationary points We ave: eore 1: a) Consider : I R suc tat '( x) 0 or any x I en ( x ) is strictly increasing on I (tat is, i x x x x or any x, x I ) ; b) Consider : I R suc tat '( x) 0 or any x I en ( x ) is strictly decreasing on I (tat is, i x x x x or any x, x I ) ; Deinition: I '( ) 0 In sort : a, ten te point, ( ) ' 0 -increasing ' 0 -decreasing ' 0 -stationary P a a is called a stationary point or ( x ) is result (and deinition) are o paraount iportance or te derivative and its applications, and it will be used in te next section as well Exaple: 4

5 Fro Cabridge AS Level Mateatics, June 01, Q Maxiu and iniu points Metods to ind axiu and iniu points Deinition: Consider : I R en: R is called a global iniu point or ( x) on I i ( x) or any x I ; R is called a local iniu point or ( x) on I i tere exist a b, a, b I suc tat ( x) or any x a, b (tat is, is a iniu point or ( x) only locally, around te point); M R is called a global axiu point or ( x) on I i ( x) M or any x I ; M R is called a local axiu point or ( x) on I i tere exist a b, a, b I suc tat ( x) M or any x a, b (tat is, M is a axiu point or ( x) only locally, around te point); eore 1: Consider : a, b R suc tat - continuous on, a global iniu on ab, ab en ( x) as bot a global axiu and (In words: A continuous unction on a closed interval adits bot a global axiu and a global iniu tere, oterwise suc extrea points are not guaranteed) In wat ollows, we will be concerned alost exclusively in inding te local ax / in points or a unction eore : I : I R '( c) 0, and, ( ) c c is a local iniu or local axiu point c I, suc tat '( c ) exists, ten So, i a point is a local ax/ in inside te interval, ten it is stationary However, we would like to ave suc a result in te reversed way (tat is, does it ollow tat i '( c) 0, wit c I and i '( ) ( x )? ) c exists, does it ollow tat, ( ) Suc a result does not old, as we can easily see ro extreu c c is a local iniu or local axiu point or :, R, ( x) x wit '(0) 0, but 0 is not a local In order to ind te local extrea or a unction, we typically use te ollowing teore: 5

6 eore : Consider : I R and c I, suc tat '( ) Local axiu i '( ) Local iniu i '( ) x canges it sign at c, ( c ) en c, ( c) c c ; x canges its sign ro positive to negative at, ( ) c c ; x canges its sign ro negative to positive at, ( ) is a point o: eore above is te ost iportant teore in te entire capter, as it gives us te etod to ind te local extrea or a unction We reduce te inding o tese extrea to deterining te sign o '( x ) o deterine te sign o '( x ), we build a sign table or '( x ) (we ind all zeros o '( x ), ten we deterine te sign o '( x) between any successive zeros by using a test point in te corresponding interval) Exaple: Fro Exercise Set 5E: probles 1,,4,5,1,14,15 and e cain rule 1) Wat is te derivative o x and o g( x) x 1 ( ) x 1? i( x) x 1? ) Wat o te derivative o ( x) x 1 5, o 10 ) Can we calculate te derivative o ( x) x 1, o 1 gx ( ) x 1 and o ( x) x 1? I or exercises 1) and ) above we can calculate te derivatives (altoug it becoes increasingly ore tedious), or exercise we need to use te deinition o te derivative and lengty algebra to calculate it A ore direct (and uc quicker ) way to calculate te derivatives o all tese unctions is to use te cain rule e cain rule states: ( u( x)) ' '( u( x)) u '( x) (1), or, in better notation: () ( ) d u( x) d u( x) du( x) dx d u x dx In exercises, we will use orula (1), wic reads : te derivative o a coposed unction is te derivative o te unction outside ( ) wit respect to its entire arguent (u ) ultiplied wit te derivative o te unction inside ( u ) Applying orula (1) to te power rule learned in 5 produces te generalized power rule: n n () 1 u( x) ' n u( x) u '( x) n n 1 We see tat te dierence ro te siple power rule : x ' nx is a siple extra ultiplication wit u'( x ) 6

7 Exercises: Using orula () or te probles in 1), ) and ) above gives: '( x) x 1 ' x 1 x 1 ' 4 x 1 wic coincides wit te result you ind ater expanding ( x) x 1 4x 4x 1, and ten calculating '( x ) Siilarly: g '( x) x 1 ' x 1 6 x 1 And: i '( x) x 1 ' 10 x 1 0 x ' 1 ' x1 x1 and Also: x x 1 1 ' x 1 ' x 1 x1 x1 and 1 '( x) x 1 ' x 1 ' x 1 x 1 Extra Practice: Look at Exaple 519 in te textbook 4 Note also tat, i V V ( r), say or a spere o radius r: V r, x ten dv dr 4 d r dr 4 r 4r, owever, i r r t and, knowing tat: V 4 r, we want to calculate: dv dt, we can use (according to te cain rule orula (1)): dv dv dr dr dt dr dt dt 4 r in our case Extra Practice: Look at Exaple 50 in te textbook ry also te ollowing past papers probles: 7

8 1 [Cabridge Ceckpoint Exa June 01 paper / Q 7] [Cabridge Ceckpoint Exa June 01 paper / Q ] Look also in te Practice Book (pages ) and do probles as needed, and in te past paper questions in te Practice Book (pages 1 7) and do all probles 57 e second derivative: For a given unction ( x ), we can calculate ' ( x ), and aterwards we can calculate d d d ( x) d ( x) ' ( x) '' ( x) soeties denoted, dx dx dx dx wic is called te second derivative o ( x ) '' ( x ) gives te instantaneous rate o cange o ' ( x ), o te slope o te tangent line to te grap o ' ( x ) is is a direct result ro te deinition o ' ( x ), in wic we replace ( x) wit ' ( x ) In a siilar extension, we see tat i ''( x) 0 on soe interval I, ten ' ( x ) is increasing on I, and i ''( x) 0 on I, ten ' ( x ) is decreasing on I Using te result stated above or te grap o te unction ( x ), we observe tat : 8

9 eore 1: a) I ''( x) 0 on an interval I, ten ( x ) is concave up unction, ' ( x ) is increasing everywere, and conversely) b) I ''( x) 0 on an interval I, ten ( x ) is concave down unction, ' ( x ) is decreasing everywere, and conversely) on I (Note tat, indeed, or a concave up on I (Note tat or a concave down Deinition 1: I '' ( x ) canges its sign at a point c I, ten c is called an inlection point or ( x ) ereore, inlection points are points were ( x ) canges its concavity Note: Sign tables or '' ( x ) deterine te concavity o te unction ( x ), tese are called concavity tables or ( x ) In conclusion, onotonicity tables or ( x ) deterine ( x ) s onotonicity (were te unction increases and decreases) and iplicitly te points o iniu and axiu or ( x ), wile concavity tables or ( x) deterine its concavity and iplicitly, its inlection points Exaple 1: Deterine te concavity and te inlection points o : R R, ( x) x x However, i a point c is a stationary point, we can use te second derivative to deterine te nature o te stationary point (local in or local ax), as stated by te next teore: eore (te second derivative test): Consider : I R and ci wit ' ( c) 0 en: a) I '' ( c) 0 (tereore ( ) is b) I '' ( c) 0 (tereore ( ) is x near c) ten, ( ) x near c) ten, ( ) c c is a point o local axiu or ( x ); c c is a point o local iniu or ( x ); c) I '' ( c) 0, ten te nature o te critical point cannot be deterined ro tis criterion, tat is c, ( c) ay be a local extreu or ( x) or it ay not be Note: is teore represents te second etod to obtain te points o in / ax or a unction ( x ) It works or establising te nature o all stationary points or ( x ), i '' ( x) 0 In all oter cases (i '( c) 0 or '' ( x) 0 or bot), te nature o te critical points needs to be establised using eore in 55 (called te irst derivative test) 9

10 Exaple: Consider ( x) x x 1x i) Find all stationary points o ( x ) ; ii) Find '' ( x ) at te stationary points and deterine ten te nature o te stationary points; iii) Wic etod do you preer or tis proble to deterine te in / ax points or ( x ): te irst derivative test or te second derivative test? Exercises: 1 Fro exercise set 5F, do probles 1 i), iii), iv) and vi, iv), v), vi) and ix),, 5 and 7 58 Applications o dierentiation: [ Cabridge Mateatics 9709/11/05/1/09] In tis section we look at practical applications in wic te goal is to ind te iniu or axiu value o a unction o practical interest, wic is called te objective unction It is iportant to ollow a ew key steps wen solving tis type o probles: 1 Understand te proble: read te proble careully until it is clearly understood Deterine te objective unction, and te variables tat tis unction depends on Draw a diagra: In ost probles it is useul to draw a diagra (i not already provided) to deterine te given and te required quantities and te relation between tese Introduce notation: Assign a sybol or te objective unction and or oter unknown quantities in te diagra (or exaple, call tese xy,,) Label te diagra wit tese sybols 4 Express in ters o tese sybols: xy,, Our goal is to express in ters o one unknown variable only 5 Use te diagra (or equations/relations stated in te proble) to ind relations between te unknown independent variables suc tat depends on one independent (unknown) variable only Set ten ( x) 10

11 6 Use a sign table or '( x ) (as studied in section 55) to ind te iniu or te axiu (as required) or ( x ) Exaple (Exaple 516 o te textbook): Solution: Draw te diagra: Note tat: A x y, is te unction tat we want to axiize, and tat: P x y 5 y 5 x A x 5 x Now, we want to axiize x 5 x Using a sign table o '( x ), we see tat A ( x) as a global axiu value or 5 x =15 eters 4 Extra practice: Read exaple 517, and ro exercise set 5G do probles, 5 6, 10, 11 and 1 Use always te sign o '( x ) (even i ''( x ) is indicated in te proble ) to deterine te nature o te stationary points 11