136 Calculus and Structures

Transcription

1 6 Calculus and Structures

2 CHAPTER 9 THE DERIVATIVE Calculus and Structures 7 Coprigt

3 Capter 9 THE DERIVATIVE 9. THE ORIGINS OF CALCULUS All great discoveries ave teir origins wit an important problem tat needs to be solved. Te problem generall as a long istor wit oter great scientists contributing to its solution. Te istor of calculus perfectl illustrates tis dnamic. Copernicus (47-54) upset religious doctrine b suggesting tat te planets revolved about te Sun rater tan te Eart as center. Galileo (564-64) found supporting evidence for tis potesis b building te first telescope and observing te moons of Saturn revolving around Saturn like a miniature solar sstem. Galileo also studied te action of gravit on balls rolling down inclined planes. Kepler (57-6) inerited te observational data of te great astronomer, Tco Brae (546-6), and in is first law potesized tat te planets traveled in ellipses around te sun. Kepler also found b empirical metods is second law tat as te planets revolved around te sun te swept out equal areas in equal times, and is most important tird law tat te square of te periods of te planets around te sun equaled te cube of teir mean distances from te sun as measured in units b wic te distance of te eart to te sun is one unit. Tis is were Isaac Newton (64-77) entered te picture. Newton was one of te co-discoverers of calculus along wit Wilelm Leibnitz.(646-76) Wen a plague raged troug England, Newton returned from Cambridge to te seaside village of is birt to avoid te ravages of te plague. Tere e spent is nd troug 4 t ears wit a great deal of time to tink about tings, and e ad te most fruitful time of is life in terms of scientific discover. He wised to find a teor b wic e could predict te eact motion of te planets as te revolved around te sun, based on te work of te great scientists tat came before im. To do tis e first potesized is tree laws of motion, most notabl is famous law, F = ma, to make te bold conjecture tat it was an inverse square gravitational force field between sun and planets tat propelled te movement of te planets about te sun. He did tis b working backward in is teor from Kepler s tird law to discover te universal law of gravitation, a single law b wic objects developed forces of attraction between eac oter due to a gravitational field. As a result, from Newton s Laws one is able to derive Kepler s tree laws b pure tougt not merel observation.. To solve is equations of motion, Newton ad to invent te subject of calculus. It is also interesting tat since oter scientists were not knowledgeable about te new calculus, Newton wrote is great book, te Principia, in te form of a book of geometr downplaing is calculus discoveries. As important as Newton s teories were to science, te were even more important to pilosop. Tis was te first time tat science was found to ave precise predictive power rater tan merel observational and model building abilities. After Newton, practitioners of ever field, be it cemistr, geolog, pscolog, social science, etc. sougt te simple underling laws of teir fields of stud. In our stud of calculus and structures, we ave seen ow te forces of gravit act on te beam and ow areas and slopes are related to te computation of sear force and bending moment. We ave also found tat our inabilit to compute te slope of tangent lines to curves and areas under curves prevented us from evaluating beams beond a certain level of compleit. In tis capter we begin to rectif tis situation b finding a metod to determine te slope of te tangent line to te grap of an function. To accomplis tis we will introduce a quantit known as te derivative wic will 8 Calculus and Structures

4 { { { Section 9. be te slope of te line tangent to te grap of a function. Te stud of derivatives is known as te differential calculus. Capters 5 and 6 will ten be devoted to a sortcut metod to compute integrals and using tem to find te signed area under curves b wat is known of as te integral calculus. We will ten return to using te calculus to evaluate te bending moment of a beam. In te process we will sow ow te effect of gravitational forces on our structures and te computation of areas lead naturall to te derivation of internal forces and moments just as te did for Newton in is discover of te motions of te planets. 9. INTRO DUCTION We ave seen in Capter tat te line is te simplest matematical element. As long as te grap of a function is smoot (no cusps or sarp edges) at least locall in te vicinit of a point te grap of te function is ver muc like a line. Matematicians ave made good use of tis propert of functions to eplore teir natures b relating te functions locall to lines. Te slope of te tangent line to te grap of a function turns out to be te ke to understanding comple functions. In tis capter we will stud ow to determine te slope of te tangent line to a curve. 9. THE DERIVATIVE: A NEW APPROACH If ou look at te grap of a smoot function, = f(), suc as te one in Figure, (no discontinuities or cusps ( "sarp edges"), ou will notice tat in te vicinit of an point te curve looks like a straigt line. = f () curve () = f ( ) line = f ( ) = Fig. Tat straigt line is called te tangent line to te curve at o. Using te point slope metod, its equation is, m( ) were f ) () line ( Since we will onl be considering points near, I will let = were is a small number and rewrite Eq. as, f ) m line ( = + Calculus and Structures 9

5 Capter 9 THE DERIVATIVE And since te value of curve is ver close to line for small, it follows tat, curve line o() curve or, f ) f ( ) f ( ) m o( ) () ( Te function o(), called te "little o function", is te small difference between curve and line near. In tis equation f( ) is usuall a big number, m is a small number (because is small), and o() is a,"ver small" number. In fact, te o() is so small tat, i o( ) () In oter words, in a race to, o() alwas beats and functions wit tis propert will be said to be "muc smaller" tan. Clearl and are o() functions. Oter o() functions are less obvious and will be discussed in Section 9.5. Te o() functions ave te following algebraic properties: i. o ) o ( ) o( ); ( ii. o ) o ( ) o( ); ( iii. ko ( ) o ( ) were k is a constant. Te proof is obvious (w?), In Eq. te slope of te tangent line to te curve, m, is called te derivative of f() at te point o and denoted b f '( o ) or sometimes df ( ) d or simpl as. In Capter 4 d d we saw tat te tangent line to a curve = f() at some point represents te rate of cange of wit respect to at. So derivatives are now also identified not onl wit slopes but also wit "rate of cange." Our job will be to determine m for a variet of functions f(). Te work will be enormousl simplified b setting o() to zero and ever time a term comes up tat is "muc smaller" tan, i.e., passes te test of Eq. (for eample,,, etc.) I also einate tat term from our equation. Terefore, I will let o() = and rewrite Eq. as, f( o + ) = f( o ) + m (4) 4 Calculus and Structures

6 Section 9. I refer to Eq. 4 as te Derivative Macine since all derivatives can be derived from tis equation b solving for m. In fact we will compute derivatives using te Derivative Macine b a si step process:. Compute f ) (. Compute f ( ). Insert tese in te Derivative Macine 4. Simplif tis epression algebraicall 5. Set to zero all terms tat are o() 6. Solve algebraicall for m To justif te eination of o() in te Derivative Macine return to Eq. witout setting o() to. Rewriting Eq., m f ( ) f ( ) o( ).We ave seen in Section 4. tat te it of te first term leads to te derivative, f ( ) wile te it of te second term vanises b definition so tat m approaces f ( ) as approaces, or ' m l f ( ) f ( ). 9.4 SOME APPLICATIONS OF THE DERIVATIVE MACHINE. Eample : Compute te derivative of f ( ). Let us appl te si step process... f ( ) f ( ) ( ). ( + ) = + m (inserting in te Derivative Macine) 4. m or + = m 5. = m (einating wic is an o() function) 6. m = (dividing bot sides b ) Calculus and Structures 4

7 Capter 9 THE DERIVATIVE Terefore f () = or was an arbitrar value of. d d were I ave replaced b since Eample : Compute te derivative for f ( ) Appling te si step process:. f ( ). f ( ). m 4. = + + m ( + ) (multipling bot sides of te equation b ( + ) 5. = + m o (simplifing and einating m ) 6. m d Terefore, f '( ) d Eample : Compute te derivative of f() = c (te constant function). f ( ) c.. f ) c (. c=c+m 4. = m 5. No o() functions to einate 6. m = 4 Calculus and Structures

8 Section 9.4 dc Terefore te derivative of te constant function is, i.e.,.. d Problem : Appl te Derivative Macine and te si step process to find te derivative of : d a) f ( ) (sow tat ) ; b) f()= + 4; c) d d ) d f ( ) (sow tat Problem : Appl te Derivative Macine to find te derivatives of te following functions: a) f() = b) f() = 5 + c) f() = 7 for all values of (i.e., te constant function) d) f() = 9.5 SOME UNUSUAL o() FUNCTIONS To find te derivative of sin, cos, e, and ln b te derivative macine requires us to recognize o() functions tat are not as obvious as or. For eample sow tat sin, cos and e o( ) I are o() functions. To do tis we must sow tat.. i suggest tat ou do tis b setting up a table in wic, as gets smaller, and smaller sin and e srink towards. Use our calculator to complete Table to sow tis. Table.... sin cos e Problem : Wic of te following functions are o() functions, i.e., wic ones pass te test of Eq.? a) 5 b) sin c) sin d) + e) sin Calculus and Structures 4

9 Capter 9 THE DERIVATIVE 9.6 APPLICATION OF THE UNUSUAL () FUNCTIONS TO FINDING THE DERIVATIVE F cos, e, and In. Eample 4: Making use of te fact tat sin, and cos - are o() functions, d cos sow tat sin. d. f ( ) cos. f ) cos( ) (. cos( + ) = cos o + m (inserting and into te Derivative Macine) 4. cos o cos - sin o sin = cos o + m (using a trig identit) or, cos o (cos - + ) - sin o (sin - + ) = cos o + m or, cos o (cos -) + cos o - sin o (sin - ) - sin o = cos o + m 5. cos o - sin = cos o + m or - sin o = m (setting o() functions to and cancelling cos o.) 6. m = - sin o Terefore, d cos sin d Eample 5: Sow tat de d. f ( ) e. f ( ) e. e e m 4. e e e m 5. e ( e ) e 6. e ( e ) e e e m e m e m (setting o() function to and doing some algebra) e m Terefore, de d e 44 Calculus and Structures

10 Section 9.6 Eample 6: Sow tat d ln d. f ( ) ln. f ( ) ln( ). ln( ) ln m ln( ) (ln m) 4. e e ln( m e ) e ln e m ( ) ( e m m ) m ( ) ( e m ) ( m ) 5. m (setting o() function to and doing some algebra) 6. m Terefore, d ln d 9.7 TANGENTS AND DERIVATIVES Consider te line, = m + b as it crases into te -ais at angle in Fig.. Te slope of te line is, rise m tan. run Now consider te tangent line to te grap of a function = f() at te point =. in Fig.. If we consider small values of ten, tan = m + b run rise Fig. Calculus and Structures 45

11 Capter 9 THE DERIVATIVE You can ceck tis approimation b taking our calculator and inputting a small angle (in radians) into te tangent function. Notice tat tan and ave nearl te same values. Since slopes are identified wit derivatives, d d (5) Wen beams ave small deflections, will be small and Eq. 5 will old true. Tis will pla a role in Capter 9 on te deflection of beams. 9.8 DERIVATIVE AS A RATE OF CHANGE Consider te Derivative macine: f ( ) f ( ) f '( ) o ( ) (6) were = f(), d f '( ), f ( ) f ( ), and d Rewriting Eq.6 in terms of tese new smbols, d d o() or d d for small. But we know from Lesson 4 tat is te average rate of cange of wit respect to over d te interval [, ] wile is te instantaneous rate of cange of wit respect to at d =. Te derivative can now be interpreted as eiter te slope of a tangent line to a curve at an point or as te instantaneous rate of cange of a function at a point in its domain. 9.9 HIGHER DERIVATIVES In Capter 4 we looked at te table of values for function, = f(), at integer values of and computed. We repeat te table for =. but also compute te cange of te cange of or ( ) and te tird cange of,. 46 Calculus and Structures

12 Section 9.9 Table Since, is an approimation to te rate of cange or derivative at eac value of. Likewise we find tat is an approimation to te rate of cange of te rate of cange. Tis d is also called te derivative of te derivative or te nd d( ) d derivative and denoted b d. d d Likewise, d.. d is an approimation to te rd derivative, denoted b d d d Since te derivative is, ten, and. B looking at tis table d d d ou will see tat tese are indeed good approimations. Now consider te table of =. Table Differentiating = we get, d d, 6 d d 4 d d, 6, 4 d d Calculus and Structures 47

13 Capter 9 THE DERIVATIVE 9. WHAT DERIVATIVES TELL US ABOUT A FUNCTION As we observed in Capter 4, positive values of indicate were a function is increasing wile negative values indicate were te function is decreasing. On te oter and, were increases, te curve is concave up, and concave down were decreases. In oter words, a curve is concave up wen is positive and concave down wen is negative. Table sows tat = decreases to = after wic it increases, wile being concave up for all. Table sows tat = increases for all values of and is concave down for negative values of, but concave up for positive values.. d Since we are identifing wit and d wit d d general statements about derivatives: we are able to make te following d a) Wen functions increase as increases. d d Wen functions decrease as increases. d d b) Wen functions are concave up. d d Wen functions are concave down. d d d Wat is te nature of te function wen? Values of were are said to be d d critical points or stationar points of te function.. At suc points te curve neiter increases nor decreases but is stationar; it as a orizontal tangent line. Fig. a sows tree possibilities for critical points: i ) te curve can eiter ave a relative maimum (a peak value as compared to nearb points) ii) a relative minimum, or iii) a orizontal plateau along a rising or falling function. d Wat is te nature of te function wen? Fig. b sows tree possibilities were d d : i) te curve canges from concave down to concave up; ii) te curve canges from d concave up to concave down; iii) te curvature does not cange. For cases i and ii te function is said to ave an inflection point. 48 Calculus and Structures

14 Section 9. ( i ) M ( i ) I d d d d ( ii ) ( ii ) d d m d d I ( iii ) ( iii ) I d d d d Fig. a Fig. b d d.one can get a picture of ow,, and interrelate b sketcing teir graps one above d d te oter. We do tis in Fig. 4 for = and =.. Calculus and Structures 49

15 Capter 9 THE DERIVATIVE = d d = d d d d = d d = Notice tat: Fig.4a Fig.4b a) = d d increases were is positive and decreases were is negative; is concave d d d up since d d positive; and as a relative minimum at = were. d. = d increases for all since is positive for all, is concave down wen < and d d concave up wen > since is negative for < ; and positive for >, as an d d inflection point at = since at = and te curve goes from concave down to d d concave up, as a flat plateau at = since at =. d 5 Calculus and Structures

16 Section LIMITS Te concept of a it pervades te subject of calculus even toug it is often out of sigt. For eample, its were idden in te o() functions of tis capter and te alwas vanised in te computation of derivatives. Still we ad to sow tat certain functions got nearer and nearer to in terms of its decimal values as got nearer to. All of our its were computed as approaced. Tis is not entirel rigorous since b tis approac, te computation of a it would ave to etend to an infinite process. However, tis wa of looking at its is enoug to satisf our interests, and it is in te spirit of te founders of calculus wo ad to wait more tan one undred ears before calculus was placed on an entirel rigorous and logical foundation. Wat about its of functions f() were approaces some number oter tan? If te values of f() gets nearer and nearer in terms of decimal values to te number L as gets nearer and nearer to = a, we sa, a f ( ) L Eample 7 Sow tat, ( ) =. Clearl as gets close to =, f() gets close to. In fact ou can actuall plug = into te function to get f() =. Tat was eas. Eample 8 Sow tat ( ) Tis is not so obvious since if ou plug = into tis function ou get.wic tells ou noting. Tere are two metods tat we can use to find te it. Metod : You can solve tis problem, as we did wit o() functions, wit a table Calculus and Structures 5

17 Capter 9 THE DERIVATIVE Table 4 f() = Tis certainl suggests tat ( ) However, wit a bit of algebra we can do even better. Metod : Let, ( )( ) f ( ) ( ) So long as does not equal we can cancel (-) from numerator and denominator. But ou will notice in te epression of a it, onl gets closer and closer to never actuall aving to equal it so we can do te canceling. Tis give te result, f ( ) Now it is clear tat as gets nearer and nearer to =, f() gets nearer and near to eactl L =, so we can sa, precisel, tat, ( ) Metod was elegant but it used a bit of algebraic tricker tat was avoided b Metod. Sometimes we want to see wat te function approaces as approaces an arbitraril large or small value, i.e.,. Eample: 9 5 Sow tat ( ) / 7 5 Calculus and Structures

18 Section 9. Metod : Make a table of values were gets large. Table 5 5 f ( ) ( ) It is apparent tat f ( ). 5 Remark : Tere are two decimal representations of te rational number / eiter.5 or Metod : a) Divide numerator and denominator b te igest power of (in tis case.), i.e., f ( ) / 5 / / / 7 / 5/ / 7 / b) Let get large approacing., ten f() clearl approaces / =.5. Metod : Since as gets ver large onl te igest order power of in te numerator and denominator matters, te oters are insignificant b comparison. For eample, is times greater tan. Terefore, f ( ). Remark : Metod is certainl te easiest wa of evaluating infinite its Eample : Evaluate te following it: Using Metod, f ) 5 ( ) 5 5 ( 5 Calculus and Structures 5

19 Capter 9 THE DERIVATIVE Clearl f ( ) Eample : Using Metod, 4 ) 5 ( f 4 ) 5 ( As, f() gets arbitraril large. Since f() does not approac an particular number we sa tat tere is no it. However, sometimes it makes sense to sa tat f () wic gives us some information about te beavior of f() as gets large. Eample : Sow tat n ( ) n n e were e =.788 an irrati onal number. For tis problem it is best to use a Table. Table 6 n f(n) = 54 Calculus and Structures ( L HOSPITAL S RULE n n ) We saw in Eample 8 tat if ou insert = into f ( ), te result is, and in 5 Eample if ou insert = into f ( ) ( ), te result is. In evaluating a 7 it, if te result is eiter or ten ou can evaluate te it b a teorem known as L Hospital s Rule. Wat ou do is to differentiate te numerator and ten differentiate te denominator of te function. If tis ratio of derivatives approaces a iting value, tat will also be te it of te function.

20 Section 9. Eample For eample, appling L Hospital s rule to f ( ) f ( ), Eample 5 Appling L Hospital s Rule to f ( ) ( ) f ( ) 4 But 6 5, so we can appl L Hopital s Rule again to get, 4 Eample 4 Evaluate: e. If te function is rewritten as f ) e Rule we ave, e e (, ten appling L Hospital s 9. CONTINUITY Te continuit of te graps of functions is ver muc connected to te concept of its.. Before giving a matematical definition of a continuous function, I will first describe functions tat are discontinuous. Tere are tree was in wic a function of one variable can be discontinuous. Eamples are sown in Fig. 5 were te discontinuit of te function occurs at = in eac case. Calculus and Structures 55

21 Capter 9 THE DERIVATIVE - i ) ii ) iii ) Case i) f() = Case ii) f ( ) -, for, for > wile f ( ). as. { a f ( ) a f ( ) Fig.5 Tis function is well-defined at = were f() = -, but it immediatel jumps to f() = wen is greater tan. For tis function, f ( ) and f ( ) were. means tat approaces from values greater tan and. means tat approaces zero from values less tan. We ave seen functions wit suc discontinuities in our evaluation of sear force for beams wit concentrated forces. In order for a function to be continuous at = a, it must be true tat, Tis function eplodes at =. B tis I mean tat as ou approac = from eiter te positive or negative side of = te function values get large beond all its. Sometimes we sa tat te value of te function at = approaces. Remark : If te function under consideration is f ( ), f ( ). as. Case iii) f ( ) Tis function as te value f() = at all values of ecept at = were it is not defined since its value is f (). So te function as a ole at =, b ut since ( ) f te ole can easil be plugged b redefining te function to wen =, i.e, f() =. Te discontinuit in tis case is said to be removable. 56 Calculus and Structures

22 Section 9. Remark 4: Clearl as no particular value. For te function in Case iii). However, 4 if our function was f ( ), ten we could sa tat 4. B te same reasoning can equal an number so we sa tat it is undefined. We now ave enoug information to define a continuous function. Definition: f() is continuous at = a if and onl if, Eample 5 a) f ( ) L, and a b) L = f(a) B tis definition f ( ) from Eample 7 certainl passes te two continuit tests and is a continuous function. Eample 6 Tis function represents te bending moment of a beam. { 5, 6 f() = 6 + 5, 6 5, Te function passes bot tests for continuit so tat it is a continuous function. Eample 7: Consider f ( ). Tis not a continuous function because it fails test a) at =. Tere is a ole in te function at = as seen in Fig. 6. However since f ( ) tis point of discontinuit can be easil remedied b defining f() =. f ( ) Fig.6 Calculus and Structures 57