Derivative at a point

Transcription

1 Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Derivative at a point Wat you need to know already: Te concept of liit and basic etods for coputing liits. Wat you can learn ere: How to use liits to define te concept of slope of a function at a point, also known as te derivative. Given any two points a, f a and b, f b on te grap of a function y f, we can always copute te slope of te line joining te by using te sdard slope forula: rise f b f a run b a Suc a line cuts troug te grap at tese two points, so tat, borrowing fro Latin, it is called te ant (cutting) line to te function at tose points. Also, tis slope can be interpreted as te average rate of cange of te function between a and b, since it gives us te ratio between te total cange in y and te total cange in over tat interval. So far, so good. Ecept tat in any teoretical and, ore iportly, applied probles te question of interest is not about an average slope or rate of cange, but about te a b inseous slope or rate of cange, tat is, te slope at a single point. But in tat case te slope forula is useless, since it gives te indeterinate for 0/0. Luckily, we now know ow to andle suc a for: by using liits! a, f a b, f b To ake te setting easier wile working wit liits, we reinterpret te ant line of te previous picture as te line joining a fied point to anoter one nearby. To be consistent wit te coonly used notation, I will now denote by c te coordinate of te point of interest, by an arbitrary sall nuber and by c+ te coordinate of a variable point near te one of interest, as sown in tis grap. c, f c Wit tis notation, we can state and ake sense of te following very iport definition. c c c, f c Differential Calculus Capter : Definition of derivative Section : Derivative at a point Page

2 Definition Te slope of te ant line joining a point on te grap of y f ( ) to a nearby point c, f c is given by: rise run f c f c c, f c Tis slope can be interpreted as te average rate of cange of y f ( ) between c and c. Eaple: y at (,) Knot on your finger In te definition of te slope of a ant line, te quantity is a function of te sall quantity 0, not of te coordinate c, wic we consider as a const. Tat real issue I just entioned is te proble of deterining te slope of te function at te point c, f c, witout aving to rely on an additional artificial point nearby. Tat is because te slope at te point, if we could copute it, tells us ow fast te function is canging at tat point, independently of our coice of te nearby point. If we apply te forula to tis case, we obtain te slope: f f Eaple: y at 0.5, We do te sae wit te given inforation: f f Tecnical fact Te slope of a function at a point can be interpreted as te inseous rate of cange of te function at tat point. Te first, natural attept at coputing te slope at a point would be to copute Well, we can try, since tere is no guarantee tat it will eist. But wen it does, te liit is indeed te answer we are looking for. In fact, it is te starting point for Can t we siplify tese epressions, or copute te for soe values of? te wole field of ateatics called calculus. And it is all based on tat We certainly can, especially since we are assuing 0, but for now tat is fundaental tool we call liits. not needed and our real interest is in anoter issue tat will coe up soon. So, old on to tat tougt for wen it will becoe relevant. Differential Calculus Capter : Definition of derivative Section : Derivative at a point Page 0. But tis cannot be done, since in tat case te denoinator is 0. However, so is te nuerator tere and we are dealing wit a 0/0 for! So, we can copute te liit!

3 Definition Te slope of a function y f ( ) at a point c, f ( c) is given by te liit: f ( c ) f ( c) li li 0 0 Te line wit tis slope troug te point c, f ( c ) is called te gent line to te function at c, f ( c ) and its equation is: y c f c would need to look uc closer to te point of gency and our eyes ay never be satisfied! Tat is wy we base te definition on an algebraic property tat can be cecked eactly and witout relying on visual acuity. Eaple: y at (,) Te slope of tis point of te function is just te liit of te ant slope as 0, tat is: li li 0 0 By factoring and cancelling, we obtain: li li li Te equation of te gent line is terefore given by te point-slope forula: y 0 y0 Te picture we saw before for te ant line ay now becoe soeting like tis: c, f c c But tis line does not look gent in te way I a used to; in fact it sees to cut troug te curve soewere near our point! Yes, and tat is fine, since we epect te property of being gent to old at te point, not in a wole interval around it. We ll coe back repeatedly to tis issue and, opefully it will becoe clearer. To be visually sure tat tis line is gent we Eaple: y at 0.5, Differential Calculus Capter : Definition of derivative Section : Derivative at a point Page As before, we copute te liit of te ant slope, tis tie by cobining fractions and ten factoring and cancelling: 0.5 li li li li li li We can conclude tat te equation of te gent line is: y 48

4 Since te concepts of a gent line and its slope are very iport, several tecnical words and epressions ave becoe associated wit te and we now need to becoe acquainted wit te. Knot on your finger Te slope of a function at a point is te sae as te slope of its gent line at tat point. Terefore we sall use te epressions slope of a function and slope of te gent line intercangeably. Definition Wen te liit tat defines te slope of a function y f ( ) c, f ( c ) eists, it is called te at a point derivative of y f ( ) at c. We sall see ore related jargon in te net tion. For now, notice tat te last tree boes can be suarized in one fact tat is te ost iport ting tat you need to reeber, for life, fro tis calculus course: Knot on your finger Derivative = Slope = Rate of cange We ll see ore of tis in te net tion. In particular we ll see tat te derivative of te position function is te velocity and te derivative of te velocity function is te acceleration. Before I sow you anoter eaple, ere is a suary of te steps we ave taken so far. Strategy for coputing te derivative of a function at a point In order to copute te derivative of a function y f ( ) c, f ( c ) : at a point. Copute f ( c ) first.. Insert suc epression in te forula for te ant slope: f c f c. Use an appropriate etod to evaluate te liit needed to obtain te gent slope, te derivative: f ( c ) f ( c) li 0 Eaple: f at 4, To copute te derivative (slope) of tis function at tis point we observe f 4 4, so tat: tat Differential Calculus Capter : Definition of derivative Section : Derivative at a point Page 4

5 f ( c ) f ( c) 4 Te required slope is terefore: f ( c ) f ( c) 4 li li 0 0 Tis can be coputed by rationalizing: li li Terefore, te slope is li and te gent line is y y 4 4 y 4, 4 : 4 You are referring to te so-called rules of differentiation. We sall eplore and use te soon and etensively. But first you need to be failiar wit te definition derivative and ow to use it, so as to fully undersd tis concept. Tis will greatly elp you aster later tecniques and ideas tat are based on it. But tat defining forula is coplicated! Only until you becoe used to it. In te eantie, to elp you eorize it, keep in ind tat it is te liit of a slope, tat is, te liit of rise over run! And to coplete tis tion, ere is an alternative defining forula tat soe teacers and students prefer and tat can be useful in soe situations. Tecnical fact Te following forulae provide equivalent and alternative definitions of te derivative of a function at a point c, f c : c f f c li c c f c f li c c Proof Tis looks like te etod of first principles! It is eactly wat in ig scool is often called te etod of first principles, since it is te etod based on te defining forula Wy can t we use tose quicker coputational etods we also saw in ig scool ten? Tey seeed uc sipler and ore practical. By perforing te substitution c, te sdard becoes te alternative: f ( c ) f ( c) f ( ) f ( c) c li li 0 c c Multiplying nuerator and denoinator by - provides te ond version. Differential Calculus Capter : Definition of derivative Section : Derivative at a point Page 5

6 Eaple: f at 4, If we use te alternative definition, we get, also by rationalizing: c f ( ) f (4) 4 li li li li li One advantage of tis alternative definition is tat it epresses te forula ore clearly as te liit of a slope: te nuerator is te rise and te denoinator is te run. Also, it involves te original independent variable,, witout requiring te additional quantity tat, at least at te beginning, ay be a bit of a ystery. However, one advantage of te original definition is tat it is coputationally easier to work out in ost of te situations we sall encounter. Tis is because in tat forula te denoinator consists of only. So, use wicever one you see fit and find ore cofortable, but better still, becoe proficient in using bot. Suary Te derivative of a function at a point is te slope of te gent line tere. To copute te derivative at a point, first copute te slope of a ant line joining te given point to a generic nearby point, ten take te liit as te nearby point approaces te given one. Te derivative can be interpreted also as te rate of cange of te dependent variable wit respect to te independent variable: DERIVATIVE=SLOPE=RATE OF CHANGE Coon errors to avoid Use proper algebra to copute bot te slope of te ant line and its liit as it becoes te gent line. Poor use of algebra will lead you to dead ends or errors. Differential Calculus Capter : Definition of derivative Section : Derivative at a point Page 6

7 Learning questions for Section D - Review questions:. Eplain wy a liit is needed to copute te slope of a function.. Describe te eaning of all parts of te definition of te derivative of a function at a point.. Eplain ow te sdard and alternative definitions of derivative at a point are related. Meory questions:. Wat is te general forula tat defines te derivative of a function y f at c?. In te forula for te derivative of a function, wat does te quantity represent?. Wat is te geoetrical eaning of te nuerator of te forula for te derivative of a function at a point? 4. Wat is te geoetrical eaning of te denoinator of te forula for te derivative of a function at a point? 5. If a function represents a relationsip between two practical quantities, wat does its derivative represent in practice? 6. Wat is te alternative definition of te derivative of f tat does not use? Coputation questions: For eac of te functions and points given in questions -8, copute: a) Te forula for te slope of te ant line b) Te derivative at te point c) Te equation of te gent line at te point.. y 5 at (, ) f ( ) at.. y at (,.5) 4. f at, 5. y 5 at (6, ) Differential Calculus Capter : Definition of derivative Section : Derivative at a point Page 7

8 6. y 7 at y at (4, 9) f 9. f 9 at 4 at. 0. f 5 at. Te liit presented in eac of questions -4 provides te derivative of a function tere. y f at c. Deterine te function f, te value c and te value of te derivative. li 0 /.. li 0 k. z li z z 4. e e li 5. Te function cos if 0 f 0 if 0 is continuous at =0, as you ay ave cecked in an earlier tion. Use te definition of derivative to copute its derivative at =0, tus sowing tat te derivative also eists tere. 6. A function f f li 0 f as a derivative of at. Use tis fact to copute. 7. Estiate te derivative of te function nuerical etod. f ( ) at by using te 8. Estiate te derivative of te function f at nuerical etod., by using te Teory questions:. Wat is te ain connection between liits and derivatives?. Te forula for te definition of derivative includes a fraction: wat is its geoetrical eaning? Differential Calculus Capter : Definition of derivative Section : Derivative at a point Page 8

9 . Wic procedure allows us to go fro te slope of te ant line to tat of te gent line? f c f c 4. Wat can we conclude if we find out tat li 0 for 0/0? is not of te 5. How is te line gent to te grap of f( ) at tose -values for wic its derivative is 0? 6. If a continuous function as a vertical gent line at =, wat can we say about its derivative tere? 7. If te derivative of a function eists at a certain point, wat kind of indeterinate for will be generated by te liit tat defines it tere? 8. Wic derivative property is used to define te nuber e? Proof questions:. Prove tat if te derivative of f at c eists, ten f is continuous at c.. Copute te derivative of any siple function at any point you dee suitable. Teplated questions:. Copute any of te slopes required in te Coputation questions by using te alternative definition. Wat questions do you ave for your instructor? Differential Calculus Capter : Definition of derivative Section : Derivative at a point Page 9

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