# 4.4 The Derivative. 51. Disprove the claim: If lim f (x) = L, then either lim f (x) = L or. 52. If lim x a. f (x) = and lim x a. g(x) =, then lim x a

Save this PDF as:

Size: px
Start display at page:

Download "4.4 The Derivative. 51. Disprove the claim: If lim f (x) = L, then either lim f (x) = L or. 52. If lim x a. f (x) = and lim x a. g(x) =, then lim x a"

## Transcription

1 Capter 4 Real Analysis Disprove te claim: If lim f () = L, ten eiter lim f () = L or a a lim f () = L. a 52. If lim a f () = an lim a g() =, ten lim a f + g =. 53. If lim f () = an lim g() = L R f (), ten lim a a a g() =. 54. If lim f () = L, ten lim(f () L) = 0. a a 55. Disprove te two claims: (a) If lim f () = L, ten f (a) = L. a (b) If f (a) = L, ten lim f () = L. a 56. Te squeeze teorem (teorem 4.3.4). In eercises 57 68, prove eac matematical statement about continuity. 57. Te constant function is continuous (from teorem 4.3.5). 58. Te sum of two continuous functions is continuous (from teorem 4.3.5). 59. Te ifference of two continuous functions is continuous (from teorem 4.3.5). 60. Te prouct of two continuous functions is continuous (from teorem 4.3.5). 61. If f is continuous at = a, ten f is continuous at = a. 62. If f is continuous at = a, ten f is continuous at = a. 63. Prove tat f () = n is continuous for all n N via inuction (an teorem 4.3.5). 64. If lim f (a + ) = f (a), ten f is continuous at = a Disprove te claim: If f an g are not continuous at = a, ten f + g is not continuous at = a. 66. Disprove te claim: If f is continuous at = a, ten f is continuous at = a. 67. Disprove te claim: If te composite function f (g()) is continuous, ten f () an g() are bot continuous. 68. Te following variation on te caracteristic function of Q is continuous at = 0: { if Q f () = 0 if Q In eercises 69 70, state bot an intuitive escription an a efinition of eac limit. 69. lim a f () = 70. lim f () = L 4.4 Te Derivative Calculus is te stuy of cange. Wile Sir Isaac Newton an Gottfrie Leibniz are bot creite for inepenently eveloping calculus in te late 1600s, matematicians a alreay been working wit erivatives for nearly a alf century. Te stuy of cange as epresse by te erivative was motivate by a siteent an seventeent century European reflection on an ultimate rejection of ancient Greek astronomy an pysics. Te European astronomers Nicolaus Copernicus, Tyco Brae, an Joannes Kepler

2 282 A Transition to Avance Matematics eac a insigts tat callenge te teories of te ancient Greeks, setting te stage for te groun-breaking work of te Italian scientist Galileo Galilei in te early 1600s. Many of te questions about a moving object (tat is, an object canging position an velocity) tat tese scientists were stuying are reaily answere by consiering lines tangent to curves. A number of matematicians from many ifferent countries mae important contributions to te question of fining te equation of a tangent line. Pierre e Fermat stuie maima an minima of curves via tangent lines, essentially using te approac stuie in contemporary calculus courses. Tis work prompte fellow Frenc matematician Josep-Louis Lagrange to assert tat Fermat soul be creite wit te evelopment of calculus! Te Englis matematician Isaac Barrow, wo was Newton s teacer an mentor, correspone regularly wit Leibniz on tese matematical ieas. As we ave mentione, neiter Newton nor Leibniz tougt of te erivative as a measure of cange in terms of our contemporary efinition involving limits. Our stuy of te erivative follows more closely te work of Fermat an Barrow from te early 1600s, in wic we tink of a tangent line as a limit of secant lines. Naturally, te contemporary presentation is informe by an unerstaning of Caucy s notion of te limit from te early 1800s. Te erivative enables te etermination of te equation of a line tangent to a given curve at a given point. Given a function y = f (), te slope of a secant line joining two points (c, f (c)) an (c +, f (c + )) is m = rise run = y = f (c + ) f (c) (c + ) c = f (c + ) f (c). In tis contet, te symbol m is te first letter in te Frenc wor montrer wic translates as to climb. To fin te slope of te line tangent to a function f at te point (c, f (c)), we take a limit of te slopes of secant lines, letting approac 0. Figure 4.13 illustrates wy tis limit process makes intuitive sense; you can see tat te slopes of te secant lines get closer an closer to te slope of te tangent line as te point (c +, f (c + )) gets closer an closer to (c, f (c)). Te efinition of te erivative reflects tese ieas. Te following efinition epresses te real number c as a variable quantity to ientify a general formula for te erivative, enabling us to etermine te slope of te line tangent to f () wenever tis slope is efine. Definition4.4.1 Let f () be a function wit omain D. Ten te erivative of f () is f f ( + ) f () (), 0 wenever tis limit eists. We say tat f () is ifferentiable at = c wen f (c) eists for c D, an tat f () is ifferentiable wen f () eists for all D. Te ratio f ( + ) f () is calle te ifference quotient of te erivative.

3 Capter 4 Real Analysis Figure4.13 A tangent line at = 2 as a limit of secant lines Recall from calculus tat many ifferent notations are use for te erivative of a function y = f (), incluing f () = f = (f ) = y = y = D (y) = ẏ. Various prases also refer to te erivative, incluing f (or y) prime, te erivative of f (or y) wit respect to, te letters f spoken iniviually, an y spoken iniviually. Most of tis notation for te erivative is attributable to Leibniz, wo gave consierable tougt to carefully ientifying a useful symbolism an is recognize as a genius in eveloping notation to make subtle concepts unerstanable. Te alternate efinition of te erivative is sometimes elpful; if te limit eists, ten f f (t) f () (). t t Te proof of te equivalence of tis alternate efinition an te one given in efinition is left for eercise 50 at te en of tis section. Eample We use te two efinitions of te erivative to etermine te equation of a line tangent to f () = 2 at (2, 4). Applying te efinition, f (+) () 2+ = Hence te slope of te tangent line at (2, 4) is f (2) = 2 2 = 4, an te equation of te line tangent to f () = 2 at (2, 4) is given by y 4 = 4( 2). Applying te alternate efinition prouces te same result: f t 2 2 () t t (t + )(t ) t + = 2. t t t Wen using te efinition (as in eample 4.4.1), we often algebraically manipulate te ifference quotient so tat appears as a factor in te numerator. Tis factor

4 284 A Transition to Avance Matematics ten cancels te enominator, simplifying te ifference quotient so te limit can be evaluate. If te original function f () is a rational function, ten fining a common enominator will simplify te ifference quotient in tis way. If f () contains a square root, multiplying by te conjugate square root function will simplify te ifference quotient. Eample4.4.2 We use te efinition of te erivative to fin te erivative of f () = Multiplying bot te numerator an te enominator of te ifference quotient by te conjugate square root function an ten simplifying yiels te following calculation. f 5 () [( + + 1) ( + 1)] 5( ) 0 = ( ) Question4.4.1 Using te efinition of te erivative, ifferentiate eac function. (a) f () = (b) g() = 7 3 (c) s() = () t() = 1 3 Wile we can use te formal efinition of te erivative to compute erivatives of a given function, teoretical applications of te efinition are more important. Using te efinition, we can prove general teorems tat ol for all erivatives, making it easy to ifferentiate many familiar functions witout eplicitly applying te efinition one function at at time. Many functions are so complicate in structure tat irectly using te ifference quotient becomes unwiely or impossible. Te net teorem states analytic properties of erivatives to facilitate suc computations. Using tese results is a common eercise in calculus courses, but you may not ave consiere te unerlying proofs tat justify tem. Tese proofs are te focus of te remainer of tis section. Teorem4.4.1 If c R an bot f an g are ifferentiable functions, ten te following ol. Te constant rule: Te scalar multiple rule: [ c ] = 0 [ c f () ] = c f ()

5 Capter 4 Real Analysis 285 Te sum rule: Te ifference rule: Te power rule: Te prouct rule: Te quotient rule: Te cain rule: [ f + g ] = f + g [ f g ] = f g [ n ] = n n 1, for n R [ f g ] = g f + f g [ ] f = g f f g g g 2, provie tat g() = 0 [ f (g()) ] = f (g()) g () A stanar goal of a calculus course is to evelop a mastery in using tese ifferentiation rules. Before iving into te proofs of various parts of tis teorem, te net eample provies te opportunity to revisit te skills you learne in calculus. Question4.4.2 Using teorem 4.4.1, ifferentiate eac function. (a) f () = (b) g() = cos (3 + 1) (c) () = () p() = ( 5 + ) tan(2) (e) q() = ln( ) sin 2 (5 + 3) (f) r() = ( ) 3 4e + 6 Te net tree eamples give te proofs of some of tese ifferentiation rules. As in te stuy of limits an continuity, we first consier te scalar multiple an sum rules, an ten iscuss a couple of ifferent approaces to proving te power rule. Eample4.4.3 We prove te scalar multiple rule from teorem 4.4.1: For any constant c R an ifferentiable function f, [ c f () ] = c f (). Proof Apply te efinition of te erivative an te limit of a scalar multiple rule. c f ( + ) c f () [ c f () ] 0 = c lim 0 f ( + ) f () 0 c [ f ( + ) f ()] = c f () Eample4.4.4 We prove te sum rule: If f an g are ifferentiable functions, ten [ f + g ] = f () + g ().

6 286 A Transition to Avance Matematics Proof Apply te efinition of te erivative an te limit of a sum rule. [ f + g ] [ f ( + ) + g( + )] [ f () + g()] 0 0 [ f ( + ) f ()] + [ g( + ) g()] f ( + ) f () g( + ) g() + lim 0 0 = f () + g () Eample4.4.5 We prove te power rule: If n R, ten [ n ] = n n 1. Proof We prove te power rule in te case of te positive integers n N by using te binomial teorem to epan te term f ( + ) = ( + ) n in te ifference quotient as follows: ( + ) n = n + n n 1 n(n 1) + n n n 1 + n. 2 Applying te efinition of te erivative, [ n ] (+) n n 0 n(n 1) [n +n n 1 + n n n 1 + n ] n 2 0 n(n 1) n n 1 + n n n 1 + n 2 0 n(n 1) [n n 1 + n 2 + +n n 2 + n 1 ] 2 0 n n 1 + n(n 1) n 2 + +n n 2 + n = n n 1. Alternatively, te power rule for n N follows by inuction (see eercise 67 in section 3.6). Te efinition of te erivative proves te base case [ ] = 1 0 = 1, an te prouct rule applies in te inuctive step (for n+1 = n ). A complete proof of te power rule must consier arbitrary real numbers n R, not just positive integers n N. Te power rule etens to te negative integers via te quotient rule, to rational powers via implicit ifferentiation, an to all real numbers via logaritmic ifferentiation. Te etails of suc a complete proof are left for your later stuies.

7 Capter 4 Real Analysis 287 Question Te following steps outline a proof of te quotient rule: If f (), g() are ifferentiable functions wit g() = 0, ten [ ] f () = g() f () f () g () g() g() 2. (a) Wat is te ifference quotient for te function f () g()? (b) Using te common enominator g() g( + ), simplify te ifference quotient from part (a). (c) In te numerator from part (b), subtract an a te term g() f (). Now split te fraction into a ifference of two ifferences, gatering togeter te two terms wit g() as a common factor an te two terms wit f () as a common factor. () Wat is te limit of te ifference of ifference quotients from part (c) as approaces 0? (e) Base on parts (a) (), craft a complete proof of te quotient rule as moele in eamples 4.4.3, 4.4.4, an Question Te following steps outline a proof of te cain rule: If f (), g() are ifferentiable functions, ten [ f [g()] ] = f [g()] g (). (a) Wat is te ifference quotient (t) () t (from te alternate efinition of te erivative) for te function () = f [ g() ]? (b) Assuming tere are no values for wic g() = g(t), multiply bot te numerator an te enominator of te ifference quotient from part (a) by g(t) g(). Factor out te resulting ifference quotient for g(). (c) Take te limit of te prouct of ifference quotients from part (b) as t approaces to obtain te cain rule formula. () Base on parts (a) (), craft a proof of te cain rule uner te assumption tat g() = g(t) as moele in eamples 4.4.3, 4.4.4, an Te assumption tat tere are no values for wic g() equals g(t) may be unreasonable; a complete proof of te cain rule tat oes not use tis assumption is outline in eercises at te en of tis section. We en tis section by consiering te relationsip between two of te most significant properties of functions stuie in tis capter: continuity an ifferentiability. Some properties of functions are completely inepenent of one anoter, as we saw in our iscussion of one-to-one an onto functions; some functions are bot, some are neiter, wile still oters ave just one of tese properties. Tis observation leas us

8 288 A Transition to Avance Matematics to ask if continuity an ifferentiability are inepenent of one anoter, or is tere a connection between tese two properties? As you may recall, every ifferentiable function is continuous, but not every continuous function is ifferentiable. We consier te teorem an its proof, along wit a countereample tat togeter justify tese assertions. Teorem4.4.2 If a function f wit omain D is ifferentiable at a (b, c) D, ten f is continuous at a. Proof By te alternate efinition of te erivative, given any ε > 0, tere eists a value δ > 0 so tat f () f (a) f (a) a < ε wenever 0 < a < δ. Multiplying bot sies by a, we see tat f () f (a) f (a)( a) < ε a. Applying te secon inequality ( y y ) from teorem in section 4.3, we ave f () f (a) f (a)( a) f () f (a) f (a)( a). Tis fact implies f () f (a) < f (a)( a) + ε a, an so f () f (a) < ( f (a) + ε) a. Te term on te rigt can be mae arbitrarily small: we restrict values of in tat term so tat a is smaller tan bot δ (so tat te first inequality ols) an ε/( f (a) + ε). Ten f () f (a) < ε, wic proves te result. Teorem asserts tat every ifferentiable function is continuous. Are tere continuous functions tat are not ifferentiable? Peraps you can recall from calculus eamples of continuous functions tat are not ifferentiable. Te net eample provies one suc countereample. Eample4.4.6 We iscuss te continuity an ifferentiability of f () = at = 0. We can sow tat y = is continuous at = 0, using te efinition. Let ε > 0 an coose δ = ε. For any suc tat 0 < δ, te following string of relations ols: f () f (0) = 0 = = < ε. By te efinition of continuity, is continuous at = 0. On te oter an, we can sow tat is not ifferentiable at = 0, using te alternate efinition of te erivative. Te ifference quotient for f () at = 0 is f () f (0) 0 = 0 =. Taking te limit of tis ifference quotient as approaces 0, lim 0 = 1 an lim 0 + = 1.

9 Capter 4 Real Analysis 289 Terefore te limit f () f (0) lim 0 0 oes not eist, an f () = is not ifferentiable at = 0. Question Give an eample of a continuous function tat is not ifferentiable at te following points: (a) = 1 (b) bot = 1 an = 1 (c) = n π, for every n Z () = 2n, for every n Z Tese results sow tat (intuitively speaking) it is more ifficult for a function to be ifferentiable tan continuous. From an informal, grapical perspective, tis fact is quite natural; at a point of iscontinuity for a grap, we cannot raw a unique tangent line. Te results also provie anoter reason for te importance of stuying continuity: te functions tat are te most well beave from te perspective of ifferential calculus are continuous. Section 4.6 will ientify an important connection between continuity an Riemann integrability. Te erivative as transforme te way matematicians tink about functions. Many questions about matematical objects an our real-worl can be prase in terms of te erivative s measure of cange. In tis way, te evelopment of te erivative set te stage for muc of te last tree centuries of investigations into function teory. From your calculus courses, you know tat tese investigations inclue fining maima an minima, an etermining increasing an ecreasing sections of curves, concavity, an points of inflection, as well as te construction of power series. In summary, te erivative flows troug function teory in a useful an meaningful way Reaing Questions for Section Define an give an eample of te slope of a line. 2. Describe an intuitive motivation for te efinition of te erivative in terms of secant lines an tangent lines to a curve. 3. State te efinition of te erivative f (). 4. State te alternative efinition of te erivative f (). 5. Give an eample of a ifferentiable function. 6. Wat is te istinction between a function being ifferentiable at a point = c an a function being ifferentiable? 7. State teorem How is tis result elpful wen stuying erivatives? 8. Give an eample of eac ifferentiation rule state in teorem Define an give an eample of a conjugate square root function. 10. State te binomial teorem. How is tis result elpful wen stuying erivatives? 11. Discuss te relationsip between continuity an ifferentiability. 12. Give two eamples of functions tat are continuous, but not ifferentiable.

10 290 A Transition to Avance Matematics Eercises for Section 4.4 In eercises 1 6, epress te slope of a secant line to eac function for te esignate -coorinates as a ifference quotient, an sketc te corresponing grap. 1. f () = at = 3 an = 4 2. f () = at = 3 an = f () = at = 3 an = f () = 3 at = 0 an = 1 5. f () = 3 at = 0 an = f () = 3 at = 0 an = In eercises 7 18, use te efinition of te erivative to compute te erivative (if it eists) of eac function. 7. f () = g() = () = j() = p() = 1/ 12. q() = r() = s() = 15. t() = u() = + 7 { 4 if v() = 2 2 if > 2 { if w() = 2 2 if > 2 In eercises 19 28, compute te erivative of eac function using te analytic ifferentiation rules from teorem 4.4.1, along wit your recollection of te erivatives of functions from calculus. 19. f () = ( ) f () = ( + 1 ) f () = ( ) (2 + 1/) 22. f () = ( 2 +1) 3 ( ) f () = sin 5 ( 3 + 2) 24. f () = ln() cos(2 + 7) 25. f () = log 3 (cot(2)) 26. f () = ln( 2 + 2) log 5 (csc() + 2) 27. f () = (k 5 + 2) 3, were k R ( f () = ) 3n, k were k, n R In eercises 29 34, etermine te eact value of (3π/4) an state te equation of te line tangent to () at = 3π/4 using te information in te following table. f () f () g() g () = 3π/ () = 7 f () sec() + π () = g() cos() 31. () = g() + f () () = tan() + π cot 2 (g()) 33. () = sin[π f ()]+cos[π g()] 34. () = f () 2 g()

11 Capter 4 Real Analysis 291 In eercises 35 38, answer eac question about f () =. 35. Using te efinition of te erivative, fin f (). 36. Using te power rule, fin f (). 37. Determine te equation of te tangent line to f () = at (9, 3). 38. Determine te equation of te tangent line to f () = tat is perpenicular to te line etermine by 2y + 8 = 16. Eercises evelop a proof tat te erivative of sin θ is cos θ. 39. Prove tat sin θ cos θ < θ < tan θ. Hint: Compare te areas of te tree neste regions in figure 4.14 an use te fact tat a pie-sape sector of te unit circle wit central angle θ (in raians) as an area of θ/ Ientify upper an lower bouns on sin θ/θ using te inequalities from eercise 39. Hint: Divie by sin θ an take reciprocals. 41. Prove tat lim θ 0 sin θ/θ = 1. Hint: Apply te squeeze teorem (see teorem from section 4.3) to te inequalities from eercise Prove tat lim θ 0 (1 cos θ)/θ = 0. Hint: Multiply bot te numerator an te enominator by 1 + cos θ an ten use bot te Pytagorean ientity sin 2 θ + cos 2 θ = 1 an te limit from eercises Prove tat te erivative of sin θ is cos θ. Hint: Working wit te efinition of te erivative, simplify te resulting ifference quotient using te limits from eercises 41 an 42 along wit te trigonometric ientity sin(u + v) = sin u cos v + sin v cos u. In eercises 44 48, erive te formulas for te erivative of te oter trigonometric functions; all but eercise 44 use te quotient rule. 44. Prove tat te erivative of cos θ is sin θ. Hint: Use te cofunction ientity cos = sin(π/2 ) an te erivative from eercises 43. (cos q, sin q) (1, tan q) Figure 4.14 Figure for eercise 39 q (0,0) (1,0)

12 292 A Transition to Avance Matematics 45. Prove tat te erivative of tan θ is sec 2 θ. 46. Prove tat te erivative of cot θ is csc 2 θ. 47. Prove tat te erivative of sec θ is sec θ tan θ. 48. Prove tat te erivative of csc θ is csc θ cot θ. In eercises 51 66, prove eac matematical statement about erivatives. 49. Te erivative of a ifferentiable function is unique. Hint: See te unique limit teorem (teorem from section 4.3). 50. Te two efinitions of te erivative are equivalent. Hint: Let = a. 51. Te constant rule from teorem Te ifference rule from teorem Te prouct rule from teorem Hint: A an subtract f ( + ) g() in te numerator of te ifference quotient for f () g(). 54. Te quotient rule from teorem Hint: See question Te cain rule from teorem Hint: See question Every polynomial is ifferentiable. 57. Te erivative of a polynomial of egree n is a polynomial of egree n Te erivative of an even function is o; tat is, if f () = f ( ), ten f () = f ( ). 59. If f is a ifferentiable function on an interval (, + ) for some R, ten te erivative f () equals lim 0 f ( + ). Hint: Apply L Hôpital s rule from calculus to te limit of te ifference quotient. 60. Appling te alternative efinition of te erivative at = 0, te following function is not ifferentiable at = 0. [ ] 1 sin if = 0 f () = 0 if = Te function f () efine as follows as erivative f (0) = 0. [ ] 1 2 sin if = 0 f () = 0 if = For every k R, te function f () efine as follows as erivative f (0) = 0. f () = { k 2 if Q 0 if Q 63. If a function f is ifferentiable on (b, c) an f (a) = 0 for a (b, c), ten it is not necessarily true tat eiter a relative maimum or relative minimum for f occurs at = a. 64. If f an g are ifferentiable functions on (a, b) wit te same erivative, ten f () g() is a constant for any (a, b).

13 Capter 4 Real Analysis If f an g are ifferentiable functions on (a, b) wit f g a constant, ten f an g ave te same erivative at any (a, b). 66. Define a function f tat is nowere ifferentiable, wile f 2 is everywere ifferentiable. Hint: Consier a variation on te caracteristic function of Q. Eercises evelop a proof of te cain rule in a fuller generality tan was iscusse in question Trougout tese eercises assume tat g() is ifferentiable at a point = a an tat f () is ifferentiable at g(a). 67. Prove tat te following function F is continuous at = 0; intuitively, we tink of F as te erivative of f wit respect to t = g(a). f [g(a) + ] f [g(a)] if = 0 f () = f [g(a)] if = Prove tat f [g(a) + ] = f [g(a)] + F() for sufficiently small values of by taking te limit of tese two epressions as approaces In a parallel way, we can efine a function G so tat G(0) = g (a) an g(a + k) = g(a) + k G(k) for sufficiently small values of k. Use tis fact, te result from eercise 68, an te coice of = g(a + k) g(a) = k G(k) to prove tat: f [g(a) + ] = f [g(a + k)] an F() = k G(k) F(k G(k)). 70. Using te two equations obtaine in eercise 69, substitute te first equation into te secon to prove tat f [g(a + k)] = f [g(a)] + k G(k) F(k G(k)). Te last term on te rigt is continuous at 0 base on te efinitions of F an G. Subtract f [g(a)] on bot sies of tis equation, ivie bot sies by k an take te limit as k approaces 0 to obtain te cain rule. 4.5 Unerstaning Infinity Te notion of infinity as been an important element in many cultures attempts to unerstan life: people refer to eternal time; an eternal spiritual afterlife; a bounless universe; an all-powerful eity. Matematics as a unique an important perspective on infinity; te insigts arising from matematics rigorous, logical approac to infinity ave a an important influence on Western society s view of te worl. But many avance matematical results on infinity (especially tose tat grew out of Georg Cantor s work in te late 1860s) are not wiely known. In tis section, we eplore a matematical unerstaning of te infinite. We ave alreay taken te first steps in tis irection in our stuy of limits. One major breaktroug in te evelopment of calculus is te arnessing of infinity in te very specific an powerful way epresse by te notion of limit to obtain te erivative (an te integral as iscusse in section 4.6). As matematicians evelope an refine teir unerstaning of limits, erivatives, an integrals in te eigteent an nineteent

### Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of

### Proof of the Power Rule for Positive Integer Powers

Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive

### 1 Derivatives of Piecewise Defined Functions

MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.

### Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation

Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here

### Instantaneous Rate of Change:

Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over

### f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.

Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,

### Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)

OpenStax-CNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStax-CNX an license

### Understanding the Derivative Backward and Forward by Dave Slomer

Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange.

### Lecture 10. Limits (cont d) One-sided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)

Lecture 10 Limits (cont d) One-sided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define one-sided its, were

### CHAPTER 8: DIFFERENTIAL CALCULUS

CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly

### 7.6 Complex Fractions

Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are

### Lecture 17: Implicit differentiation

Lecture 7: Implicit ifferentiation Nathan Pflueger 8 October 203 Introuction Toay we iscuss a technique calle implicit ifferentiation, which provies a quicker an easier way to compute many erivatives we

### Math 229 Lecture Notes: Product and Quotient Rules Professor Richard Blecksmith richard@math.niu.edu

Mat 229 Lecture Notes: Prouct an Quotient Rules Professor Ricar Blecksmit ricar@mat.niu.eu 1. Time Out for Notation Upate It is awkwar to say te erivative of x n is nx n 1 Using te prime notation for erivatives,

### ACT Math Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as

### Math 113 HW #5 Solutions

Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten

### MAT1A01: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules

MAT1A01: Differentiation of Polynomials & Exponential Functions + te Prouct & Quotient Rules Dr Craig 17 April 2013 Reminer Mats Learning Centre: C-Ring 512 My office: C-Ring 533A (Stats Dept corrior)

### This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.

Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.

### Differentiable Functions

Capter 8 Differentiable Functions A differentiable function is a function tat can be approximated locally by a linear function. 8.. Te derivative Definition 8.. Suppose tat f : (a, b) R and a < c < b.

### Lecture 13: Differentiation Derivatives of Trigonometric Functions

Lecture 13: Differentiation Derivatives of Trigonometric Functions Derivatives of the Basic Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the

### The Derivative. Not for Sale

3 Te Te Derivative 3. Limits 3. Continuity 3.3 Rates of Cange 3. Definition of te Derivative 3.5 Grapical Differentiation Capter 3 Review Etended Application: A Model for Drugs Administered Intravenously

### Derivatives Math 120 Calculus I D Joyce, Fall 2013

Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te

### Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and

Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric

### Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function

Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between

### SAT Subject Math Level 1 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses

### 2 Limits and Derivatives

2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line

### Exponential Functions: Differentiation and Integration. The Natural Exponential Function

46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential

### Introduction to Integration Part 1: Anti-Differentiation

Mathematics Learning Centre Introuction to Integration Part : Anti-Differentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction

### CHAPTER 7. Di erentiation

CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.

### 2.1: The Derivative and the Tangent Line Problem

.1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position

### CHAPTER 5 : CALCULUS

Dr Roger Ni (Queen Mary, University of Lonon) - 5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective

### arcsine (inverse sine) function

Inverse Trigonometric Functions c 00 Donal Kreier an Dwight Lahr We will introuce inverse functions for the sine, cosine, an tangent. In efining them, we will point out the issues that must be consiere

### Tangent Lines and Rates of Change

Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims

### Inverse Trig Functions

Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that

### Optimal Pricing Strategy for Second Degree Price Discrimination

Optimal Pricing Strategy for Second Degree Price Discrimination Alex O Brien May 5, 2005 Abstract Second Degree price discrimination is a coupon strategy tat allows all consumers access to te coupon. Purcases

### f(a + h) f(a) f (a) = lim

Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )

### 2 HYPERBOLIC FUNCTIONS

HYPERBOLIC FUNCTIONS Chapter Hyperbolic Functions Objectives After stuying this chapter you shoul unerstan what is meant by a hyperbolic function; be able to fin erivatives an integrals of hyperbolic functions;

### 1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution

1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis

### Surface Areas of Prisms and Cylinders

12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of

### 6. Differentiating the exponential and logarithm functions

1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose

### M3 PRECALCULUS PACKET 1 FOR UNIT 5 SECTIONS 5.1 TO = to see another form of this identity.

M3 PRECALCULUS PACKET FOR UNIT 5 SECTIONS 5. TO 5.3 5. USING FUNDAMENTAL IDENTITIES 5. Part : Pythagorean Identities. Recall the Pythagorean Identity sin θ cos θ + =. a. Subtract cos θ from both sides

### New Vocabulary volume

-. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding

### The Derivative as a Function

Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)

### f(x) = a x, h(5) = ( 1) 5 1 = 2 2 1

Exponential Functions an their Derivatives Exponential functions are functions of the form f(x) = a x, where a is a positive constant referre to as the base. The functions f(x) = x, g(x) = e x, an h(x)

### 1. [2.3] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions. Indeterminate Form-Eliminate the Common Factor

Review for the BST MTHSC 8 Name : [] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions Evaluate cos 6 Indeterminate Form-Eliminate the Common Factor Find the

### Chapter 11. Limits and an Introduction to Calculus. Selected Applications

Capter Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem Selected Applications

### Finite Difference Approximations

Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip

### Trapezoid Rule. y 2. y L

Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,

### M147 Practice Problems for Exam 2

M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work

### Average and Instantaneous Rates of Change: The Derivative

9.3 verage and Instantaneous Rates of Cange: Te Derivative 609 OBJECTIVES 9.3 To define and find average rates of cange To define te derivative as a rate of cange To use te definition of derivative to

### MOOCULUS. massive open online calculus C A L C U L U S T H I S D O C U M E N T W A S T Y P E S E T O N A P R I L 1 0,

MOOCULUS massive open online calculus C A L C U L U S T H I S D O C U M E N T W A S T Y P E S E T O N A P R I L 0, 2 0 4. 2 Copyright c 204 Jim Fowler an Bart Snapp This work is license uner the Creative

### The EOQ Inventory Formula

Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of

### Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

### The Inverse Trigonometric Functions

The Inverse Trigonometric Functions These notes amplify on the book s treatment of inverse trigonometric functions an supply some neee practice problems. Please see pages 543 544 for the graphs of sin

### Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123

Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from

### The Quick Calculus Tutorial

The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,

### Math 230.01, Fall 2012: HW 1 Solutions

Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The

### Contents. 6 Graph Sketching 87. 6.1 Increasing Functions and Decreasing Functions... 87. 6.2 Intervals Monotonically Increasing or Decreasing...

Contents 6 Graph Sketching 87 6.1 Increasing Functions and Decreasing Functions.......................... 87 6.2 Intervals Monotonically Increasing or Decreasing....................... 88 6.3 Etrema Maima

### 4.1 Right-angled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53

ontents 4 Trigonometry 4.1 Rigt-angled Triangles 4. Trigonometric Functions 19 4.3 Trigonometric Identities 36 4.4 pplications of Trigonometry to Triangles 53 4.5 pplications of Trigonometry to Waves 65

### Given three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);

1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the

### MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of

### 7.4 Trigonometric Identities

7.4 Trigonometric Identities Section 7.4 Notes Page This section will help you practice your trigonometric identities. We are going to establish an identity. What this means is to work out the problem

### Math Warm-Up for Exam 1 Name: Solutions

Disclaimer: Tese review problems do not represent te exact questions tat will appear te exam. Tis is just a warm-up to elp you begin studying. It is your responsibility to review te omework problems, webwork

### Solutions to modified 2 nd Midterm

Math 125 Solutions to moifie 2 n Miterm 1. For each of the functions f(x) given below, fin f (x)). (a) 4 points f(x) = x 5 + 5x 4 + 4x 2 + 9 Solution: f (x) = 5x 4 + 20x 3 + 8x (b) 4 points f(x) = x 8

### sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2

. Problem Show that using an ɛ δ proof. sin() lim = 0 Solution: One can see that the following inequalities are true for values close to zero, both positive and negative. This in turn implies that On the

### Supporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Calculus: Module 15. The calculus of trigonometric functions

Supporting Australian Mathematics Project 3 4 5 6 7 8 9 0 A guie for teachers Years an Calculus: Moule 5 The calculus of trigonometric functions The calculus of trigonometric functions A guie for teachers

### f(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q =

Lecture 6 : Derivatives and Rates of Cange In tis section we return to te problem of finding te equation of a tangent line to a curve, y f(x) If P (a, f(a)) is a point on te curve y f(x) and Q(x, f(x))

### n-parameter families of curves

1 n-parameter families of curves For purposes of this iscussion, a curve will mean any equation involving x, y, an no other variables. Some examples of curves are x 2 + (y 3) 2 = 9 circle with raius 3,

### Area of Trapezoids. Find the area of the trapezoid. 7 m. 11 m. 2 Use the Area of a Trapezoid. Find the value of b 2

Page 1 of. Area of Trapezoids Goal Find te area of trapezoids. Recall tat te parallel sides of a trapezoid are called te bases of te trapezoid, wit lengts denoted by and. base, eigt Key Words trapezoid

### 19.2. First Order Differential Equations. Introduction. Prerequisites. Learning Outcomes

First Orer Differential Equations 19.2 Introuction Separation of variables is a technique commonly use to solve first orer orinary ifferential equations. It is so-calle because we rearrange the equation

### ME422 Mechanical Control Systems Modeling Fluid Systems

Cal Poly San Luis Obispo Mecanical Engineering ME422 Mecanical Control Systems Modeling Fluid Systems Owen/Ridgely, last update Mar 2003 Te dynamic euations for fluid flow are very similar to te dynamic

### An Interest Rate Model

An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal

### Chapter 2 Review of Classical Action Principles

Chapter Review of Classical Action Principles This section grew out of lectures given by Schwinger at UCLA aroun 1974, which were substantially transforme into Chap. 8 of Classical Electroynamics (Schwinger

### Similar interpretations can be made for total revenue and total profit functions.

EXERCISE 3-7 Tings to remember: 1. MARGINAL COST, REVENUE, AND PROFIT If is te number of units of a product produced in some time interval, ten: Total Cost C() Marginal Cost C'() Total Revenue R() Marginal

### ACTIVITY: Deriving the Area Formula of a Trapezoid

4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any

### Writing Mathematics Papers

Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not

### Rules for Finding Derivatives

3 Rules for Fining Derivatives It is teious to compute a limit every time we nee to know the erivative of a function. Fortunately, we can evelop a small collection of examples an rules that allow us to

### Notes on tangents to parabolas

Notes on tangents to parabolas (These are notes for a talk I gave on 2007 March 30.) The point of this talk is not to publicize new results. The most recent material in it is the concept of Bézier curves,

### Verifying Numerical Convergence Rates

1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and

### 1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion

Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctions-intro.tex October, 9 Note tat tis section of notes is limitied to te consideration

### Hyperbolic functions (CheatSheet)

Hyperbolic functions (CheatSheet) 1 Intro For historical reasons hyperbolic functions have little or no room at all in the syllabus of a calculus course, but as a matter of fact they have the same ignity

### Learning Objectives for Math 165

Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

### Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

### Exam 2 Review. . You need to be able to interpret what you get to answer various questions.

Exam Review Exam covers 1.6,.1-.3, 1.5, 4.1-4., and 5.1-5.3. You sould know ow to do all te omework problems from tese sections and you sould practice your understanding on several old exams in te exam

### In other words the graph of the polynomial should pass through the points

Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form

### Projective Geometry. Projective Geometry

Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,

### Solution Derivations for Capa #7

Solution Derivations for Capa #7 1) Consider te beavior of te circuit, wen various values increase or decrease. (Select I-increases, D-decreases, If te first is I and te rest D, enter IDDDD). A) If R1

### Theoretical calculation of the heat capacity

eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: Dulong-Petit, Einstein, Debye models Heat capacity of metals

### Section 3.1 Worksheet NAME. f(x + h) f(x)

MATH 1170 Section 3.1 Worksheet NAME Recall that we have efine the erivative of f to be f (x) = lim h 0 f(x + h) f(x) h Recall also that the erivative of a function, f (x), is the slope f s tangent line

### Answers to the Practice Problems for Test 2

Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan

### CHAPTER TWO. f(x) Slope = f (3) = Rate of change of f at 3. x 3. f(1.001) f(1) Average velocity = 1.1 1 1.01 1. s(0.8) s(0) 0.8 0

CHAPTER TWO 2.1 SOLUTIONS 99 Solutions for Section 2.1 1. (a) Te average rate of cange is te slope of te secant line in Figure 2.1, wic sows tat tis slope is positive. (b) Te instantaneous rate of cange

### MEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:

MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an

### 22.1 Finding the area of plane figures

. Finding te area of plane figures a cm a cm rea of a square = Lengt of a side Lengt of a side = (Lengt of a side) b cm a cm rea of a rectangle = Lengt readt b cm a cm rea of a triangle = a cm b cm = ab

### Lagrangian and Hamiltonian Mechanics

Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying

### How to Avoid the Inverse Secant (and Even the Secant Itself)

How to Avoi the Inverse Secant (an Even the Secant Itself) S A Fulling Stephen A Fulling (fulling@mathtamue) is Professor of Mathematics an of Physics at Teas A&M University (College Station, TX 7783)

### Preliminary Questions 1. Which of the lines in Figure 10 are tangent to the curve? B C FIGURE 10

3 DIFFERENTIATION 3. Definition of te Derivative Preliminar Questions. Wic of te lines in Figure 0 are tangent to te curve? A D B C FIGURE 0 Lines B an D are tangent to te curve.. Wat are te two was of

### Chapter 7 Numerical Differentiation and Integration

45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea

### Differentiability of Exponential Functions

Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an