Over the course of the next two chapters 12 (Ch 2 and 3), we will make precise, what we
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1 Lecture : Tangents Functions Te word Tangent means toucing in Latin. Te idea of a tangent to a curve at a point P, is a natural one, it is a line tat touces te curve at te point P, wit te same direction as te curve. However tis description is somewat vague, since we ave not indicated wat we mean by te direction of te curve. In Euclidean Geometry, te notion of a tangent to a circle at a point P on its circumference, is precise; it is defined as te unique line troug te point P tat intersects te circle once and only once. 6 P 4 Tis definition gworks ( x) = 3 ( x for ) + a circle, owever in te case of te curve sown below, te object we wis to 0 use as te tangent line intersects te curve more tan once. In oter cases tere may not be a unique line toucing te curve at a point. We must terefore make tis intuitive definition of te tangent more 8 precise. 6 4 P f( x) = x Over te course of te next two capters (C and 3), we will make precise, wat we mean by te slope or direction of a curve at a point P. We will accomplis a definition, and 4 a metod of measuring te slope, wit te aid of te concept of a limit. We will ten use our measure of te slope of te curve at a point P (wen 6 it exists) to define te tangent at te point P as te line troug P wit te same slope as te curve at tat point. 8 Altoug te process of defining te slope and learning to calculate slopes (derivatives) for a wide range of functions will take some time, we can see 0 te concept in action immediately wit some examples.
2 Example Find te equation of te tangent line to te curve y = x at te point were x = (at te point P (, )). Tis means, we need to find te slope of te tangent line toucing te curve drawn in te picture P(,) f( x) = x We ave only one point on te tangent line, P (, ), wic is not enoug information to find te slope.. However we can approximate te slope of tis line using te slope of a line segment joining P (, ) to a point Q on te curve near P. Let us consider te point Q(.5,.5), wic is on te grap of te function f(x) = x Q(.5,.5 ). P(, ) y =.5 - f( x) = x 0.8 x =.5 - =.5 Tangent 0.6 f( x) = x 0.4 Q(.5,.5) 0. P(,) Since Q is on te curve y = x, te slope of te line segment joining te points P and Q (secant), m P Q = y.5 x = m P Q te cange in elevation on te curve y = x between te points P and Q divided by te cange in te value of x, y (see diagram on rigt ). If we tink of te curve y = x as a ill and imagine x we are walking up te ill from left to rigt, m P Q agrees wit our intuitive idea of te average slope or incline on te ill between te points P and Q..4 Because, te point Q is so close to P, and because te curve y = x does not deviate too far from.6 its tangent near P,.8 slope of tangent at te point P m P Q =. ( x) = ( x ) + If we coose a point Q on te curve y = x wic is closer to te point P and calculate.4.6
3 te slope of te line segment P Q, m P Q, we get a better estimate (in tis case) for te slope of te tangent line to te curve at P. Fill in te table below to see wat appens to te slopes of te secants P Q as te point Q moves closer to P slope of secant(q = Q(x, x)) x y x m P Q = x Cange in y (from P to Q) x = Cange in x (from P to Q) x x = = = = = = Fill in te last few lines of te table and complete te following sentence: As x approaces, te values of m P Q approac 3
4 We can reprase te sentence on te previous page using x and y : As x approaces 0, te values of m P Q approac / or : As x approaces 0, te values of y x approac / We can rewrite tis statement (to wic we will assign a precise meaning later) in a number of ways, all of wic will be used in te course. We say: or or lim m P Q = / x lim m P Q = / Q P y lim x 0 x = /. From te picture above, we can see tat te slopes of te line segments P Q approac te slope of te tangent we seek, as Q approaces P. Hence it is reasonable to define te slope of te tangent to be tis limit of te slopes of te line segments P Q as Q approaces P. Tis will be called te derivative of te function f(x) = x at x = later and will be denoted by f (). Hence te slope of te tangent to te curve y = x at te point P (, ) is / and te equation of te tangent to te curve y = 4 x at tis point is Equation of te tangent at P is y = (x ) or y = x We will also make eavy use of te following notation: Wen calculating te slope of a secant, 3 instead of using x to denote te small cange in te value of x (between P and Q), we use. For P.8 and Q sown in te diagrams below, tis translates to. m P Q = = Q(.5,.5 ) P(, ) x =.5 - =.5 f( x) = x y = P(, ) =.5 - =.5 Q(.5,.5 ) f( + ) - f() = Wen we rewrite our table replacing x by, 0.4we see tat we can reprase our statement about te 0. limiting value of te slope of te secants as
5 As approaces 0, te values of m P Q = or in te language of limits : + approac / lim m P Q = lim = / slope of secant(q = Q(x, x)) f( + ) f() x m P Q = + = Cange in y (from P to Q) Cange in x (from P to Q) = = Te slope of te tangent to a curve at a point gives us a measure of te instantaneous rate of cange of te curve at tat point. Tis measure is not new to us, in a car, te odometer tells us te distance te car as travelled (under its own steam) since it rolled off te assembly line. Tis a function D of time, t. Te speedometer on a car gives us te instantaneous rate of cange of te function D(t), wit respect to time, t, at any given time. Wen you are driving a car, you see tat te speed of te car is usually canging from moment to moment. Tis reflects te fact tat te instantaneous rate of cange of D(t) or slope of te tangent to te curve y = D(t) varies from moment to moment. Te following Wolfram demonstration sows te above process in action for a wider variety of examples: Increasing/Decreasing Functions Wen a function is increasing, we get a positive slope for te tangent and wen a function is decreasing, we get a negative slope for te tangent. D(t) above never decreases, reflecting te fact tat te speedometer always reads 0 or someting positive. 5
6 Example A Buzz Ligtyear toy is dropped (no initial velocity) from te top of te Willis Tower in Cicago, wic is 44 m tall. We will denote te distance fallen by te toy after t seconds by s(t) meters. According to Gallileo s law, te distance fallen by any free falling object is proportional to te square of te time it as been falling. Hence, s(t) = kt. Let us assume tat te only force acting on te toy is te force of gravity (no air resistance or wind) wic causes te speed of te toy to increase as it falls wit an acceleration of 9.8m/s or 3ft/s. Later we will see tat tis translates to saying tat distance fallen by te toy after t seconds is s(t) = 4.9t meters Te velocity or speed of te toy at any given time is te instantaneous rate of cange of te function s(t) at tat time. (a) How far as te toy travelled after t = 3 seconds? (b) How long does it take for te toy to reac te ground? (c) Wat is te average speed of te toy on its way to te ground? (d) Use te table below to estimate te velocity of te toy after 3 seconds? Time Interval Average velocity = s t (measured in m/s) 3 t 4 3 t 3. 3 t t t
7 EXTRAS: Attempt te following questions before you look at te solutions provided In te following examples, we ave empirical data about a function, rater tan a formula for te function. However te metods of estimation outlined above rely only on knowing te values of te function at a finite number of points. Terefore te same metod can be used to estimate instantaneous rate of cange from a finite set of data: Example Te following data sows te position of a sprinter, s(t) = meters travelled after t seconds. t (seconds) s(t) (meters) (a) Find te average velocity of te sprinter over te time periods [, 8], [, 3] and [, ]. (b) Wic of te above averages gives te best estimate of te instantaneous velocity of te sprinter wen t =? (c) Can you tink of any oter way to estimate te instantaneous velocity of te sprinter wen t =? It may elp to consider te grap of te data Example Te following data sows te world population in te 0t century. Estimate te rate of population growt in 90 and compare it wit an estimate of te rate of population growt in 980. (Note you ave a number of coices for your estimates). 7
8 Solutions Example Te following data sows te position of a sprinter, s(t) = meters travelled after t seconds. t (seconds) s(t) (meters) (a) Find te average velocity of te sprinter over te time periods [, 8], [, 3] and [, ]. Te average velocity from t = to t = 8 is given by s(8) s() 8 = 74 7 Te average velocity from t = to t = 3 is given by = 0.8 m/s. s(3) s() 3 = 9 Te average velocity from t = to t = is given by = 8.5 m/s. s() s() = 0 = 8 m/s. (b) Wic of te above averages gives te best estimate of te instantaneous velocity of te sprinter wen t =? Te average velocity for te sprinter over te time period [, ] is most likely to give us te best estimate for te instantaneous velocity of te sprinter wen t =, since te cange in velocity over tat period is likely to be less because it is te smallest time interval. (c) Can you tink of any oter way to estimate te instantaneous velocity of te sprinter wen t =? Anoter good estimate for te instantaneous velocity at t = is te average velocity between t = 0 and t = ; s() s(0) 0 = 0 = m/s. Also anoter good estimate for te instantaneous velocity at t = is te average velocity between t = 0 and t = ; s() s(0) 0 = 0 0 = 5 m/s. Example Te following data sows te world population in te 0t century. Estimate te rate of population growt in 90 and compare it wit an estimate of te rate of population growt in 980. (Note you ave a number of coices for your estimates). To estimate te rate of growt in 90, we take te average rate of growt per year between 90 and 930: rate of growt in 90 = 6 million/year
9 To estimate te rate of growt in 980, we take te average rate of growt per year between 970 and 990: rate of growt in 980 = 78.5 million/year
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