Calculus and Vectors. Day Lesson Title Math Learning Goals Expectations

Transcription

1 Unit Eploring Derivatives Calculus and Vectors Lesson Outline Day Lesson Title Mat Learning Goals Epectations 1 Key caracteristics of instantaneous rates of cange (TIPS4RM Lesson) Determine intervals in order to identify increasing, decreasing, and zero rates of cange using grapical and numerical representations of polynomial functions Describe te beaviour of te instantaneous rate of cange at and between local maima and minima A.1 Patterns in te Derivative of Polynomial Functions (TIPSRM Lesson) 3 Derivatives of Polynomial Functions (Sample Lesson Included) 4 Patterns in te Derivative of Sinusoidal Functions (Sample Lesson Included) 5 Patterns in te Derivative of Eponential Functions (Sample Lesson Included) 6 Identify e (Sample Lesson Included) 7 Relating f()= ln() and f ( ) = e (Sample Lesson Included) Use numerical and grapical representations to define and eplore te derivative function of a polynomial function wit tecnology, Make connections between te graps of te derivative function and te function A. Determine, using limits, te algebraic representation of te derivative of polynomial functions at any point A.3 Use patterning and reasoning to investigate connections grapically and numerically between te graps of f() = sin(), f() = cos(), and teir derivatives using tecnology Determine te grap of te derivative of f() = a using tecnology Investigate connections between te grap of f() = a and its derivative using tecnology Investigate connections between an eponential function wose grap is te same as its derivative using tecnology and recognize te significance of tis result Make connections between te natural logaritm function and te function f ( ) = e Make connections between te inverse relation of f() = ln() and f ( ) = e A.4 A.5 A.6 A.7 Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 1 of 37

2 Day Lesson Title Mat Learning Goals Epectations 8 Verify derivatives of eponential functions (Sample Lesson Verify te derivative of te eponential function f()=a is f ()=a ln a for various values of a, using tecnology Included) 9 Jazz Day / Summative Assessment (Sample Assessment Included) 10, Power Rule Verify te power rule for functions of te form f() = n (were n is a natural number) 11 Verify te power rule applies to functions wit rational eponents Verify numerically and grapically, and read and interpret proofs involving limits, of te constant, constant multiple, sums, and difference rules 1 Solve Problems Determine te derivatives of polynomial functions Involving Te algebraically, and use tese to solve problems Power Rule involving rates of cange A.8 A3.1, A3. A3.4 A3.3 13, 14, 15 Eplore and Apply te Product Rule and te Cain Rule Verify te cain rule and product rule Solve problems involving te Product Rule and Cain Rule and develop algebraic facility were appropriate A3.4 A3.5 16, 17 Connections to Rational and Radical Functions (Sample Lessons Included) Use te Product Rule and Cain Rule to determine derivatives of rational and radical functions Solve problems involving rates of cange for rational and radical functions and develop algebraic facility were appropriate A3.4 A3.5 18, 19 Applications of Derivatives 0 Jazz Day 1 Summative Assessment Pose and solve problems in contet involving instantaneous rates of cange A3.5 Note: TIPS4RM Lesson refers to a lesson developed by writing teams funded by te Ministry of Education. Tese lessons are not included wit tis package. Tey will be available at a later date. Details will be posted on te OAME web site. ( Note: Te assessment on day 9 is available from te member area of te OAME website and from te OMCA website ( Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page of 37

3 Unit : Day 3 Algebraic Representation of te Derivative of Polynomial Functions Minds On: 5 Action: 55 Consolidate:10 Learning Goal: Determine te derivatives of polynomial functions by simplifying te algebraic epression f ( + )! f ( ) epression as '' approaces zero. and ten taking te limit of te simplified MCV4U Materials Graping calculators BLM.3.1 BLM.3. Total=75 min Assessment Opportunities Minds On Pairs Tink/Pair/Sare Students ave 1 minute to tink about and record teir prior knowledge of average rates of cange and approimations of instantaneous rates of cange and ow tey relate to secant lines and tangent lines. Tey sould ten pair wit a partner and sare, editing teir list during te discussion. Action! Consolidate Debrief Wole Class Discussion Ask select students to sare one point teir partner ad. Groups Investigation In eterogeneous groups of 3 or 4, students complete BLM.3.1. Wole Class Debrief Have students eplain, in teir own words, te relationsip between te grap of a function and te grap of its derivative. Summarize te student tougts on te blackboard. Process Epectation/Oral Questions/Mental Note As students eplain teir reasoning and respond to questions, assess teir ability to make connections. Matematical Process Focus: Connecting Wole Class Teacer Led Discussion Using BLM.3. as a guide, demonstrate ow te first principles definition, f ( + )! lim " 0 functions. f ( ), can determine te derivative function for polynomial Groups Discussion In eterogeneous groups of 3 or 4, students discuss te meaning of a derivative at a given -value. Students debate te most appropriate tecnique for calculating derivatives of polynomial functions (difference of squares for even powered polynomials, difference of cubes for polynomials wit powers tat are multiples of 3, binomial epansion). Students can be assigned varying values of for comparison, or coose teir own. By storing te varying '' values in te different lists in te graping calculators, te calculations can be done very quickly. Concept Practice Home Activity or Furter Classroom Consolidation 1) Determine te derivatives of f() = 3, 4, 5 using first principles by factoring and by epanding. ) Determine te derivative of 4 3 using te most appropriate metod. 3) If f() = 3 5, find f / () and eplain its meaning. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 3 of 37

4 .3.1 Connections Between te Grap of f() and f / (). 1) Complete te following table. f( + )! f() ) Before completing te final column, vary te value of '' in te epression ten complete te table by ypotesizing a value for f / () as '' approaces zero. and f() = (, f()) f(+) f( + )! f() f / () = f( + ) lim " 0! f() ) Grap f() =. 4) How do te values in te final column of te table relate to te grap of f()? 5) Sketc te tangents to f() at eac point plotted on te grap. 6) By using te -coordinates given, and te entries in te final column of te table, grap te derivative of f() on a separate grid. 7) Find te equation of f / () from te grap. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 4 of 37

5 .3. Algebraic connection between f() and f / () In order to verify your conclusion from.3.1 algebraically, consider: 1) f() =. ) f(+) =. 3) so, witout simplifying f( + ) lim " 0! f() =. 4) Now, simplify te epression by: a) factoring te difference of squares in te numerator, simplifying and ten reducing. f( + ) lim " 0! f() = = = b) epanding te numerator, simplifying te numerator and ten reducing. f( + )! f() lim " 0 = = = 5) If f() = 3 : a) find f / (). b) Wat is te value of f / (4). c) Wat does f / (4) mean grapically? Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 5 of 37

6 Unit : Day 4: Patterns in te Derivative of Sinusoidal Functions Minds On: 10 Action: 50 Consolidate:15 Total=75 min Minds On Learning Goal: Determine, troug investigation using tecnology, te grap of te dy derivative f / () or of a given sinusoidal function. (A.4) d Wole Class Discussion Using graping calculators students review ow to find te derivative of a polynomial function grapically and algebraically. Sare te procedure to find te slope at a point: 1) Enter y = into y 1 = ) Use a standard window 3) Grap y = 4) Trace! 3 enter dy 5) Calc!! 3 enter, repeat for 4 and 5 d MCV4U Materials cart papergraping calculator BLM.4.1 BLM.4. Assessment Opportunities Action! Consolidate Debrief Students discuss wat te graping calculator tells tem about steps 4 and 5 for y =. (e.g., you ave now generated te slopes at = 3, 4 and 5). Small Groups Eperiment In eterogenous groups of 3 or 4 students will use graping calculators to complete BLM.4.1. As students work on BLM.4.1, tey sould consider te guiding question Are tere any patterns or similarities in te values of y = sin(), y =cos(), and teir derivatives? Process Epectation/Observation/Ceckbric Observe students as tey work and assess teir ability to reason and prove as tey make connections between te grap and te derivative. Matematical Process Focus: Reasoning and Proving, Selecting Tools and Computational Strategies Small Group Summarizing Wit teir group mates, students prepare a summary of teir response to te guiding question on cart paper. Wole Class Presentation Groups sare teir summaries. Students can redraw te graps from te BLM s and/or plot points A- M on te grap. Students continue to add to teir understanding of trigonometric graps and radian measures and relationsips between slopes and instantaneous rates of cange. Application Home Activity or Furter Classroom Consolidation Complete BLM.4. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 6 of 37

7 .4.1 Looking for te Derivative of Sine and Cosine 1. Te grap of f() = sin() is sown. a) Complete te following cart (correct to 3 decimal places). y = sin() 1.5 y 1.0 b) Sketc te derivative of f() = sin() on te grap. 0.5!"!5"/6!"/3!"/!"/3!"/6 "/6 "/3 "/ "/3 5"/6 " 7"/6 4"/3 3"/ 5"/3 11"/6 "!0.5!1.0!1.5!.0 A B C D E F G H I J K L M (radians) 0! 6! 3!! 3 5! 6! 7! 6 4! 3 3! 5! 3 11! 6! F () F / () c) How does te grap of f () compare to te grap of f()? Describe all similarities and differences tat you observe. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 7 of 37

8 .4.1 Looking for te Derivative of Sine and Cosine (Continued). Te grap of f() = cos() is sown. y = cos().0 y a) Complete te following cart (correct to 3 decimal places) b) Draw te derivative of f() = cos() on te grap.!"/!"/3!"/6 "/6 "/3 "/ "/3 5"/6 " 7"/6 4"/3 3"/ 5"/3 11"/6 " 13"/6 7"/3 5"/!0.5!1.0!1.5!.0 A B C D E F G H I J K L M (radians) 0! 6! 3!! 3 5! 6! 7! 6 4! 3 3! 5! 3 11! 6! f () F / () c) How does te grap of f () compare to te grap of f()? Describe all similarities and differences tat you observe. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 8 of 37

9 .4.1 Looking for te Derivative of Sine and Cosine (Continued) 3. Wat is te derivative of f() = sin()? 4. Wat is te derivative of f() = cos()? 5. Summarize te grapical and numeric connections you found between te two functions. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 9 of 37

10 .4.: Sinusoidal Problems 1. Determine and interpret f(1.037) and f / (1.037) 3! 3!. Determine and interpret f( ) and f / ( ) An object moves so tat at time t its position s is found using s(t) = 5 cos(t). a) For wat values of 't' does te object cange direction? b) Wat is its maimum velocity? c) Wat is its minimum distance from (0,1)? 4. Are tere any numbers, 0 " "!, for wic tangents to f() = sin() and f() = cos() are parallel? If so, find te values. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 10 of 37

11 Unit : Day 5: Patterns in te Derivative of Eponential Function MCV4U Minds On: 5 Action: 35 Consolidate:15 Total=75 min Minds On Action! Learning Goals: Investigate connections between te grap of f()=a and its derivative using tecnology Eplore te ratio of f ()/f(). Pairs Activity Students work in pairs to complete BLM.5.1. Wole Class Discussion Pairs sare teir solutions to BLM.5.1 Ask all students to compare te equation of a polynomial function f()= a to te equation of te eponential function f()=a and consider similarities and differences. Ask for suggestions about possible ways to differentiate te eponential function. Review te process of finding te derivative using limits. Small Groups Guided Eploration In eterogeneous groups of 3 or 4, students complete BLM.5... Students may refer to BLM.5.4 for graping calculator instructions. Curriculum Epectations/Oral Questions/Mental Note Assess student understanding of unit epectations wit oral questions. Address common misconceptions immediately and during te consolidation time. Materials graping calculators BLM Computer and data projector BLM.5.4 (optional) Assessment Opportunities Notes: A common misconception is to consider te polynomial and eponential functions to be similar. An alternative lesson is a wole class guided eploration, using TI Emulator or te GSP diagram R.5.a numeric.gsp). Teacer notes for BLM.5. are included on BLM.5.3. Matematical Process Focus: Connecting, Representing Practice Consolidate Debrief Wole Class Discussion, Saring, Completing Notes Generalize student findings from te investigation activities to develop a metod to calculate te slope of te tangents to te grap of f() at a given point. Apply te student generated metod to some eamples and revise as necessary. Home Activity or Furter Classroom Consolidation Select practice questions from (teacer generated list). Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 11 of 37

12 .5.1 Revisiting te Eponential Function Function defined by an equation f() = a, were a is a real positive number, is called an Eponential Function. Tere is a significant difference between te graps and te properties of te function wen a>1 and wen 0<a<1. Case 1: f()=a, a>1. Domain: Range: Asymptote: Te function is over te wole domain Case : f()=a, 0<a<1. Domain: Range: Asymptote: Te function is over te wole domain Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 1 of 37

13 .5.: Eponential Derivatives Use a graping calculator to complete tese tasks. For eac of te functions below: 1. Grap te function from its equation.. Create a table of values for, f(), and f (), for at least tree values of (-1, 0, 1 is a suggested but not a mandatory coice). f!( ) 3. Calculate te ratio. f ( ) 4. Draw te tangent to te grap of f() at =0. Determine te slope of te tangent. 5. Summarize your findings. f()= 4 Slope of te tangent at =0:. f (0)=.. Conclusion: f()=.5 Slope of te tangent at =0:. f (0)=.. Conclusion: Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 13 of 37

14 .5.: Eponential Derivatives (Continued) f()= 0. Slope of te tangent at =0:. f (0)=.. Conclusion: In General: Record wat you noticed about eac of te following: 1. Te derivative of te eponential function f()=a. Te ratio f!( ) f ( ) 3. Te relationsip between f() and f () 4. Te slope of te tangent at = 0 Eamples: 1. Calculate te slope of te tangent to f()=0. at =1.5, using ONLY te data from te activity. f!( ). For an eponential function f() te ratio =1.946, determine te equation of te tangent f ( ) line to te grap of te function at te point (0.5,.646). Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 14 of 37

15 .5.3: Eponential Derivatives (Teacer Notes) Minds On: Recall te Limit Definition of te Derivative: f "( ) = lim # 0 f "( + )! f ( ) Apply to te eponential function, b f "( ) = lim # 0 b b! b = lim # 0 b ( b! 1) = lim # 0 = b +! b b! 1 lim # 0 f ( ) = b So te derivative of te eponential function f "( ) = b 1 b! lim # 0 f ( ) = b is te factor lim " 0 b! 1 multiplied by te original function. 0 b! 1 Note tat f "(0) = b lim and 0 b! 1 b b = 1, so f "(0) = lim. In oter words lim # 0 # 0 " 0 slope of te curve at te y-intercept Notes: 1. Based on te results from te table, suggest a function tat represents f (). EXPONENTIAL or f ()=c.a +b.! 1 is te 1. Use te data from te table to calculate te ratio f ()/f(): X f ()/f() Wat do you observe? THE RATIO IS CONSTANT AND EQUALS THE VALUE OF f (0). In General: (possible responses) 1. Te derivative of te eponential function f()=a is an eponential function.. For eac eponential function f(), te ratio f ()/f() is constant, ence, te derivative function f () relates to f() according to te equation f ()=f().const. 3. Te slope of te tangent at =0 equals te ratio f ()/f() (or te constant from te equation) and depends on te base of te function, a. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 15 of 37

16 .5.4 HOW TO Hints for TI Graping Calculators Calculate f () for a specific value of 1. Enter te equation of f() in te Y= editor.. Grap te function. 3. From CALC menu ( ND, TRACE), select 6: dy/d. 4. Type in te value of, were you need te value of te tangent s slope. 5. Press ENTER. Plot points (, f ()) 1. Press STAT -> EDIT->1, ENTER. Enter te values of in L1 and te values of f () in L.. In STAT PLOT ( ND, Y=), turn PLOT1 ON and coose te first TYPE (scatter plot) to represent te points making sure Xlist is L 1 and YList is L 3.. Plot te points by pressing GRAPH. Use te regression modelling feature to find a function tat approimately represents f (). 1. After te points are plotted, enter te STAT CALC menu (STAT->CALC).. Select an appropriate regression model, ten press ENTER. Its name appears on te screen. 3. Following te name of te model, enter te name of te Xlist (L1), Ylist (L), and te function (Y) were te regression equation is to be stored (VARS->Y-VARS, 1:Function, ENTER, :Y). 4. Te regression equation can be seen in te Y= editor. To grap te regression function, make sure te = sign for Y in te Y= editor is igligted, ten press GRAPH. Construct table of values from an equation of f() 1. Enter te equation of f() in te Y= editor.. Enter te TABLE SETUP menu ( ND, WINDOW). Set te initial value (TblStart) and te increment value (ΔTbl) for te independent variable. Leave te parameters Indpnt and Depend to Auto. 3. Press ND, GRAPH to construct te table of values. Draw a tangent to te grap of f() at a given point 1. Enter te equation of f() in te Y= editor.. Grap te function. 3. Enter te DRAW menu ( ND, PRGM) and select 5:Tangent. On te grap of te function, a point appears, and its coordinates are sown below te grap. To select anoter point, just type in te desired value of. Te tangent line and its equation appear on te screen. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 16 of 37

17 Unit : Day 6: Identifying e Minds On: 15 Action: 55 Consolidate: 5 Total=75 min Mat Learning Goals Investigate connections between an eponential function wose grap is te same as its derivative using tecnology Determine an approimation of te number e and its significance MCV4U Materials Graping Calculators Computer and data projector BLM BLM.6.4 (Optional) Minds On Wole Class Discussion Demonstrate using te GSP sketc te notion of te derivative of an eponential function being an eponential function wit te same base as te original multiplied by a factor. Assessment Opportunities Students could investigate in pairs. Pairs Drawing Tangents Students complete BLM.6.1 in pairs cecking teir accuracy wit a partner. Tey sould sketc te graps and draw te tangent lines as accurately as possible. Action! Consolidate Debrief Wole Class Discussion Discuss te derivative of te eponential function using te limit definition. and making connections to BLM.6.1. Pairs Evaluating Limits Students complete te cart on BLM.6. in pairs cecking teir metod wit a partner. Wole Class Discussion Discuss te approimation of te value of te base of te eponential function aving a slope of 1 at te y-intercept. Pairs Evaluating Limits Students complete of te conclusions on BLM.6. in pairs, discussing teir responses wit teir partner. Learning Skills/Observation/Anecdoctal Observe students in order to assess teir teamwork and work abits. Provide anecdotal feedback as appropriate. Matematical Process Focus: Reflecting, Reasoning and Proving, Connecting Wole Class Discussion Sow te grap of y = e empasizing te fact tat te curve is as steep as te y-value at any point. Higligt te many applications of tis beaviour, including population growt, investments, appreciation and depreciation of value, radioactive decay, and cooling and warming. *Te limit definition of te derivative is required in A.8. Its discussion ere makes a good connection. By adjusting te base, students sould determine tat tere is a particular base for wic te derivative is identical to te original function, and its value is appro..7. BLM.6.3 provides some etensions on e. BLM.6.4 provides teacer notes. Application Home Activity or Furter Classroom Consolidation Create transformations of. f ( ) = e and draw te corresponding grap. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 17 of 37

18 .6.1 Introducing te number e Problem: How steep is te tangent line to te grap of an eponential function at its y- intercept? For eac function: a) draw te grap b) draw te tangent line to te grap at te y-intercept of te grap c) use your calculator to determine te slope of te tangent line to te grap at te y- intercept. 1. y = dy d = 0 =. y = 3 dy d = 0 = Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 18 of 37

19 .6. Te value of te limit b -1 lim! 0 Test te limit on various values of b using your calculator or a spreadseet: a) Let b = and evaluate! 1 lim " 0 H b! 1 b) Let b = 3 and 3! 1 evaluate lim " 0 H b! c) Guess a value for b so tat te limit is as close to 1 as possible. b!1 b = lim " 0 b! 1 Conclusions: a) If f ( ) = ten f!() = b) If f ( ) = 3 ten f!() = f ( ) = ten f!() = & c) If ( ) Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 19 of 37

20 .6.3 Etensions Using a scientific calculator, evaluate tese epressions: i) coose a large value and substitute it into te epression = & 1 # $ 1 +! = % " & 1! # $ 1 + % " ii) coose a value of very close to 0 and substitute it into te epression ( ) = = ( 1 ) iii) at least 7 terms of te series !! 3! number of terms: sum: Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 0 of 37

21 .6.4 Etensions (Teacer Notes)! 1 Te value of b tat makes lim = 1 is called Euler s Number and is represented by te 0 b " symbol e. Tis special, irrational number is equivalent to te following fundamental limits and te infinite series: i) 1 # e = $ 1 +! ( '& % " lim ii) ( ) e = lim 1+! 0 Using a scientific calculator, evaluate tese epressions: i) coose a large value and substitute it into te epression = 100! 1 " # 1+ =.7048 $ % & iii) e = !! 3! & 1! # $ 1 + % " ii) coose a value of very close to 0 and substitute it into te epression ( ) = ( ) 1 1+ = iii) at least 7 terms of te series !! 3! number of terms: 10 sum: Euler determined tat te value of e = wic does not repeat and does not terminate. You can use your calculator to sow tis value wit te e key.. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 1 of 37

22 Unit : Day 7: Te Natural Logaritmic Function Minds On: 10 Action: 50 Consolidate: 15 Total = 75 min Learning Goal: Make connections between te natural logaritm function and te function f()=e. Make connections between te inverse relation of f() = ln and f() = e including compositions of te two functions. Minds On Individual Investigation Ask students to create a function and determine its inverse using a graping calculator or oter metod of coice. Wole Class Discussion Students sould sare teir function and inverse wit te class and discuss ow te grap of a function and te grap of its inverse function are related. Pose te guiding question for tis lesson: Does te inverse of te natural eponential function eist and, if yes, wat is it? Have students sare teir immediate reactions to tis question. Action! Pairs Investigation Students complete BLM.7.1. Matematical Process Focus: Connecting, Reflecting Process Epectations/Observation/Mental Note Observe students ability to reflect and connect during tis activity. Higligt strong skills during te debrief. Wole Class Discussion Have students sare teir response to te guiding question. Troug discussion of student ideas, confirm tat log e, or ln is te inverse of e, just as log a was determined to be te inverse of a in MHF4U. Pairs Consolidating concepts Students work on BLM.7.. MCV4U Materials Graping calculators Ruler BLM.7.1 BLM.7. Assessment Opportunities Consolidate Debrief Wole class Discussion Have students summarize wat tey ave learned from te activities using a grapic organizer. 1 Application Concept Practice Home Activity or Furter Classroom Consolidation Complete BLM.7.3. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page of 37

23 .7.1 It Is So Natural! 1. A table of values is given for f()=e. On te grid, grap te following: (i) f()=e (ii) f() = (te line y = ) (iii) te reflection of te grap of f()=e in te line y =. (Hint: For eac point (a, b) on te grap of f(), te point (b, a) is its reflection image on te line y =.) Coose appropriate scales for te aes. X e (to te nearest undredt) Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 3 of 37

24 .7.1 It Is So Natural! (Continued) 3. Does te grap of te reflection appear to represent a function? Eplain. 4. Denote grap of te reflection g() and label te graps accordingly. Complete te table below to compare te two graps: f() g() Domain Range Asymptote -intercept(s) y-intercept(s) Intervals were te function increases Intervals were te function increases 5. Does te inverse of te natural eponential function eist and, if yes, wat is it? Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 4 of 37

25 .7.: Te Inverse Relationsip: e and ln! 1. Use te Inverse Function feature of te graping calculator to sow tat f() = e and g() = ln are inverse functions. Eac partner needs a calculator. Partner 1 Partner Enter f() = e as Y1 in te Y= editor. Enter f() = ln() as Y1 in te Y= editor. Grap it. Grap it. From te DRAW menu ( ND, PRGM), From te DRAW menu ( ND, PRGM), select 8: DrawInv. select 8: DrawInv. Enter te parameter Y1 Enter te parameter Y1 (VARS->Y-VARS; 1: (VARS->Y-VARS; 1: Function; 1:Y1). Function; 1:Y1). Press ENTER Press ENTER Compare your graps wit your partner. Eplain to your partner ow you constructed your grap. How do te graps of f() = e and its inverse and g() = ln and its inverse compare?. Since f() = e and g() = ln are inverse functions, ten f(g()) = g(f()) =. Tis is called te Identity Property for inverse functions. a) Wat is e ln? On wat domain is e ln defined? b) Wat is ln (e )? On wat domain is ln (e )? defined? 3. Verifying te Identity properties using a graping calculator. ln a) Enter te function f ( ) = e in te Y= editor. Grap it. How does te function relate to te function f() =? (Hint: consider te domain) b) Repeat part a) for f ( ) = ln( e ) Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 5 of 37

26 .7.3 It Runs in te Family! Te natural eponential function f() =e belongs to te family of eponential functions f() =a, were a>1. Similarly, te natural logaritmic function f() =ln belongs to te broader family of logaritmic functions f ( ) = log were a>1. In fact, 1. ln = log. ln = log e = 1 e e e a 3. y = ln is te inverse of = e y te same way as y = log is te inverse of =a y. As a consequence, e ln = lne =. Laws of Logaritms applied to Natural Logaritms Recall tat for any real numbers a, b,, y>0, te following LOG RULES old: Log of a Product loga ( y) = loga + loga y Rule! " log Log of a Quotient a $ % = loga # loga y & y ' Rule y loga = yloga Log of a Power Rule a Complete te table wit tese rules wen a=e: LN of a Product Rule LN of a Quotient Rule LN of a Power Rule 1. Calculate (if possible) ln for =3; 0.31; -; 0. Round to te nearest tousandt.. Compare te values of te epressions A= ln e!, B= ln( e! ), C= ln( e + ) and D = ln( e ). Specify wic of te Natural Logaritm Rules are applicable in eac case. Evaluate te following witout tecnology. 1. Find ln( e ). ' 1 $ ln Find ln% " + e! 3ln e. & e # 3. Simplify te following epressions: ln a. ln( e! 5 e ) b. ln(ln e) e!!007 ln1 c. ln ( e ) 008 Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 6 of 37

27 Unit : Day 8: Verify Derivatives of Eponential Functions Minds On: 15 Action: 45 Consolidate:15 Total=75 min Learning Goal: Verify, using tecnology, tat te derivative of te eponential function f() = a is f / () = a lna for various values of a. Assessment Opportunities Minds On Pairs Brainstorm Using cart paper, students will create a grapical organizer (e.g., mind map, concept circle) wic communicates te grapical and algebraic connections between f () = a and its derivative, and between f () = e and its derivative. Upon completion, tese sould be posted around te room. MCV4U Materials Graping Calculators Computer and data projector BLM.8.1 Cart paper Action! Consolidate Debrief Concept Practice Application Wole Class Gallery Walk Students sould circulate and review at least two oter grapical organizers.. Small Groups Investigation In eterogeneous groups of 3 or 4, students use te graping calculator and BLM.8.1 to eplore te caracteristics of an eponential function and its derivative. Different values of can be assigned to different groups to determine if varying as an effect on te outcome. Curriculum Epectations/Observation/Cecklist Assess student understanding of te concept of derivative as developed to tis point in te unit. Matematical Process Focus: Reasoning and Proving Wole Class Discussion Have students sare teir conclusions. Ask students ow tey migt verify tese conclusions using te First Principles definition. Use GSP sketc MCV_U3L8_GSP.gsp to verify te derivative for eponential functions. Summarize te derivative of an eponential function and its properties. Home Activity or Furter Classroom Consolidation 1) Prepare a summary of te concepts learned so far in te unit for te summative assessment net class ) Find te derivative of f () = 5, f () = (5), f () = 5(5) 3) If f () = 4, find f / (4) and eplain its relation to te function. 4) Verify numerically using te First Principles Definition tat te derivative of f () = a is f / () = a ln(a). Students can also find f / () by using te nderiv( function on te graping calculator in te MATH menu. Te synta is: nderiv(epression, variable, value) i.e. nderiv(, X, -) Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 7 of 37

28 .8.1 Verifying Derivatives of Eponential Functions (Note: te general equation of an eponential function is f() = a(b ).) 1. Complete te cart for f () =. X f() f ( + ) f ( + )! f ( ) / f ( + ) " f ( ) f ( ) = lim! 0 f / ( ) f ( ) Wat is te significance of f / ( ) f ( )? 3. Looking at your table, discuss te relationsip between f () and f / (). 4. Wat does te relationsip between f () and f / () tell you about te derivative of f ()? 5. Is tere a relationsip between f / ( ) and ln? Eplain. f ( ) 6. Make a conclusion about te derivative of f () =. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 8 of 37

29 .8.1 Verifying Derivatives of Eponential Functions (Continued) 1. Use te graping calculator to complete te table for f() = e. X Y 1 Y Y 3 f () nderiv(y 1, X, ) / f ( ) Y = f ( ) Y Wat is te significance of f / ( ) f ( )? 3. Looking at your table, discuss te relationsip between f() and f / (). 4. Wat does te relationsip between f () and f / () tell you about te derivative of f()? 5. Is tere a relationsip between f / ( ) and ln e? Eplain. f ( ) 6. Wat is te derivative of f() = e? Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 9 of 37

30 Unit : Day 16: Connections to Rational Functions Minds On: 10 Action: 35 Consolidate:30 Learning Goal: Use te Product Rule and te Cain Rule to determine derivatives of rational functions. Solve problems involving rates of cange for rational functions and develop algebraic facility were appropriate. MCV4U Materials Graping calculators and Viewscreen BLM.16.1 BLM.16. Total=75 min Assessment Opportunities Minds On Groups Grafitti Prepare cart paper wit Product Rule and Cain Rule: written in te center. Eac group gets a different coloured marker. In eterogeneous groups of 4 or 5 eac group writes all tey know about te term on teir paper. After one minute te papers are passed. Post and discuss. Action! Pairs A Coaces B Distribute BLM.16.1 and one graping calculators per pair.. For eac part of te activity A completes te question and B coaces. Alternate roles after eac part. Wole Class Teacer Led Demonstration/Note-Taking Ask students wo ave used different processes wen tey applied te product rule (i.e., apply te rule first or simplify first) to sare teir solutions. Have students discuss te merits of eac strategy. Engage students in a discussion based on teir responses to Question 4 of BLM As students sare, use te TI-83/84 to demonstrate te properties. Students will create teir own summary note during tis discussion. Use nderiv(y1,x,x) to grap te derivative of Y1 Curriculum Epectations/Oral Questions/Mental Note As students sare teir responses, ask probing questions to assess teir understanding. Use of derivative rules is te focus but reflect on furter eamples and instruction if algebraic manipulation of te functions appears weak. Matematical Process Focus: Connecting Consolidate Debrief Individual Practice Students will apply te cain rule and product rule to find te derivatives of rational functions on BLM.16.. Concept Practice Home Activity or Furter Classroom Consolidation Create tree rational functions and determine te derivative of eac using te cain rule and product rule. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 30 of 37

31 .16.1 Te Product Rule Again + 1) Rewrite eac rational function as a product (e.g.,! = ( + 3)( 1 ) ). State any restrictions on te domain and use te product rule to find te derivative. Function As a Product Restrictions Derivative + 3 f ( ) = 3 3! g( ) = ( ) = ! 6 y = t! 16 v = t + 4 ) Matc te grap to te equation for eac rational function. Consider restrictions on te domain. i) y = 3! 4 ii) y =!! iii) y = + 1 A B C 3) Use nderiv(y1,x,x) to grap te derivative of eac rational function in. Sketc te derivative on te graps above. 4) Summarize your findings of te derivatives of rational functions. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 31 of 37

32 .16.: Derivatives of Rational Functions 1. Apply te product rule and te cain rule to differentiate eac function. Do not simplify yet. Verify your result by using te nderiv() function on your graping calculator.! a) y = ( + 1) 7 b) y = 4 3 c) 3 y = d) y = 4! 3 e) y = f) y = + 1 (3! ). Find an equation of te tangent to te curve at te given point. a) 4! 1 f ( ) =,(,8) b) f ( ) =,(1,0)! 1 b) f ( ) =,(! 1,! ) + 1 Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 3 of 37

33 .16.: Derivatives of Rational Functions (Continued) 3. Mandy likes to coast to a stop on er bicycle. Se always stops pedalling at 15 seconds before se reaces er garage door. Te distance, d, in metres from er bike to er 4(15! t) garage door at ome as a function of time t, in seconds, is d( t) = for 0! t! 15. t + 3 a) Wen does er bike it te garage door? b) Find d '( t ). Wat does tis represent? c) At wat velocity does er bike strike te garage door? d) Wat was te velocity of te bike wen se stopped pedalling? 4. Te radius of a circular juice blot on a paper towel t seconds after te juice was spilled is 1+ t modeled by te function r( t) = were r is measured in centimetres. 1 + t a) Find te radius of te blot wen it was first spilled. b) Find te time wen te radius was 1.5 cm. c) Wat is te rate of increase of te blot wen te radius was 1.5 cm? According to tis model, will te radius ever reac cm? Eplain your answer. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 33 of 37

34 Unit : Day 17: Radical Rationals Minds On: 10 Action: 35 Learning Goal: Use te Product Rule and te Cain Rule to determine derivatives of radical functions. Solve problem involving rates of cange for radical functions and develop algebraic facility were appropriate. MCV4U Materials Graping Calculators BLM.17.1 BLM.17. Consolidate:30 Total=75 min Assessment Opportunities Minds On Pairs Pair Sare Have students eac select one of teir tree rational functions from Day 16 s Home Activity.. Student A coaces Student B as (s)e determines te derivative of te selected function. Roles are reversed for Student B s function.. Action! Wole Class Discussion Ask students to complete a Frayer Model of rational eponents. Have tem conjecture wat radical functions may look like. Pairs Investigation Students will complete BLM Matematical Process Focus: Connecting, Reflecting Wole Class Discussion Ask students to reflect on te derivative rules and teir application to radical functions. Summarize teir comments on te board. Have students apply teir understanding by determining te derivative of select simple and comple radical functions. Encourage students to sare different metods used. Students may wis to use te graping calculator to demonstrate relationsips between graps of functions and teir derivatives. In particular, students sould note tat a function wit a vertical tangent as an asymptote at tat point in its derivative. Te Frayer Model is found in Tink Literacy! Grades Consolidate Debrief Individual Practice Students complete BLM.17. to consolidate understanding. Concept Practice Curriculum Epectation/Observation/Cecklist Circulate as students complete BLM.17. to assess teir ability to determine a derivative using te product rule and cain rule. Home Activity or Furter Classroom Consolidation Complete practice questions from te selection as appropriate.. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 34 of 37

35 .17.1 Introducing Radical Functions Remember Rational Eponents?! Rewrite using rational eponents and simplify if possible (81 ) (16 ) All about te Power n n 1 If f ( ) = ten f '( ) = n!. Describe tis rule in words. Off on anoter tangent Compare te values of te slopes of te tangents to te given function wit te values of te derivative function determined using te power rule. Coose 1 monomial function wit rational eponents for te second table. Use te TANGENT function on your graping calculator to determine te slope of te tangent at eac point. Were are te tangents orizontal or vertical? Wat can you conclude about derivatives of radical functions? 3 = f ( ) =? f ( ) Slope of Tangent f () Slope of Tangent f () Different Differentiating f ( ) = Consider te function ( ) a) Determine te derivative using te cain rule. b) Determine te derivative by simplifying te function first and ten applying te product rule. c) Compare and contrast te two metods. Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 35 of 37

36 .17. It s Radical 1. Determine te derivative of eac function. Verify your answers. 1 3 e) y =! 6 d) y = ( + 1) f) y 3 = (! 5) g) 4 y =! + 1 ) y = + 1 i) y =! 5 5! Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 36 of 37

37 .17. It s Radical (Continued) 5. Many materials, suc as metals, form an oide coating (rust) on teir surfaces tat increases in tickness,, in centimetres, over time, t, in years, according to te equation 1 = kt. a) Determine te growt rate, d G =, as a function of time. dt b) If k = 0.0, calculate te growt rate after 4 years. c) Using a graping calculator or graping software, grap bot and G for k=0.0. Wat appens to te tickness and growt rate as t increases? d) Wy is it very important to protect te new metal on cars from rusting (oidizing) as soon as possible? Calculus and Vectors: MCV4U Unit : Eploring Derivatives Page 37 of 37