Chapter 1. Rates of Change. By the end of this chapter, you will

Transcription

1 Capter 1 Rates of Cange Our world is in a constant state of cange. Understanding te nature of cange and te rate at wic it takes place enables us to make important predictions and decisions. For eample, climatologists monitoring a urricane measure atmosperic pressure, umidit, wind patterns, and ocean temperatures. Tese variables affect te severit of te storm. Calculus plas a significant role in predicting te storm s development as tese variables cange. Similarl, calculus is used to analse cange in man oter fields, from te psical, social, and medical sciences to business and economics. B te end of tis capter, ou will describe eamples of real-world applications of rates of cange, represented in a variet of was describe connections between te average rate of cange of a function tat is smoot over an interval and te slope of te corresponding secant, and between te instantaneous rate of cange of a smoot function at a point and te slope of te tangent at tat point make connections, wit or witout graping tecnolog, between an approimate value of te instantaneous rate of cange at a given point on te grap of a smoot function and average rate of cange over intervals containing te point recognize, troug investigation wit or witout tecnolog, grapical and numerical eamples of limits, and eplain te reasoning involved make connections, for a function tat is smoot over te interval a a, between te average rate of cange of te function over tis interval and te value of te epression f( a) f( a), and between te instantaneous rate of cange of te function at a and te f value of te limit lim ( a ) f ( a ) 0 compare, troug investigation, te calculation of instantaneous rate of cange at a point (a, f (a)) for polnomial functions, wit and witout simplifing te epression f ( a ) f ( a ) before substituting values of tat approac zero generate, troug investigation using tecnolog, a table of values sowing te instantaneous rate of cange of a polnomial function, f (), for various values of, grap te ordered pairs, recognize tat te grap represents a function called te derivative, f () or d, and make d connections between te graps of f () and f ( ) or and d d determine te derivatives of polnomial functions b simplifing te algebraic epression f ( ) f( ) and ten taking te limit of te simplified epression as approaces zero 1

2 Prerequisite Skills First Differences 1. Complete te following table for te function 3 5. a) Wat do ou notice about te first differences? b) Does tis tell ou anting about te sape of te curve? First Difference Slope of a Line. Determine te slope of te line tat passes troug eac pair of points. a) (, 3) and (4, 1) b) (3, 7) and (0, 1) c) (5, 1) and (0, 0) d) (0, 4) and (9, 4) Slope-Intercept Form of te Equation of a Line 3. Rewrite eac equation in slope-intercept form. State te slope and -intercept for eac. a) 4 7 b) c) d) Write te slope-intercept form of te equations of lines tat meet te following conditions. a) Te slope is 5 and te -intercept is 3. b) Te line passes troug te points (5, 3) and (1, 1). c) Te slope is and te point (4, 7) is on te line. d) Te line passes troug te points (3, 0) and (, 1). Epanding Binomials 5. Use Pascal s triangle to epand eac binomial. a) (a b) d) (a b) 4 b) (a b) 3 e) (a b) 5 c) (a b) 3 f) (a b) 5 Factoring 6. Factor. a) 1 b) c) 3 1 d) e) 4 Factoring Difference Powers f) t 3 t 3t 7. Use te pattern in te first row to complete te table for eac difference of powers. Difference of Factored Form Powers a) a n b n (a b)(a n1 a n b a n3 b a b n3 ab n b n1 ) b) a b c) (a b)(a ab b ) d) a 4 b 4 e) a 5 b 5 f) ( ) n n Epanding Difference of Squares 8. Epand and simplif eac difference of squares. a) ( )( ) b) ( 1 )( 1 ) c) ( 1 1)( 1 1) d) ( 3( ) 3)( 3( ) 3) Simplifing Rational Epressions 9. Simplif. a) c) 1 1 b) 1 1 d) ( ) MHR Calculus and Vectors Capter 1

3 Function Notation 10. Determine te points (, f ()) and (3, f (3)) for eac given function. a) f () 3 1 b) f () 5 1 c) f () For eac function, determine f (3 ) in simplified form. a) f () 6 b) f () 3 5 c) f () For eac function, determine f ( ) f ( ) in simplified form. a) f () 6 b) f () 3 c) f ( ) 1 d) f ( ) 4 Domain of a Function 13. State te domain of eac function. a) f () 3 5 b) 8 8 c) Q() 4 4 d) e) 6 f) D ( ) 9 Representing Intervals 14. An interval can be represented in several was. Complete te missing information in te following table. Interval Notation Inequalit Number Line (3, 5) 3 5 (3, 5] [3, ) (, ) Graping Functions Using Tecnolog 15. Use a graping calculator to grap eac function. State te domain and range of eac using set notation. a) 5 3 b) c) d) PROBLEM CHAPTER Alicia is considering a career as eiter a demograper or a climatologist. Demograpers stud canges in uman populations wit respect to birts, deats, migration, education level, emploment, and income. Climatologists stud bot te sort-term and long-term effects of cange in climatic conditions. How are te concepts of average rate of cange and instantaneous rate of cange used in tese two professions to analse data, solve problems, and make predictions? Prerequisite Skills MHR 3

4 1.1 Rates of Cange and te Slope of a Curve Te speed of a veicle is usuall epressed in terms of kilometres per our. Tis is an epression of rate of cange. It is te cange in position, in kilometres, wit respect to te cange in time, in ours. Tis value can represent an average rate of cange or an instantaneous rate of cange. Tat is, if our veicle travels 80 km in 1, te average rate of cange is 80 km/. However, tis epression does not provide an information about our movement at different points during te our. Te rate ou are travelling at a particular instant is called instantaneous rate of cange. Tis is te information tat our speedometer provides. In tis section, ou will eplore ow te slope of a line can be used to calculate an average rate of cange, and ow ou can use tis knowledge to estimate instantaneous rate of cange. You will consider te slope of two tpes of lines: secants and tangents. Secants are lines tat connect two points tat lie on te same curve. Secant PQ Q P Tangents are lines tat run parallel to, or in te same direction as, te curve, toucing it at onl one point. Te point at wic te tangent touces te grap is called te tangent point. Te line is said to be tangent to te function at tat point. Notice tat for more comple functions, a line tat is tangent at one -value ma be a secant for an interval on te function. P Tangent to f () at te tangent point P P Q 4 MHR Calculus and Vectors Capter 1

5 Investigate Wat is te connection between slope, average rate of cange, and instantaneous rate of cange? Imagine tat ou are sopping for a veicle. One of te cars ou are considering sells for $ 000 new. However, like most veicles, tis car loses value, or depreciates, as it ages. Te table below sows te value of te car over a 10-ear period. Tools grid paper ruler Time (ears) Value ($) A: Connect Average Rate of Cange to te Slope of a Secant 1. Eplain w te car s value is te dependent variable and time is te independent variable.. Grap te data in te table as accuratel as ou can using grid paper. Draw a smoot curve connecting te points. Describe wat te grap tells ou about te rate at wic te car is depreciating as it ages. 3. a) Draw a secant to connect te two points corresponding to eac of te following intervals, and determine te slope of eac secant. i) ear 0 to ear 10 ii) ear 0 to ear iii) ear 3 to ear 5 iv) ear 8 to ear 10 b) Reflect Eplain w te slopes of te secants are eamples of average rate of cange. Compare te slopes for tese intervals and eplain wat tis comparison tells ou about te average rate of cange in value of te car as it ages. 4. Reflect Determine te first differences for te data in te table. Wat do ou notice about te first differences and average rate of cange? B: Connect Instantaneous Rate of Cange to te Slope of a Tangent 1. Place a ruler along te grap of te function so tat it forms a tangent to te point corresponding to ear 0. Move te ruler along te grap keeping it tangent to te curve. a) Reflect Stop at random points as ou move te ruler along te curve. Wat do ou tink te tangent represents at eac of tese points? b) Reflect Eplain ow slopes can be used to describe te sape of a curve. 1.1 Rates of Cange and te Slope of a Curve MHR 5

6 . a) On te grap, use te ruler to draw a tangent troug te point corresponding to ear 1. Use te grap to find te slope of te tangent ou ave drawn. b) Reflect Eplain w our calculation of te slope of te tangent is onl an approimation. How could ou make tis calculation more accurate? C: Connect Average Rate of Cange and Instantaneous Rate of Cange 1. a) Draw tree secants corresponding to te following intervals, and determine te slope of eac. i) ear 1 to ear 9 ii) ear 1 to ear 5 iii) ear 1 to ear 3 b) Wat do ou notice about te slopes of te secants compared to te slope of te tangent ou drew earlier? Make a conjecture about te slope of te secant between ears 1 and in relation to te slope of te tangent. c) Use te data in te table to calculate te slope for te interval between ears 1 and. Does our calculation support our conjecture?. Reflect Use te results of tis investigation to summarize te relationsip between slope, secants, tangents, average rate of cange, and instantaneous rate of cange. Eample 1 Determine Average and Instantaneous Rates of Cange From a Table of Values A decorative birtda balloon is being filled wit elium. Te table sows te volume of elium in te balloon at 3-s intervals for 30 s. 1. Wat are te dependent and independent variables for tis problem? In wat units is te rate of cange epressed?. a) Use te table of data to calculate te slope of te secant for eac of te following intervals. Wat does te slope of te secant represent? i) 1 s to 30 s ii) 1 s to 7 s iii) 1 s to 4 s b) Reflect Wat is te significance of a positive rate of cange in te volume of te elium in te balloon? t(s) V (cm 3 ) a) Grap te information in te table. Draw a tangent at te point on te grap corresponding to 1 s and calculate te slope of tis line. Wat does tis grap illustrate? Wat does te slope of te tangent represent? b) Reflect Compare te secant slopes tat ou calculated in question to te slope of te tangent. Wat do ou notice? Wat information would ou need to calculate a secant slope tat is even closer to te slope of te tangent? 6 MHR Calculus and Vectors Capter 1

7 Solution 1. In tis problem, volume is dependent on time, so V is te dependent variable and t is te independent variable. For te rate of cange, V is epressed wit respect to t, or V. Since te volume in tis problem is t epressed in cubic centimetres, and time is epressed in seconds, te units for te rate of cange are cubic centimetres per second (cm 3 /s).. a) Calculate te slope of te secant using te formula ΔV V V 1 Δ t t t 1 i) Te endpoints for te interval 1 t 30 are (1, ) and (30, ). ΔV Δ t ii) Te endpoints for te interval 1 t 7 are (1, ) and (7, 305.1). ΔV Δ t CONNECTIONS Te smbol indicates tat an answer is approimate. iii) Te endpoints for te interval 1 t 4 are (1, ) and (4, 143.6). ΔV Δ t Te slope of te secant represents te average rate of cange, wic in tis problem is te average rate at wic te volume of te elium is canging over te interval. Te units for tese solutions are cubic centimetres per second (cm 3 /s). b) Te positive rate of cange during tese intervals suggests tat te volume of te elium is increasing, so te balloon is epanding. 3. a) Volume of Helium in a Balloon V 500 Volume (cm 3 ) P(1, 1436) 500 Q(16.5, 500) Time (s) t 1.1 Rates of Cange and te Slope of a Curve MHR 7

8 Tis grap illustrates ow te volume of te balloon increases over time. Te slope of te tangent represents te instantaneous rate of cange of te volume at te tangent point. To find te instantaneous rate of cange of te volume at 1 s, sketc an approimation of te tangent passing toug te point P(1, 1436). Coose a second point on te line, Q(16.5, 500), and calculate te slope. ΔV Δ t At 1 s, te volume of te elium in te balloon is increasing at a rate of approimatel 08 cm 3 /s. b) Te slopes of te tree secants in question are 305.6, 69.4, and Notice tat as te interval becomes smaller, te slope of te secant gets closer to te slope of te tangent. You could calculate a secant slope tat was closer to te slope of te tangent if ou ad data for smaller intervals. << >> KEY CONCEPTS Average rate of cange refers to te rate of cange of a function over an interval. It corresponds to te slope of te secant connecting te two endpoints of te interval. Instantaneous rate of cange refers to te rate of cange at a specific point. It corresponds to te slope of te tangent passing troug a single point, or tangent point, on te grap of a function. An estimate of te instantaneous rate of cange can be obtained b calculating te average rate of cange over te smallest interval for wic tere is data. An estimate of instantaneous rate of cange can also be determined using te slope of a tangent drawn on a grap. However, bot metods are limited b te accurac of te data or te accurac of te sketc. Communicate Your Understanding C1 Wat is te difference between average rate of cange and instantaneous rate of cange. C Describe ow points on a curve can be cosen so tat a secant provides a better estimate of te instantaneous rate of cange at a point in te interval. C3 Do ou agree wit te statement Te instantaneous rate of cange at a point can be found more accuratel b drawing te tangent to te curve tan b using data from a given table of values? Justif our response. 8 MHR Calculus and Vectors Capter 1

9 A Practise 1. Determine te average rate of cange between eac pair of points. a) (4, 1) and (, 6) b) (3., 6.7) and (5, 17) c) 4 3, and, Complete te following eercises based on te data set a) Determine te average rate of cange of over eac interval. i) 3 1 ii) 3 3 iii) 1 7 iv) 1 5 b) Estimate te instantaneous rate of cange at te point corresponding to eac -value. i) 1 ii) 1 iii) 3 iv) 5 3. Estimate te instantaneous rate of cange at te tangent point indicated on eac grap. a) b) c) B Connect and Appl 4. For eac grap, describe and compare te instantaneous rate of cange at te points indicated. Eplain our reasoning. a) A 8 6 G b) A 10 8 E 4 D C E 6 D 4 B C B F i) B, D, and F ii) A and G i) B and C ii) A, B, and E iii) C and G iv) A and E iii) C and D iv) A and C 1.1 Rates of Cange and te Slope of a Curve MHR 9

10 5. As air is pumped into an eercise ball, te surface area of te ball epands. Te table sows te surface area of te ball at -s intervals for 30 s. Connecting Reasoning and Proving Representing Problem Solving Time (s) Surface Area (cm ) Communicating Selecting Tools Reflecting a) Wic is te dependent variable and wic is te independent variable for tis problem? In wat units sould our responses be epressed? b) Determine te average rate of cange of te surface area of te ball for eac interval. i) te first 10 s ii) between 0 s and 30 s iii) te last 6 s c) Use te table of values to estimate te instantaneous rate of cange at eac time. i) s ii) 14 s iii) 8 s d) Grap te data from te table, and use te grap to estimate te instantaneous rate of cange at eac time. i) 6 s ii) 16 s iii) 6 s e) Wat does te grap tell ou about te instantaneous rate of cange of te surface area? How do te values ou found in part d) support tis observation? Eplain. 10 MHR Calculus and Vectors Capter 1 6. Wic interval gives te best estimate of te tangent at 3 on a smoot curve? a) 4 b) 3 c) d) cannot be sure 7. a) For eac data set, calculate te first differences and te average rate of cange of between eac consecutive pair of points. i) ii) b) Compare te values found in part a) for eac set of data. Wat do ou notice? c) Eplain our observations in part b). d) Wat can ou conclude about first differences and average rate of cange for consecutive intervals? 8. Identif weter eac situation represents average rate of cange or instantaneous rate of cange. Eplain our coice. a) Wen te radius of a circular ripple on te surface of a pond is 4 cm, te circumference of te ripple is increasing at 1.5 cm/s. b) Niko travels 550 km in 5. c) At 1 P.M. a train is travelling at 10 km/. d) A stock price drops 0% in one week. e) Te water level in a lake rises 1.5 m from te beginning of Marc to te end of Ma. 9. Te grap sows te Temperature of Water Being Heated temperature of water C being eated in an 100 electric kettle. Temperature (ºC) a) Wat was te initial temperature of te water? Wat appened after 3 min? b) Wat does te Time (s) grap tell ou about te rate of cange of te temperature of te water? Support our answer wit some calculations. 75 t

11 10. Capter Problem Alicia found data sowing Canada s population in eac ear from 1975 to 005. Year Canadian Population Source: Statistics Canada, Estimated Population of Canada, 1975 to Present (table). Statistics Canada Catalogue no XIE. Available at ttp://www. statcan.ca/englis/freepub/ xie/pop.tm. a) Determine te average rate of cange in Canada s population for eac interval. i) 1975 to 005 ii) 1980 to 1990 iii) consecutive 10-ear intervals beginning wit 1975 b) Compare te values found in part a). Wat do ou notice? Eplain. Estimate te instantaneous rate of cange of population growt for 1983, 1993, and 003. c) Use Tecnolog Grap te data in te table wit a graping calculator. Wat does te grap tell ou about te instantaneous rate of cange of Canada s population? d) Make some predictions about Canada s population based on our observations in parts a), b), and c). e) Pose and answer a question tat is related to te average rate of cange of Canada s population. Pose and answer anoter related to instantaneous rate of cange. Acievement Ceck 11. a) Describe a grap for wic te average rate of cange is equal to te instantaneous rate of cange for te entire domain. Describe a reallife situation tat tis grap could represent. b) Describe a grap for wic te average rate of cange between two points is equal to te instantaneous rate of cange at i) one of te two points ii) te midpoint between te two points c) Describe a real-life situation tat could be represented b eac of te graps in part b). 1. Wen electricit flows troug a certain kind of ligt bulb, te voltage applied to te bulb, in volts, and te current flowing troug it, in amperes, are as sown in te grap. Te instantaneous rate of cange of voltage wit respect to current is known as te resistance of te ligt bulb. Resistance of a Ligtbulb a) Does te V resistance increase or 10 decrease as te voltage 90 is increased? Justif our answer. b) Use te grap to determine te resistance of te ligt bulb at a C voltage of 60 V. Current (A) Voltage (V) 1.1 Rates of Cange and te Slope of a Curve MHR 11

12 C Etend and Callenge 13. An offsore oil platform develops a leak. As te oil spreads over te surface of te ocean, it forms a circular pattern wit a radius tat increases b 1 m ever 30 s. a) Construct a table of values tat sows te area of te oil spill at -min intervals for 30 min, and grap te data. b) Determine te average rate of cange of te area during eac interval. i) te first 4 min ii) te net 10 min iii) te entire 30 min c) Wat is te difference between te instantaneous rate of cange of te area of te spill at 5 min and at 5 min? d) W migt tis information be useful? 14. Te blades of a particular windmill sweep in a circle 10 m in diameter. Under te current wind conditions, te blades make one rotation ever 0 s. A ladbug lands on te tip of one of te blades wen it is at te bottom of its rotation, at wic point te ladbug is m off te ground. It remains on te blade for eactl two revolutions, and ten flies awa. a) Draw a grap representing te eigt of te ladbug during er time on te windmill blade. b) If te blades of te windmill are turning at a constant rate, is te rate of cange of te ladbug s eigt constant or not? Justif our answer. c) Is te rate of cange of te ladbug s eigt affected b were te blade is in its rotation wen te ladbug lands on it? 15. a) How would te grap of te eigt of te ladbug in question 14 cange if te wind speed increased? How would tis grap cange if te wind speed decreased? Wat effect would tese canges ave on te rate of cange of te eigt of te ladbug? Support our answer. b) How would te grap of tis function cange if te ladbug landed on a spot 1 m from te tip of te blade? Wat effect would tis ave on te rate of cange of te eigt? Support our answer. 16. Te table sows te eigt, H, of water being poured into a cone saped cup at time, t. a) Compare te following in regard to te eigt of water in te cup. i) Average rate of cange in te first 3 seconds and last 3 seconds. ii) Instantaneous rate of cange at 3 s and 9 s. b) Eplain our results in part a). t (s) H (cm) c) Grap te original data and grapicall illustrate te results ou found in part a). Wat would tese graps look like if te cup was a clinder? d) Te eigt of te cup is equal to its largest diameter. Determine te volume for eac eigt given in te table. Wat does te volume tell ou about te rate at wic te water is being poured? 17. Mat Contest If and are real numbers suc tat 8 and 1, determine te value of Mat Contest If 5 3 g, find te value of log 9 g. 1 MHR Calculus and Vectors Capter 1

13 1. Rates of Cange Using Equations Te function (t) 4.9t 4t models te position of a starburst fireworks rocket fired from m above te ground during a Jul 1st celebration. Tis particular rocket bursts 10 s after it is launced. Te protecnics engineer needs to be able to establis te rocket s speed and position at te time of detonation so tat it can be coreograped to music, as well as coordinated wit oter fireworks in te displa. In Section 1.1, ou eplored strategies for determining average rate of cange from a table of values or a grap. You also learned ow tese strategies could be used to estimate instantaneous rate of cange. However, te accurac of tis estimate was limited b te precision of te data or te sketc of te tangent. In tis section, ou will eplore ow an equation can be used to calculate an increasingl accurate estimate of instantaneous rate of cange. Investigate How can ou determine instantaneous rate of cange from an equation? An outdoor ot tub olds 700 L of water. Wen a valve at te bottom of te tub is opened, it takes 3 for te water to completel drain. Te volume of water in te tub is modelled b te function V() t 1 1 (180 t), were V is te volume of water in te ot tub, in litres, and t is te time, in minutes, tat te valve is open. Determine te instantaneous rate of cange of te volume of water at 60 min. A: Find te Instantaneous Rate of Cange at a Particular Point in a Domain Metod 1: Work Numericall 1. a) Wat is te sape of te grap of tis function? b) Epress te domain of tis function in interval notation. Eplain w ou ave selected tis domain. c) Calculate V(60). Wat are te units of our result? Eplain w calculating te volume at t 60 does not tell ou anting about te rate of cange. Wat is missing? 1. Rates of Cange Using Equations MHR 13

14 . Complete te following table. Te first few entries are done for ou. Tools graping calculator CONNECTIONS To see ow Te Geometer s Sketcpad can be used to determine an instantaneous rate of cange from an equation, go to calculus1 and follow te links to Section 1.. Tangent Point P Time Increment (min) Second Point Q (60, 100) 3 (63, ) Slope of Secant PQ (60, 100) 1 (61, ) (60, 100) 0.1 (60.1, 1198) (60, 100) 0.01 (60, 100) (60, 100) a) W is te slope of PQ negative? b) How does te slope of PQ cange as te time increment decreases? Eplain w tis makes sense. 4. a) Predict te slope of te tangent at P(60, 100). b) Reflect How could ou find a more accurate estimate of te slope of te tangent at P(60, 100)? Metod : Use a Graping Calculator 1. a) You want to find te slope of te tangent at te point were 60, so first determine te coordinates of te tangent point P(60, V(60)). b) Also, determine a second point on te function, Q(, V()), tat corresponds to an point in time,.. Write an epression for te slope of te secant PQ. 3. For wat value of is te epression in step not valid? Eplain. 4. Simplif te epression, if possible. 5. On a graping calculator, press Y= and enter te epression for slope from step into Y1. 6. a) Press ND WINDOW to access TABLE SETUP. b) Scroll down to Indpnt. Select Ask and press ENTER. c) Press ND GRAPH to access TABLE. 14 MHR Calculus and Vectors Capter 1

15 7. a) Input values of tat are greater tan but ver close to 60, suc as 61, 60.1, 60.01, and b) Input values of tat are smaller tan but ver close to 60, suc as 59, 59.9, 59.99, and c) Wat do te output values for Y1 represent? Eplain. d) How can te accurac of tis value be improved? Justif our answer. B: Find te Rate at An Point 1. Coose a time witin te domain, and complete te following table. Let a represent te time for wic ou would like to calculate te instantaneous rate of cange in te volume of water remaining in te ot tub. Let represent a time increment tat separates points P and Q. Tangent Point P(a, V(a)) Time Increment (min) Second Point Q((a ), V(a )) Slope of Secant Va ( ) Va ( ) ( a) a. a) Predict te slope of te tangent at P(a, V(a)). b) Verif our prediction using a graping calculator. 3. Reflect Compare te metod of using an equation for estimating instantaneous rate of cange to te metods used in Section 1.1. Write a brief summar to describe an similarities, differences, advantages, and disadvantages tat ou notice. 4. Reflect Based on our results from tis Investigation, eplain ow te formula for slope in te table can be used to estimate te slope of a tangent to a point on a curve. Wen te equation of function f () is known, te average rate of cange over an interval a b is determined b calculating te slope of te secant: Q(b, f (b)) Δ fb () fa () Secant Δ b a PQ f (b) f (a) P(a, f (a)) b a 1. Rates of Cange Using Equations MHR 15

16 If represents te interval between two points on te -ais, ten te two points can be epressed in terms of a: a and (a ). Te two endpoints of te secant are (a, f (a)) and ((a ), f (a )). Q((a), f (a)) f (a ) f (a) P(a, f (a)) a a Te Difference Quotient Te slope of te secant between P(a, f (a)) and Q(a, f (a )) is Δ fa ( ) fa ( ) Δ ( a) a fa ( ) fa ( ), 0 Tis epression is called te difference quotient. Instantaneous rate of cange refers to te rate of cange at a single (or specific) instance, and is represented b te slope of te tangent at tat point on te curve. As becomes smaller, te slope of te secant becomes an increasingl closer estimate of te slope of te tangent line. Te closer is to zero, te more accurate te estimate becomes. Q((a), f (a)) f (a ) f (a) P(a, f (a)) a a P(a, f(a)) Q((a), f(a)) a a 16 MHR Calculus and Vectors Capter 1

17 Eample 1 Estimate te Slope of a Tangent b First Simplifing an Algebraic Epression Amed is cleaning te outside of te patio windows at is aunt s apartment, wic is located 90 m above te ground. Amed accidentall kicks a flowerpot, sending it over te edge of te balcon. 1. a) Determine an algebraic epression, in terms of a and, tat represents te average rate of cange of te eigt above ground of te falling flowerpot. Simplif our epression. b) Determine te average rate of cange of te flowerpot s eigt above te ground in te interval between 1 s and 3 s after it fell from te edge of te balcon. c) Estimate te instantaneous rate of cange of te flowerpot s eigt at 1 s and 3 s.. a) Determine te equation of te tangent at t 1. Sketc a grap of te curve and te tangent at t 1. b) Use Tecnolog Verif our results in part a) using a graping calculator. Solution Te eigt of a falling object can be modelled b te function s(t) d 4.9t, were d is te object s original eigt above te ground, in metres, and t is time, in seconds. Te eigt of te flowerpot above te ground at an instant after it begins to fall is s(t) t. 1. a) A secant represents te average rate of cange over an interval. Te epression for estimating te slope of te secant can be obtained b writing te difference quotient Δ fa ( ) fa ( ) for Δ s(t) t. Δ [ ( a) ] ( a) Δ ( aa) a a 98. a a a a Rates of Cange Using Equations MHR 17

18 b) To calculate te rate of cange of te flowerpot s eigt above te ground over te interval between 1 s and 3 s, use a 1 and (i.e., te -s interval after 1 s). Δ Δ 983.() 49.() Between 1 s and 3 s, te flowerpot s average rate of cange of eigt above te ground was 19.6 m/s. Te negative result in tis problem indicates tat te flowerpot is moving downward. c) As te interval becomes smaller, te slope of te secant approaces te tangent at a. Tis value represents te instantaneous rate of cange at tat point. a Slope of Secant 9.8a (1) 4.9(0.01) (1) 4.9(0.001) (3) 4.9(0.01) (3) 4.9(0.01) Flowerpot s Heigt Above te Ground s(t) Heigt (m) t Time (s) From te available information, it appears tat te slope of te secant is approacing 9.8 m/s at 1 s, and 9.4 m/s at 3 s.. a) To determine te equation of te tangent at Flowerpot s Heigt Above t 1, first find te tangent point b substituting te Ground s(t) into te original function. s(1) (1) 85.1 Te tangent point is (1, 85.1). From question 1, te estimated slope at tis point is 9.8. Substitute te slope and te tangent point into te equation of a line formula: 1 m( 1 ). s (t 1) s 9.8t 94.9 Te equation of te tangent at (1, 85.1) is s 9.8t b) Verif te results in part a) using te Tangent operation on a graping calculator. Cange te window settings as sown before taking te steps below. Heigt (m) t Time (s) 18 MHR Calculus and Vectors Capter 1

19 Enter te equation Y Press GRAPH. Press ND PRGM. Coose 5:Tangent(. Enter te tangent point, 1. Te grap and equation of te tangent verif te results. Tecnolog Tip Te standard window settings are [10, 10] for bot te -ais and -ais. Tese window variables can be canged. To access te window settings, press WINDOW. If non-standard window settings are used for a grap in tis tet, te window variables will be sown beside te screen capture. Window variables: [0, 0], [0, 100], Yscl 5 << >> KEY CONCEPTS For a given function f (), te instantaneous rate of cange at a is estimated b calculating te slope of a secant over a ver small interval, a a, were is a ver small number. fa ( ) fa ( ) Te epression, 0 is called te difference quotient. It is used to calculate te slope of te secant between (a, f (a)) and ((a ), f (a )). It generates an increasingl accurate estimate of te slope of te tangent at a as te value of comes closer to 0. A graping calculator can be used to draw a tangent to a curve wen te equation for te function is known. Communicate Your Understanding C1 Wic metod is better for estimating instantaneous rate of cange: an equation or a table of values? Justif our response. C How does canging te value of in te difference quotient bring te slope of te secant closer to te slope of te tangent? Do ou tink tere is a limit to ow small can be? Eplain. C3 Eplain w cannot equal zero in te difference quotient. 1. Rates of Cange Using Equations MHR 19

20 A Practise 1. Determine te average rate of cange from 1 to 4 for eac function. a) b) c) 3 d) 7. Determine te instantaneous rate of cange at for eac function in question Write a difference quotient tat can be used to estimate te slope of te tangent to te function f () at Write a difference quotient tat can be used to estimate te instantaneous rate of cange of at Write a difference quotient tat can be used to obtain an algebraic epression for estimating te slope of te tangent to te function f () 3 at Write a difference quotient tat can be used to estimate te slope of te tangent to f () 3 at Wic statements are true for te difference quotient 41 ( ) 3 4? Justif our answer. Suggest a correction for te false statements. a) Te equation of te function is 4 3. b) Te tangent point occurs at 4. c) Te equation of te function is d) Te epression is valid for 0. B Connect and Appl 8. Refer to our answer to question 4. Estimate te instantaneous rate of cange at 3 as follows: a) Substitute 0.1, 0.01, and into te epression and evaluate. b) Simplif te epression, and ten substitute 0.1, 0.01, and and evaluate. c) Compare our answers from parts a) and b). Wat do ou notice? W does tis make sense? 9. Refer to our answer to question 3. Suppose f () 4. Estimate te slope of te tangent at 4 b first simplifing te epression and ten substituting 0.1, 0.01, and and evaluating. 10. Determine te average rate of cange from 3 to for eac function. a) 3 b) 1 c) 7 4 d) Determine te instantaneous rate of cange at for eac function in question a) Epand and simplif eac difference quotient, and ten evaluate for a 3 and i) ( a) a ii) ( a ) 3 a3 iii) ( a ) 4 a4 b) Wat does eac answer represent? Eplain. 13. Compare eac of te following epressions fa ( ) fa ( ) to te difference quotient, identifing i) te equation of f () ii) te value of a iii) te value of iv) te tangent point (a, f (a)) a) ( 401. ) 16 b) ( ) c) 3 ( 0. 9 ) (.) 16 d) MHR Calculus and Vectors Capter 1

21 14. Use Tecnolog A soccer ball is kicked Representing into te air suc tat Problem Solving its eigt, in metres, Connecting Communicating after time t, in seconds, can be modelled b te function s(t) 4.9t 15t 1. a) Write an epression tat represents te average rate of cange over te interval 1 t 1. b) For wat value of is te epression not valid? Eplain. c) Substitute te following -values into te epression and simplif. i) t 5 ii) t 10 iii) t 15 iv) t 0 Reasoning and Proving Selecting Tools Reflecting i) 0.1 ii) 0.01 iii) iv) d) Use our results in part c) to predict te instantaneous rate of cange of te eigt of te soccer ball after 1 s. Eplain our reasoning. e) Interpret te instantaneous rate of cange for tis situation. f) Use a graping calculator to sketc te curve and te tangent. 15. An oil tank is being drained. Te volume V, in litres, of oil remaining in te tank after time t, in minutes, is represented b te function V(t) 60(5 t), 0 t 5. a) Determine te average rate of cange of volume during te first 10 min, and ten during te last 10 min. Compare tese values, giving reasons for an similarities and differences. b) Determine te instantaneous rate of cange of volume at eac of te following times. Compare tese values, giving reasons for an similarities and differences. c) Sketc a grap to represent te volume, including one secant from part a) and two tangents from part b). 16. As a snowball melts, its surface area and volume decrease. Te surface area, in square centimetres, is modelled b te equation S 4πr, were r is te radius, in centimetres. Te volume, in cubic centimetres, is modelled 17. b te equation V 4 π r3, were r is te 3 radius, in centimetres. a) Determine te average rate of cange of te surface area and of te volume as te radius decreases from 5 cm to 0 cm. b) Determine te instantaneous rate of cange of te surface area and te volume wen te radius is 10 cm. c) Interpret te meaning of our answers in parts a) and b). A dead branc breaks off a tree located at te top of an 80-m-ig cliff. After time t, in seconds, it as fallen a distance d, in metres, were d(t) 80 5t, 0 t 4. a) Determine te average rate of cange of te distance te branc falls over te interval [0, 3]. Eplain wat tis value represents. b) Use a simplified algebraic epression in terms of a and to estimate te instantaneous rate of cange of te distance fallen at eac of te following times. Evaluate wit i) t 0.5 ii) t 1 iii) t 1.5 iv) t v) t.5 vi) t 3 c) Wat do te values found in part b) represent? Eplain. 1. Rates of Cange Using Equations MHR 1

22 18. a) Complete te cart for f () 3 and a tangent at te point were 4. b) Use a graping calculator to grap te curve and te tangent at 6. Tangent Point (a, f (a)) Side Lengt Increment, Second Point (a, f (a )) Slope of Secant fa ( ) fa ( ) b) Wat do te values in te last column indicate about te slope of te tangent? Eplain. 19. Te price of one sare in a tecnolog compan at an time t, in ears, is given b te function P(t) t 16t 3, 0 t 16. a) Determine te average rate of cange of te price of te sares between ears 4 and 1. b) Use a simplified algebraic epression, in terms of a and, were 0.1, 0.01, and 0.001, to estimate te instantaneous rate of cange of te price for eac of te following ears. i) t ii) t 5 iii) t 10 iv) t 13 v) t 15 c) Grap te function. 0. Use Tecnolog Two points, P(1, 1) and Q(, ), lie on te curve. a) Write a simplified epression for te slope of te secant PQ. b) Calculate te slope of te secants wen 1.1, 1.01, 1.001, 0.9, 0.99, 0.99, and c) From our calculations in part b), guess te slope of te tangent at P. d) Use a graping calculator to determine te equation of te tangent at P. e) Grap te curve and te tangent. 1. Use Tecnolog a) For te function, determine te instantaneous rate of cange of wit respect to at 6 b calculating te slopes of te secant lines wen 5.9, 5.99, and 5.999, and wen 6.1, 6.01, and Capter Problem Alicia did some researc on weater penomena. Se discovered tat in parts of Western Canada and te United States, cinook winds often cause sudden and dramatic increases in winter temperatures. A world record was set in Spearfis, Sout Dakota, on Januar, 1943, wen te temperature rose from 0 C (or 4 F) at 7:30 A.M. to 7 C (45 F) at 7:3 A.M., and to 1 C (54 F) b 9:00 A.M. However, b 9:7 A.M. te temperature ad returned to 0 C. a) Draw a grap to represent tis situation. b) Wat does te grap tell ou about te average rate of cange in temperature on tat da? c) Determine te average rate of cange of temperature over tis entire time period. d) Determine an equation tat best fits te data. e) Use te equation found in part d) to write an epression, in terms of a and, tat can be used to estimate te instantaneous rate of cange of te temperature. f) Use te epression in part e) to estimate te instantaneous rate of cange of temperature at eac time. i) 7:3 A.M. ii) 8 A.M. iii) 8:45 A.M. iv) 9:15 A.M. g) Compare te values found in part c) and part f). Wic value do ou tink best represents te impact of te cinook wind? Justif our answer. MHR Calculus and Vectors Capter 1

23 3. As water drains out of a 50-L ot tub, te amount of water remaining in te tub is represented b te function V(t) 0.1(150 t), were V is te volume of water, in litres, remaining in te tub, and t is time, in minutes, 0 t 150. C Etend and Callenge 4. Use Tecnolog For eac of te following functions, i) Determine te average rate of cange of wit respect to over te interval from 9 to 16. ii) Estimate te instantaneous rate of cange of wit respect to at 9. iii) Sketc a grap of te function wit te secant and te tangent. a) b) 4 c) 7 d) 5 5. Use Tecnolog For eac of te following functions, i) Determine te average rate of cange of wit respect to over te interval from 5 to 8. ii) Estimate te instantaneous rate of cange of wit respect to at 7. iii) Sketc a grap of te function wit te secant and te tangent. a) b) 1 c) d) 1 6. Use Tecnolog For eac of te following functions, i) Determine te average rate of cange of wit respect to θ over te interval from θ π 6 to θ π 3. ii) Estimate te instantaneous rate of cange of wit respect to θ at θ π 4. a) Determine te average rate of cange of te volume of water during te first 60 min, and ten during te last 30 min. b) Use two different metods to determine te instantaneous rate of cange in te volume of water after 75 min. c) Sketc a grap of te function and te tangent at t 75 min. iii) Sketc a grap of te function wit te secant and te tangent. a) sin θ b) cos θ c) tan θ 7. a) Predict te average rate of cange for te function f () c, were c is an real number, for an interval a b. b) Support our prediction wit an eample. c) Justif our prediction using a difference quotient. d) Predict te instantaneous rate of cange of f () c at a. e) Justif our prediction. 8. a) Predict te average rate of cange of a linear function m b for an interval a b. b) Support our prediction wit an eample. c) Justif our prediction using a difference quotient. d) Predict te instantaneous rate of cange of m b at a. e) Justif our prediction. 9. Determine te equation of te line tat is perpendicular to te tangent to 5 at, and wic passes troug te tangent point. 30. Mat Contest Solve for all real values of given tat Mat Contest If ab 135 and log 3 a log 3 b 3, determine te value of log 3 ( a b ). 1. Rates of Cange Using Equations MHR 3

24 1.3 Limits Te Greek matematician Arcimedes (c B.C.) developed a proof of te formula for te area of a circle, A πr. His metod, known as te metod of eaustion, involved calculating te area of regular polgons (meaning teir sides are equal) tat were inscribed in te circle. Tis means tat te were drawn inside te circle suc tat teir vertices touced te circumference, as sown in te diagram. Te area of te polgon provided an estimate of te area of te circle. As Arcimedes increased te number of sides of te polgon, its sape came closer to te sape of a circle. For eample, as sown ere, an octagon provides a muc better estimate of te area of a circle tan a square does. Te area of a eadecagon, a polgon wit 16 sides, would provide an even better estimate, and so on. Wat about a mriagon, a polgon wit sides? Wat appens to te estimate as te number of sides approaces infinit? Arcimedes metod of finding te area of a circle is based on te concept of a limit. Te circle is te limiting sape of te polgon. As te number of sides gets larger, te area of te polgon approaces its limit, te sape of a circle, witout ever becoming an actual circle. In Section 1., ou used a similar strateg to estimate te instantaneous rate of cange of a function at a single point. Your estimate became increasingl accurate as te interval between two points was made smaller. Using limits, te interval can be made infinitel small, approacing zero. As tis appens, te slope of te secant approaces its limiting value te slope of te tangent. In tis section, ou will eplore limits and metods for calculating tem. Investigate A How can ou determine te limit of a sequence? Tools graping calculator Optional Fatom CONNECTIONS An infinite sequence sometimes as a limiting value, L. Tis means tat as n gets larger, te terms of te sequence, t n, get closer to L. Anoter wa of saing as n gets larger is, as n approaces infinit. Tis can be written n. Te smbol does not represent a particular number, but it ma be elpful to tink of as a ver large positive number. 1. Eamine te terms of te infinite sequence 1, , 1 100, 1000, ,. Te general term of tis sequence is t n 1. Wat appens to te value 10 n of eac term as n increases and te denominator becomes larger?. Wat is te value of lim tn (read te limit of t n as n approaces n infinit. )? W can ou ten sa tat its limit eists? 1 3. Plot ordered pairs, n, n, tat correspond to te sequence. Describe 10 ow te grap confirms our answer in step. 4. Reflect Eplain w lim n reaced. tn epresses a value tat is approaced, but not 5. Reflect Eamine te terms of te infinite sequence 1, 4, 9, 16, 5, 36,, n,. Eplain w lim t does not eist for tis sequence. n n 4 MHR Calculus and Vectors Capter 1

25 In te development of te formula A πr, Arcimedes not onl approaced te area of te circle from te inside, but from te outside as well. He calculated te area of a regular polgon tat circumscribed te circle, meaning tat it surrounded te circle, wit eac of its sides toucing te circle s circumference. As te number of sides to te polgon was increased, its sape and area became closer to tat of a circle. Arcimedes approac can also be applied to determining te limit of a function. A limit can be approaced from te left side and from te rigt side, called left-and limits and rigt-and limits. To evaluate a left-and limit, we use values tat are smaller tan, or on te left side of te value being approaced. To evaluate a rigt-and limit we use values tat are larger tan, or on te rigt side of te value being approaced. In eiter case, te value is ver close to te approaced value. Investigate B How can ou determine te limit of a function from its equation? 1. a) Cop and complete te table for te function b) Eamine te values in te table tat are to te left of 3, beginning wit. Wat value is approacing as gets closer to, or approaces, 3 from te left? c) Beginning at 4, wat value is approacing as approaces 3 from te rigt? d) Reflect Compare te values ou determined for in parts b) and c). Wat do ou notice?. a) Grap. b) Press te TRACE ke and trace along te curve toward 3 from te left. Wat value does approac as approaces 3? c) Use TRACE to trace along te curve toward 3 from te rigt. Wat value does approac as approaces 3? d) Reflect How does te grap support our results in question 1? 3. Reflect Te value tat approaces as approaces 3 is te limit of te function as approaces 3, written as lim( ). Does it 3 make sense to sa, te limit of eists at 3? Tools graping calculator Optional Fatom computer wit Te Geometer s Sketcpad CONNECTIONS To see an animation of step of tis Investigate, go to calculus1, and follow te links to Section Limits MHR 5

26 It was stated earlier tat te limit eists if te sequence approaces a single value. More accuratel, te limit of a function eists at a point if bot te rigt-and and left-and limits eist and te bot approac te same value. lim f ( ) eists if te following tree criteria are met: a 1. lim f ( ) eists a. lim f ( ) eists a 3. lim f ( ) lim f ( ) a a Investigate C How can ou determine te limit of a function from a given grap? CONNECTIONS To see an animated eample of two-sided limits, go to www. mcgrawill.ca/links/calculus1, and follow te links to Section a) Place our fingertip on te grap at 6 and trace te grap approacing 5 from te left. State te -value tat is being approaced. Tis is te left-and limit. b) Place our fingertip on te grap corresponding to 9, and trace te grap approacing 5 from te rigt. State te -value tat is being approaced. Tis is te rigt-and limit. c) Reflect Wat does te value f (5) represent for tis curve? Reflect Trace te entire curve wit our finger. W would it make sense to refer to a curve like tis as continuous? Eplain w all polnomial functions would be continuous. 3. Reflect Eplain te definition of continuous provided in te bo below A function f () is continuous at a number a if te following tree conditions are satisfied: 1. f (a) is defined. lim f ( ) eists a 3. lim f ( ) fa ( ) a 6 MHR Calculus and Vectors Capter 1

27 A continuous function is a function tat is continuous at, for all values of. Informall, a function is continuous if ou can draw its grap witout lifting our pencil. If te curve as oles or gaps, it is discontinuous, or as a discontinuit, at te point at wic te gap occurs. You cannot draw tis function witout lifting our pencil. You will eplore discontinuous functions in Section 1.4. Eample 1 Appl Limits to Analse te End Beaviour of a Sequence For eac of te following sequences, i) state te limit, if it eists. If it does not eist, eplain w. Use a grap to support our answer. ii) write a limit epression to represent te end beaviour of te sequence. a) 1 3, 1397,,,,,3 n, b) n,,,,,,, n 1 Solution a) i) Eamine te terms of te sequence 1, ,,,,, 3 n,. Since te terms are increasing and not converging to a value, te sequence does not ave a limit. Tis fact is verified b te grap obtained b plotting te points ( nt, n ) : 1, 1, ( 1, ), ( 33, ), ( 49, ), ( 57, ), ( 681, ),. 3 ii) Te end beaviour of te sequence is represented b te limit epression lim 3n. Te infinit smbol indicates tat te terms n of te sequence are becoming larger positive values, or increasing witout bound, and so te limit does not eist. b) i) Eamine te terms of te sequence n,,,,,,, n 1 t n Convert eac term to a decimal, te sequence becomes 0.5, 0.67, 0.75, 0.8, 0.83,. Te net tree terms of te sequence are 6 7,, and 8 0.8, or 0.857, 0.875, and Toug te terms are increasing, te are not increasing witout bound 0.4 te appear to be approacing 1 as n becomes larger. n 0. Tis can be verified b determining tn for large values n of n. t and t Te value t n n n 1.3 Limits MHR 7

28 of tis function will never become larger tan 1 because te value of te numerator for tis function is alwas one less tan te value of te denominator. ii) Te end beaviour of te terms of te sequence is represented b te n limit epression lim n n 1 1. Eample Analse a Grap to Evaluate te Limit of a Function Determine te following for te grap below. a) lim f ( ) 4 b) lim f ( ) 4 c) lim f ( ) 4 d) f (4) f( ) Solution a) lim f ( ) refers to te limit as approaces 4 from te left. Tracing 4 along te grap from te left, ou will see tat te -value tat is being approaced at 4 is. b) lim f ( ) refers to te limit as approaces 4 from te rigt. Tracing 4 along te grap from te rigt, ou will see tat te -value tat is being approaced at 4 is. c) Bot te left-and and rigt-and limits equal, tus, lim f ( ). 4 Also, te conditions for continuit are satisfied, so f () is continuous at. d) f (4). 8 MHR Calculus and Vectors Capter 1