Tutorial 1 Differentiation

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1 Tutorial 1 Differentiation

2 What is Calculus? Calculus 微積分 Differential calculus Differentiation 微分 y lim 0 f f The relation of very small changes of ifferent quantities f f y y Integral calculus Integration 積分 y = ò f ( )= N Why we nee these? æ N Aing a large amount of small quantities to fin the sum limçå f ( )( - ) i i+1 i è i=0 ö ø

3 Limit

4 Consier the functions f g h What are the values of the functions when = 0? We cannot simply substitute = 0 into the functions because in all three cases, this gives 0/0, which is unefine In fact the functions are unefine at = 0 y 5 4 y y 3 y

5 However, we can still iscuss what the functions ten to when approaches 0 Notice that when 0, the functions are equivalent to 1 f g h So when 0, = infinity f 0 g h y 5 4 y 1 y 3 y

6 There are times when we nee to iscuss the value a fraction a/b tens to when both a an b approach zero This form of 0/0 can be any value, epening on how fast a an b approach zero. 0/0 is calle an ineterminate form For eample faster slower same rate 0 f g h Finite non-zero number slower faster

7 The limit of a function refers to the value that the function approaches, not the actual value (if any) Eample: lim f lim f f 1 5

8 If lim f lim f a a we say that the limit of the function at = a eists an efine lim f lim f lim f a a a Eample: 1 1 f Not efine 4 Hence lim f lim f lim f

9 Most of the techniques of calculus require that functions be continuous. A function is continuous if you can raw it in one motion without picking up your pencil. A function f is continuous at = a if (1) f(a) is efine () the limit at = a eists (3) the limit equals f(a) If the function is continuous at every point insie a certain interval of, we say that the it is a continuous function in that interval

10 Eercise: Is the function continuous at = 1, =, an = 3?

11 Properties of Limits Limits can be ae, subtracte, multiplie, multiplie by a constant, ivie, an raise to a power If Then lim f P, lim g Q a a a a a lim f g P Q lim f g P Q lim f g PQ f P lim if Q 0 a g Q Note: Similar rules also hol for one-sie limits

12 The Squeeze (Sanwich) Theorem If g f h for all c in some interval aroun c an lim g lim h L, then lim f L. c c c Eample: Given that Then 0 0 lim h lim g 0 lim f h() f() g()

13 The Squeeze (Sanwich) Theorem The theorem also hols when c is an en point of the interval In this case, the limits become the one-sie limits Eample: Given that 0 0 lim h lim g h() Then 0 lim f 0 f() g()

14 Eample: f sin when 0 This function is efine everywhere ecept at = 0 When = 0, it is an ineterminate form What is the limit of the function when 0? First, notice that f is an even function ( f() = f(-) ) Hence, the left-han limit equals the right-han limit, if it eists To fin the limit when 0 from the positive sie, let us use a geometric metho

15 Consier a sector of a unit circle subtening an angle at the center: Area of the sector =/ (Note that this is true only when is measure in raian) Area of the small (blue) triangle = (sin cos/ Area of the large (blue + green) triangle = (tan/ 1 sinq cosq < 1 q < 1 tanq Þ cosq < Þ cosq < q sinq < 1 cosq q sinq < secq 1 sin cos tan

16 1 cosq > sinq q > cosq Because 1 lim cos 1 lim 1 cos 0 0 Hence, by the sanwich theorem sin lim sec sin cos

17 Since it is an even function sin sin lim 1 lim So: sin lim 1 0

18 Eercise:

19 Differentiation an Derivatives

20 For a function y = f() Derivative an Differentiation The slope of the tangent at point P = tan α Taking a neighboring point Q, we have

21 Let the point Q moves towars P. In the limit Q coincies with P, the angle α is equal to α The erivative at is efine by An we have

22 y oes not mean times! y oes not mean times y! y oes not mean the fraction y over! (ecept when it is convenient to think of it as ivision)

23 y is better interprete as a short han of refers to the operation of fining the erivative of the following function w.r.t y

24 Obtaining the erivative irectly starting from the efinition is calle erivation from first principle Eample: Obtain the erivative of Answer: y lim y lim from first principle f f 0 0 y f 1 1 1

25 Solution: y lim 0 lim 0 lim 0 f f lim 0 0 1

26 If the erivative of a function is its slope, then for a constant function, the erivative must be zero c 0 Eample: y 3 y 0 The erivative of a constant is zero Proof: Let y f c y y f f c c lim lim lim

27 Eercise: Use the first principle metho to calculate:

28 Differentiation Rules All the ifferentiation rules can be prove by the first principle You DON T nee to remember the proof BUT you nee to know how to use the rules

29 Differentiation Rules u() an v are two functions of 1 cu c u constant multiple rule u v u v uv u v v u u v v u u v v sum an ifference rules prouct rule quotient rule

30 1 Proof: Let y f y lim y lim lim lim 1 f f

31 constant multiple rule: Eamples: cu c u c c c Proof: cu lim 0 0 u u lim c 0 c lim cu cu u u u c

32 sum an ifference rules: u v u v (Each term is treate separately) Eamples: y 1 y 1 y y Proof: u v lim 0 0 u v u v lim u u v v u u v v lim lim u v 0 0

33 prouct rule: v u Notice that this is not just the uv u v prouct of two erivatives This is sometimes memorize as: Eample: uv u v v u

34 Proof: u v u v u v lim 0 u uv v uv lim 0 uv uv vu uv uv lim 0 v u uv lim u lim v lim v u v u u lim v lim lim u or lim v v u v u u v lim u lim or lim vlim v u u v

35 quotient rule: u v v u u v v u v u u v v v or Eample:

36 Proof: 0 u v u u v v u / v lim 1 v u u u v v lim 0 v v v 1 vu uv lim 0 vv v 1 u v lim lim v u 0v v v 0 1 u vlim u u v v u v v 0 0 lim v

37 y We saw that if, y This is part of a pattern n n n1 Eamples: 3 f y 8 power rule 3 f y 8 7 n is integer -> prove by inuction n is real -> prove by eponential an logarithm

38 Derivatives of Trigonometric Functions

39 Consier the function y sin We coul make a graph of the slope: Now we connect the ots! The resulting curve is a cosine curve 0 slope sin cos

40 Proof: From first principle Recall the ientity sin sin sin lim 0 A B A B sin A sin B cos sin sin sin cos sin sin sin cos lim 0 cos sin sin / lim cos / 0 / sin / cos lim / 0 / = 1 sin lim 1 0

41 We can o the same thing for y cos slope The resulting curve is a sine curve that has been reflecte about the -ais. 1 0 cos sin

42 Proof: From first principle Recall the ientity Hence cos cos cos cos lim 0 AB AB cos A cos B sin sin cos cos sin sin sin sin lim 0 sin / lim sin / 0 / sin / sin lim / 0 / cos sin

43 We can fin the erivative of tangent by using the quotient rule tan cos sin cos sin cos 1 cos cos cos sin sin cos tan sec sec

44 Derivatives of trigonometric functions: sin cos cot csc cos sin sec sec tan tan sec csc csc cot

45 Remember that all these results are base on sin lim 1 0 which is correct only when the angle is measure in raian In calculus, always use raian to measure angles

46 Differentiation of Inverse Functions

47 As an eample, consier y f y 0 Since the function is one-to-one, the inverse function eists: 1 f y y y Use the power rule: 0 y y 1 1/ 1 y y y

48 Notice that 8 6 y Slopes are reciprocals y y y 4 y Derivative Formula for Inverses: y 1 y /

49 (NOT important) Proof: y lim lim 1 y y lim lim y0 0 y y0y y0 y / We assume the function y() is one-to-one an continuous of

50 西貢 We can use this rule to fin the erivatives of inverse trigonometric functions

51 Arcsine 1.5 y sin 1 1 y sin sin y 1 1 / y / sin y sin 1 1 1

52 Proof: y y y sin sin y 1 y 1 sin y y cos 1 cos y y 1 1 sin y sin ycos y 1 cos y 1sin cos y But so 1 sin y y y cos y is positive. y 1 1 cos y 1 sin y

53 We coul use the same technique to fin the erivatives of other inverse trigonometric functions: sin cos tan cot sec csc 1

54 The Chain Rule

55 It is unesirable to obtain the erivative of every function by first principle With the rules we learne, we now have a pretty goo list of shortcuts to fin erivatives of simple functions We will now learn another very powerful rule to calculate erivative of composite functions

56 Consier a simple composite function: y u y then u y 610 y u u y 6 y u u y y u u

57 an another: y 5u where u 3t then y 5 3t y 5 3t y 5u u 3t y 15t y 15 t y u 5 u t y y u t u t

58 Chain Rule: y y u u Eample: Fin:

59 Define f u = sin u an u = 4, we have f = f u = sin 4 u Chain rule:

60 After you become familiar with the rule, you can skip some steps: y sin 4 y cos 4 4 y cos 4 Differentiate the outsie function... then the insie function

61 Another eample: cos 3 cos 3 cos 3 cos 3 It looks like we nee to use the chain rule again! erivative of the outsie function erivative of the insie function

62 Another eample: cos 3 cos 3 cos 3 cos 3 cos 3 sin 3 3 cos 3 sin 3 3 6cos 3 sin 3 The chain rule can be use more than once. (That s what makes the chain in the chain rule!)

63 NOT important Proof: y y y u y u lim lim lim lim 0 0 u 0 u 0 y u y u lim lim u0u 0 u u() is a continuous function of

64 u y u y

65 Summary sin cos cot csc 1 cu c u cos tan sin sec sec sec tan csc csc cot u v u v uv u v v u u v v u u v v y y 1 y / y u u

66 Eercise: Fin the erivative of the following functions:

67 幻彩詠香江 Higher Orer Derivatives

68 The erivative of a function y = f() is also calle the first erivative: y f Since it is still a function of, one can ifferentiate it further y The secon erivative is obtaine when a function is ifferentiate twice: Eample: y y 3 5 y 6 y 6 6

69 Notation: æ ç è y ö = y ø = y ( ) = f ( )

70 Eample: Consier the function y y y y 10 0 y All further higher erivatives are zero: 10 y Slope of y n 0 3 y n 10 0 y Slope of y 30

71 Application: Maimum or Minimum of a function f() For a given function f(), we can fin the (local) maimum or minimum by inspecting the first an secon orer erivative respectively. minimum maimum f () = f () = f () =1-4

72 f () = f () = f () =1-4 f () f () 0 <0 f() is maimum 0 >0 f() is minimum 0 =0 f() is a stationary point

73 Application in physics

74 Rates of Change Spee, Velocity an Acceleration A pilot ejecte from his USAF Thunerbir aircraft ue to an accient uring an airshow The pilot was not injure Greg Kelly, Hanfor High School, Richlan, Washington

75 Consier a function y f The rate of change of y with respect to (w.r.t.) is Average rate of change = f f Instantaneous rate of change = f f f lim 0 These efinitions are true for any function ( oes not have to represent time ) The first erivative y is the rate of change of y w.r.t.

76 Eample: The area of a circle is given by A r The rate of change of A w.r.t. r is A r r which is the circumference of the circle Rate of change increases with r For larger circle, the area changes faster with respect to the same amount of change in r

77 Velocity s A t t B s Average velocity can be foun by taking: v change in position change in time av s t s t s t t s t t The instantaneous velocity at t is the erivative obtaine when B A vt s t s t t s t lim t 0 t The velocity at one moment in time

78 Velocity is the rate of change of isplacement with respect to time vt s t Velocity has both magnitue an irection Spee is the absolute value of velocity Spee has no irection Spee v

79 Eample: Suppose the position of a particle is given by Then the velocity is 3 s vt 3t 1t 15 t s t t 6t 15t st vt t

80 At t = 10, the velocity is v The tangent at t = 10, whose slope is 195, is shown by the green line in the figure Now if you only know the velocity at t = 10, but not the position function s(t), can you preict the change in position s after a very small time interval t? ys t A B s t

81 As shown in the graph, if t is small enough, a goo approimation to s is s s t vt t The smaller the t, the better the approimation 900 y s 800 B A s t t s t t

82 Important trick: When h is small, we have In general, we have

83 Eample: Calculate tan 46 without using a calculator. Answer: tan 46 tan 45 1 tan Consier the function f tan f / 4 1 sec f f / 4 sec / 4 1 /4 f f / 4 /180 / 90 f() A /180 tan 46 tan / 4 /180 tan / 4 / 90 1 / B f f

84 Eample: Given that the instantaneous spee (km/h) of a car varies with time t (h) as v(t) = 3t. At t = 5h, the car is heaing towars a gas station at a istance of 3km Estimate its istance from the station after 10 minutes. Answer: The instantaneous spee at t = 5h is 15 km/h After t = 1/6, it has travelle approimately 1 s 15.5 km 6 Hence it will be approimately at a istance of 0.5 km from the station

85 Acceleration In general, the instantaneous velocity obtaine is also a function of time Acceleration means how fast the velocity is changing Acceleration is the rate of change of velocity with respect to time

86 After you obtain the instantaneous velocity at all t, you can plot a graph of v vs. t Average acceleration can be foun by taking: B v A v change in velocity change in time v t t t a av v t v t t v t t The instantaneous acceleration at t is the erivative obtaine when B A at v t v t t v t lim t 0 t The acceleration at one moment in time

87 Acceleration is the erivative of velocity a v s s t t t t Acceleration is the secon erivative of isplacement Acceleration has both magnitue an irection

88 Eample: Free Fall Equation An object falling freely near the surface of the earth has a position function given by s 1 g t 0 Here g is a constant: g 9.8 ms s +ve Photo courtesy Wikimeia Commons

89 Velocity is the erivative of isplacement: s t t 1 v gt gt Acceleration is the erivative of velocity: v a gt g t t The constant g is calle the acceleration ue to gravity All freely-falling objects near the earth s surface have the same constant ownwar acceleration of g 9.8 ms

90 Non-uniform Acceleration (SHM) In general, the acceleration may not be constant For eample, consier an object moving on the -ais, with position function given by t Asin kt where A an k are constants The acceleration is a t t Asin kt k Asin kt t t which is not a constant (Simple harmonic motion) -A A

91 Non-uniform Acceleration (circular motion) A particle moving accoring to the parametric equations (t) = cos t, y(t) = sin t will move counterclockwise aroun the unit circle at one raian per secon beginning at the point (1,0). Fin the velocity an acceleration of its at time time t. We use the vector notation Ԧr t = cos t i Ƹ + sin t jƹ Velocity Ԧv t = Ԧr t Acceleration Ԧa t = v t = sin t i Ƹ + cos t jƹ an Ԧr t Ԧv t = 0 = cos t iƹ sin t j Ƹ = Ԧr t

92 Displacement, Velocity an Acceleration Recall: erivative of Displacement Velocity erivative of Velocity Acceleration

93 Displacement, Velocity an Acceleration v s v is the slope of the s-t graph s v t t a v a is the slope of the v-t graph a v t t

94 It is important to unerstan the relationship between the graph of isplacement, velocity an acceleration: v > 0 & ecreasing a < 0 v < 0 & increasing a > 0 velocity zero v < 0 & ecreasing a < 0 v > 0 & increasing a > 0 t

95 Displacement s(t) Velocity v(t) Acceleration a(t)

Chapter 2 Derivatives

Chapter 2 Derivatives Chapter Derivatives Section. An Intuitive Introuction to Derivatives Consier a function: Slope function: Derivative, f ' For each, the slope of f is the height of f ' Where f has a horizontal tangent line,