The derivative function

Transcription

1 Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative of a function Wat you can learn ere: How to define te derivative of a function, ow to compute it by using suc definition and ow to denote it according to standard practice. Despite wat many people may tink, one of te main goals of matematics is to find te simplest possible way to solve a given problems. In particular: Knot on your finger A goal common to all areas of matematics is te construction of a formula tat can be used repeatedly to solve a certain type of problem, by using te known values of te key quantities involved. Eample: Many centuries ago, matematicians in several countries, working independently of eac oter, developed several ways to solve certain equations of te type tat we now call quadratic. Most of tese metods worked only in special cases, so it was desirable to find a general and simple way to solve every quadratic equation. Wen te metod of completing te square was developed in a clear way, it was apparent tat it could be used to find a general metod of solution for quadratic equations. Tis gave rise to wat now is one of te most famous formulae in matematics, te quadratic formula. It s a single, simple formula tat allows us to solve every quadratic equation. If you are interested in seeing ow completing te square leads to te quadratic equation, ceck te note on Completing te square. Cool, but wy are you saying tis now? In te previous section we looked at ow to compute te slope of a function at a point. But wy sould we go troug all tat work just to find te slope at one point? If we could find a single way to find te derivative of a function at every point, it would be more efficient, no? But to do tat we need to develop a metod tat works eactly in te kind of general way I just mentioned. It turns out tat in our case suc general formula is very simple to obtain. All we ave to do is use te same metod we used to get te slope at a point, but leave te coordinate variable. Differential Calculus Capter : Definition of derivative Section : Te derivative function Page 1

2 Definition Given a function y f ( ), its derivative function or simply its derivative, is defined as te function: f ( ) f ( ) f( ) lim 0 Tis new function is identified verbally as f-prime. Altoug te definition is eactly te same as tat for te derivative at a point, only wit c canged to, an important difference separates te two definitions. Knot on your finger Te derivative of a function at a point provides a single number tat represents te slope, or rate of cange, of te function tere. Te derivative function is a wole new function wose output values represent te slope, or rate of cange, of te original function at every point were suc function as a slope. So, do we compute te derivative function wit te same strategy as for te derivative at a point? Yes, ecept for te use of te variable instead of te constant. Eample: y In te previous section we saw ow to compute te slope of tis function at 1, 1. But wy waste all tat energy on one point only wen wit te same effort we can get ALL slopes at all points were tere is a slope? So, let s do it more generally. We start by computing f ( ) f ( ) We ten include it in te defining formula and compute te limit: f lim lim 0 0 lim lim lim So, if we need to know te slope of tis function at any point cc,, we just need to compute c. Very efficient! But don t we get to face more complicated calculations because of using instead of a number? Yes, but, as you are probably aware we sall soon develop muc faster ways to compute tis new function, metods tat do not require te computation of complicated limits. However, you still need to understand te general definition and apply it to relatively simple functions, suc as tis one. g Eample: To determine te derivative of tis function, we apply te definition: f f g lim lim 0 0 Differential Calculus Capter : Definition of derivative Section : Te derivative function Page

3 We cange te numerator of te fraction by combining its own fractions: lim lim 1 1 lim lim Terefore te derivative of g is g So, tis is also te metod called first principles. Yes, altoug I still prefer to describe it as te metod based on te definition, bot because it is not clear wat first principles we are referring to, and to remind you tat tis metod relies on wat te derivative is, rater tan on sort cuts tat may obscure te meaning of wat we are doing. Because of te importance of derivatives, a wole set of words ave been developed to describe particular aspects of tem. We ll see most of tem in later sections, but some of tem need to be seen, used and spelled correctly rigt away. f 1 Eample: To compute te derivative of tis function, we use te definition: We ten rationalize: lim 0 f lim lim lim lim Terefore te slope of tis function at 1 it is f / 5. is f, wile at Definition If te derivative of f( ) eists at c, f c tat f( ) is differentiable at tat point., we say If f ( ) eists at every point in te domain of f( ) we simply say tat f( ) is differentiable. Te process of computing derivatives is referred to as differentiation. To engage in suc process is described by te verb to differentiate. Te property for a function to be differentiable is called differentiability. By using tese words, we can state in a sort way a very important property of derivatives tat will be used very often in later sections. Differential Calculus Capter : Definition of derivative Section : Te derivative function Page

4 Tecnical fact If a function is differentiable at a point, ten it is continuous tere: Differentiability implies continuity It may be stated simply, but I am not sure of wat it means! To make sense of tis sort sentence, remember tat if te derivative at a point eists (differentiability), ten te function as a slope tere (slope and derivative are te same ting) and ence a tangent line. But in tat case te function cannot ave a discontinuity (a break) oterwise ow could we construct suc tangent line? An asymptote or a removable discontinuity would not allow us to attac te tangent line, wile a jump would require a sudden step, not a gradual increase. We can also clarify tis concept by using te defining formula for te derivative, as follows. f ( c ) f ( c) In te indeterminate form f( c) lim te denominator 0 approaces 0, so te numerator must also approac 0, or we would get a #/0 form tat leads to infinity, not to a finite value. Tis means tat: lim f ( c ) f ( c) 0 lim f ( c ) f ( c) 0 0 If we let c tis becomes lim f f ( c), wic is te condition c for continuity. Terefore, te presence of differentiability implies te presence of continuity. Please notice tat tis may be a convincing algebraic argument, but it ides some subtle tecnical details tat matematicians ave cecked, so you sould not take it as a proof, but you can trust its substance. On te oter and, isn t tis fact rater redundant? Isn t it also true tat every continuous function is differentiable? Absolutely NOT! Tis is an easy mistake to make, to te point tat some early matematicians assumed it, but it is not true. Eample: f Te grap of te absolute value function is sown ere and you can see tat it consists of two alf-lines joining at te origin. Te grap as no breaks and you can ceck formally tat it is continuous everywere, including at te origin. But you can clearly see (and maybe ceck formally) tat it as no clearly defined slope at te origin; ence it is not defined tere. Tecnical fact A function may be continuous at a point witout being differentiable tere. Continuity does NOT imply differentiability In te learning activities you will ave a cance to discover oter simple functions tat are continuous, but not differentiable at all points. On a different topic, wy is te prime notation used for derivatives? Besides its visible connection to te function from wic it comes, it is sort and it can be used and generalized effectively wen we start playing wit more advanced concepts, suc as iger derivatives. However, tere are oter notations for te derivative, tat can be used, are used and, in certain situations, are more effective tat te standard prime notation. Here are te most common and used ones. Differential Calculus Capter : Definition of derivative Section : Te derivative function Page 4

5 Definition Te derivative of a differentiable function y f ( ) may be denoted by using: Te functional notation: y f ( ) Te sort notation: Te fractional notation: y dy d or df d d Te operator notation: f ( ) d Te fluion notation: Tat is cool, but confusing! Wo came up wit tem and wy? Some famous and important matematicians came up wit tem and usually for some very good reason tat we sall not discuss ere, altoug you will see some of tose reasons soon. And just to drop some names: Te functional and sort notations are due to J.L. Lagrange. Te fractional notation is dues to G. Leibniz and is in fact known as te Leibniz notation. y Te operator notation is a variant of one tat goes back to J. Bernoulli. And te fluion notation was invented by te man imself, Isaac Newton, but is now rarely used, togeter wit te fluions terminology. You may find it in pysics wen te independent variable is time. f 1 Eample: f We ave seen tat for tis function functional notation, but we can also write: Sort notation: Leibniz notation: y 1 dy df d d 1 Operator notation: Newton s notation: d f d y Tis is in te We sall stop ere for now: work on understanding te concept and te associated terminology and notation. You can also try a few more computations by using te definition, but we sall soon develop and use muc better computational metods. Differential Calculus Capter : Definition of derivative Section : Te derivative function Page 5

6 Te derivative of a function corresponding values of. f is a new function, denoted by f Summary, wose output values represent te slope, or rate of cange, of te original function at te f ( ) f ( ) Te defining formula, f( ) lim is a very important formula tat must be understood and memorized. 0 Common errors to avoid A key step in te computation of te derivative function is to correctly obtain an epression for f ( ). If tis is not done rigt, all oter subsequent steps, and te conclusion, may be incorrect. Learning questions for Section D - Review questions: 1. Eplain te relation between a function and its derivative.. Eplain wat differentiability implies continuity means.. Present a reasonable argument of wy differentiability implies continuity, but not te reverse. Memory questions: 1. In te standard formula for te derivative of a function, wat does te quantity represent?. Wic verb describes te activity of computing derivatives?. Wat is te correct Leibniz notation for te derivative of a function y f ( ) 4. Wat is te sort notation for te derivative of a function y f ( )?? 5. Wat is te operator notation for te derivative of a function y f ( ) 6. If a function is differentiable, is it also continuous? 7. If a function is continuous, is it also differentiable?? 8. Wic tecnical noun identifies te process of computing derivatives? Make sure to spell it correctly! Differential Calculus Capter : Definition of derivative Section : Te derivative function Page 6

7 Computation questions: Use te defining formula to compute te derivative of te functions presented in questions f ( ) f ( ) 1 7 f ( ) y 9. f 10. f f ( ) f( ) 6 f ( ) f 5. 7 y 1 6. y 1 7. f f 1. y 5 1. f y y 16. y f 1. f. f 1 1 Differential Calculus Capter : Definition of derivative Section : Te derivative function Page 7

8 In questions -5, te grap of a function y f is given. Sketc a grap of y f on te same coordinate frame, and eplain ow you obtained suc grap. Y Y Y X X X Teory questions: 1. Is it possible for a function to be continuous and for its derivative to ave a jump?. Is it possible for a continuous function to ave a vertical tangent line?. Wat do te -intercepts of te derivative represent for te original function? 4. Does te derivative function represent te tangent line to a function? 5. Can two different functions ave te same derivative? 6. If a function as a jump at =c, wat can we say about its derivative tere? 7. Wat features of te grap of f ( ) 5 justify te fact tat te grap of f '( ) is entirely in te first quadrant? 8. Wat can we say about te derivative of a function at a point were its grap is going down? 9. Are tere more functions tat are continuous at = or more functions tat are differentiable tere? 10. If a function as a removable discontinuity at =c, is it differentiable at =c? Differential Calculus Capter : Definition of derivative Section : Te derivative function Page 8

9 Proof questions: /5 1. Use te definition of derivative to sow tat te derivative of f does not eist at 0. Templated questions: 1. Use te defining limit, but no differentiation rules, to compute te derivative of any simple function.. Determine te domain of any derivative you compute and compare it to te domain of te original function. Wat questions do you ave for your instructor? Differential Calculus Capter : Definition of derivative Section : Te derivative function Page 9

10 Differential Calculus Capter : Definition of derivative Section : Te derivative function Page 10