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1 Unit Applying Properties of Derivatives Calculus and Vectors Lesson Outline Day Lesson Title Math Learning Goals Epectations 1,,, The Second Derivative (Sample Lessons Included) 4 Curve Sketching from information 5, 6 Curve Sketching from an Equation Define the second derivative Investigate using technology to connect the key properties of the second derivative to the first derivative and the original polynomial or rational function (increasing and decreasing intervals, local maimum and minimum, concavity and point of inflection) Determine algebraically the equation of the second derivative f () of a polynomial or simple rational function f(), and make connections, through investigation using technology, between the key features of the graph of the function and those of the first and second derivatives Describe key features of a polynomial function and sketch two or more possible graphs of a polynomial function given information from first and second derivatives eplain why multiple graphs are possible Etract information about a polynomial function from its equation, and from the first and second derivative to determine the key features of its graph Organize the information about the key features to sketch the graph and use technology to verify B11, B1, B1 B14 B15 7 Jazz Day 8 Unit Summative (Sample Assessment Included) Note: The assessment on day 8, and an assessment for a jazz day, is available from the member area of the OAME website and from the OMCA website (wwwomcaca) Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 1 of 19

2 Unit : Day 1: Concavity of Functions and Points of Inflection Minds On: 15 Action: 45 Consolidate:15 Total=75 min Learning Goals: Recognize points of inflection as points on a graph of continuous functions where the concavity changes Sketch the graph of a (generic)derivative function, given the graph of a function that is continuous over an interval, and recognize points of inflection of the given function (ie, points at which the concavity changes) Minds On Pairs or Whole Class Discussion If enough computers are available, have students work in pairs with GSP to complete BLM 11 or, use the GSP Sketch PtsOfInflectiongsp as a demonstration and BLM 11 as a guide for discussion Key understanding: When the first derivative of a function changes direction, there is a point of inflection, and the concavity changes: MCV4U Materials Graphing Calculators BLM11 BLM1 BLM1 Computer lab or computer with data projector Chart paper and markers Assessment Opportunities Observe students working with GSP sketch Listen to their use of function terminology Action! Small Groups Investigation In heterogeneous groups of or 4, students complete BLM 1 in order to investigate points of inflection for different polynomials Mathematical Process: Connecting, Representing Consolidate Debrief Small Groups Class Sharing In heterogeneous groups of three, and using chart paper, students write their observations from the investigations Call on each group to share observations with the class Groups post the chart paper with their observations from the investigations on the wall Invite students to record information from the chart paper in their notebooks Home Activity or Further Classroom Consolidation Concept Practice Reflection Complete BLM 1 Journal Entry: Do all graphs have points of inflection? Eplain Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page of 19

3 11 Concavity and Point(s) Of Inflection 1 For each of the following graphs of functions: a) determine (approimately) the interval(s) where the function is b) determine (approimately) the interval(s) where the function is c) estimate the coordinates of any point(s) of inflection 1, Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page of 19

4 1 Investigate Points of Inflection PART A 1 i) Using technology, graph f ( ) =, the first derivative f ( ) = f! ( ) = 6 on the same viewing screen ii) Copy and label each graph on the grid below!, and the second derivative iii) Identify the point of inflection of ( ) f = using the TABLE feature of the graphing calculator iv) Show algebraically that the point where the first derivative,!( ), of the function, ( ) equal to 0 is (0, 0) v) For the interval # (!", 0) f =, the first derivative!( ) of the function ( ) f = ais and is For the interval # (!", 0) function For the interval " ( 0,!), the function ( ) f =, is f = is above the - f = is a(n), the first derivative!( ) of the function ( ) f = ais and is For the interval " ( 0,!) function, the function ( ) f = is also above the - f = is a(n) vi) Show algebraically that the point where the second derivative, f! ( ) = 6, of the function, ( ) is equal to 0 is (0, 0) vii) For the interval # (!", 0), the second derivative f! ( ) = 6 of the function ( ) ais and is For the interval # (!", 0) For the interval " ( 0,!), the function ( ) f =, f = is below the - f = is, the second derivative f! ( ) = 6 of the function ( ) ais and is For the interval " ( 0,!), the function ( ) f = is above the - f = is viii) Describe the behaviour of the second derivative of the function to the left and to the right of the point where the second derivative of the function f ( ) = is equal to 0 Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 4 of 19

5 1 Investigate Points of Inflection (Continued) PART B 1 i) The graph of ( ) = f + is the graph of ( ) f = translated units Using technology, create the graphs of the function f ( ) = ; and the function f ( ) = + derivative of the function!( ), and the second derivative of the function f ( ) = 6 f = viewing screen ii) Copy and label each graph on the grid below, the first! in the same iii) Identify the point of inflection of ( ) = + f iv) Show algebraically that the point where the first derivative, f!( ) =, of the function, ( ) = + is equal to 0 is (0, 0) v) For the interval # (!", 0), the first derivative!( ) of the function ( ) = + f = ais and is For the interval # (!", 0) function For the interval " ( 0,!), the function ( ) = + f, f is above the - f is a(n), the first derivative f!( ) = of the function ( ) = + the -ais and is For the interval " ( 0,!) function vi) Show algebraically that the point where the second derivative, f ( ) = 6 f ( ) = +, is equal to 0 is (0, 0) vii) For the interval # (!", 0) f is also above, the function ( ) = + f is a(n)!, of the function,, the second derivative f! ( ) = 6 of the function ( ) = + the -ais and is For the interval # (!", 0) For the interval " ( 0,!) f is below, the function ( ) = + f is, the second derivative f! ( ) = 6 of the function ( ) = + -ais and is For the interval " ( 0,!), the function ( ) = + f is above the f is viii) Describe the behaviour of the second derivative of the function to the left and to the right of the point where the second derivative of the function f ( ) = + is equal to 0 What effect does translating a function vertically upwards have on the location of a point of inflection for a given function? Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 5 of 19

6 1 Sketch the Graph Of the Derivative Of A Function 1 a) Determine the first and second derivatives of the function ( ) =! + f b) Sketch the first and second derivatives of the function f ( ) =! + the graph of the function ( ) =! + f on the grid below, and use these graphs to sketch f =! + a) Determine the first and second derivative of each of the function ( ) ( ) =! + b) Sketch the graphs of the first and second derivatives of the function ( ) ( ) graphs to sketch the graph of the function ( ) ( ) f =! + on the grid below f, and use these Eplain how you can determine the point of inflection for a function by studying the behaviour of the second derivative of the function to the left and to the right of the point where the second derivative of the function is equal to Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 6 of 19

7 Unit : Day : Determining Second Derivatives of Functions Minds On: 15 Action: 50 Consolidate:15 Total=75 min Learning Goals: Define the second derivative of a function Determine algebraically the equation of the second derivative f!!( ) of a polynomial or simple rational function f () Minds On Pairs Investigation Students work in pairs to complete BLM 1 Establish the definition of the second derivative The definition should include rate of change and tangent line MCV4U Materials Graphing Calculators BLM1 BLM BLM BLM4 Large sheets of graph paper and markers Assessment Opportunities Action! Consolidate Debrief Small Groups Guided Eploration In heterogeneous groups of four students complete BLM and BLM Mathematical Process: Selecting tools, Communicating Individual Journal Students write a summary of findings from BLM in their journal Use a checklist or checkbric to note whether students can apply the Product Rule Application Home Activity or Further Classroom Consolidation Use BLM 4 to consolidate knowledge of and practise finding first and second derivatives using differentiation Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 7 of 19

8 1 Find the Derivatives of Polynomial and Rational Functions 1 Find the derivative of each function Determine the derivative of the derivative function a) f ( ) = b) f ( ) =! =! c) f ( ) d) f ( ) =! 1 Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 8 of 19

9 The First and Second Derivatives of Polynomial and Rational Functions 1 Complete the following chart: Function, f ( ) First Derivative, f!( ) Second Derivative, f! ( ) a) f ( ) = 6 b) f ( ) =! + c) f ( ) = + 5! 7 d) f ( ) = (! ) ( + 1) e) f ( ) = f) f ( ) =! 6 +! g) f ( ) =! 4 + 6! + Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 9 of 19

10 The First and Second Derivatives of Polynomial and Rational Functions (Continued) Complete the following chart: [LEAVE YOUR ANSWERS IN UNSIMPLIFIED FORM] a) f ( ) Function, f ( ) First Derivative, f!( ) Second Derivative, f! ( ) + 1 =! b) f ( ) =! 4 c) f ( ) =! 9 d) f ( ) =! 1 ( + 1) 5 Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 10 of 19

11 Working Backwards from Derivatives 1 The first derivative f () is given Determine a possible function, f() First derivative, f!( ) Possible function, ( ) f? ' a) f ( ) = + 4 f ( ) = Note: Any constant could be added or subtracted here b) f '( ) = 7 c) f '( ) = ( + 1) d) f '( ) = f ' =!( + 1)! e) ( ) Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 11 of 19

12 4 Home Activity: First and Second Derivatives 1 Complete the following chart Function, f ( ) First Derivative, f!( ) Second Derivative, f! ( ) a) f ( ) = + 7! 5 f = ( + ) b) ( ) 4 c) f ( ) =! 5 + 4! 8 + d) f ( ) + =! 5 e ) f ( ) =!16 Create a polynomial function of degree Assume that your function (f ()) is the derivative of another function (f()) Determine a possible function (f()) Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 1 of 19

13 Unit : Day : The Second Derivative, Concavity and Points of Inflection MCV4U Minds On: 10 Action: 50 Consolidate:15 Learning Goals: Determine algebraically the equation of the second derivative f () of a polynomial or simple rational function f(), and make connections, through investigation using technology, between the key features of the graph of the function and those of the first and second derivatives Materials graphing calculators Computer and data projector BLM 1 BLM BLM BLM 4 Total=75 min Minds On Pairs Pair Share Students coach each other as they complete a problem similar to the work from the previous class (A coaches B, and B writes, then reverse roles) Assessment Opportunities Each pair of students has only one piece of paper and one writing instrument Whole Class Discussion Using the GSP sketch PtsOfInflectiongsp, review points of inflection, and the properties of the first and second derivatives with respect to concavity and points of inflection Action! Pairs Guided Eploration Curriculum Epectation/Observation/Mental note: Circulate, listen and observe for student s understanding of this concept as they complete BLM 1 and Students work in pairs to complete the investigation on BLM1, BLM As students engage in the investigation, circulate to clarify and assist as they: determine the value(s) of the first and second derivatives of functions for particular intervals and -values use a graphing calculator to determine the local maimum and/or minimum point(s) and point(s) of inflection of a polynomial function understand the connection between the sign of the second derivative and the concavity of a polynomial function use the second derivative test to determine if a point if a local maimum point or a local minimum point for a polynomial functions Mathematical Process: Reflecting, representing Consolidate Debrief Whole Class Discussion Lead a discussion about the information provided by the second derivative Students should be able to articulate their understanding about the properties of the second derivative Eploration Application Home Activity or Further Classroom Consolidation Complete BLM in order to consolidate your understanding of first derivatives, second derivatives, local ma/minimum points and points of infection for a simple rational function Complete a Frayer Model for the Second Derivative on BLM 4 See pages -5 of THINK LITERACY : Cross - Curricular Approaches, Grades 7-1 for more information on Frayer Models Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 1 of 19

14 1 The Properties of First and Second Derivatives Investigate: 1 a) Graph each of the following functions on the same viewing screen of a graphing calculator f ( ) =!! ( ) =! f " ( ) = f! b) Classify each of the functions in part a) as a quadratic function, a linear function, or a constant function ( ) =!! f function ( ) =! f " function ( ) = f! function c) How are the first and second derivative functions related to the original function? d) Determine the coordinates of the verte of the function ( ) =!! f e) Determine if the verte of the function ( ) =!! minimum point f is a local maimum or a local f) Does the function ( ) =!! f have a point of inflection? Eplain g) Use a graphing calculator to determine the value of the first derivative of the function f =!! for each -value ( ) = 1 "(!1) = 1 f!( 1) = = f!( ) = f = Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 14 of 19

15 1 The Properties of First and Second Derivatives (continued) h) Complete the following chart by circling the correct response for each interval Interval #("!,1) The first derivative ( ) =! f " The function, f ( ) =!! The second derivative f! ( ) = is The function, f ( ) =!! = 1 - is - is - is - is increasing - has a local ma - has a local min - is decreasing - is - is - is - is - has a pt of inflection - is - is - is - is - is increasing "( 1,! ) - has a local ma - has a local min - is decreasing - is - is - is - is - has a pt of inflection - is - is - is - is - is increasing - has a local ma - has a local min - is decreasing - is - is - is - is - has a pt of inflection - is The first derivative, f "( ) =!, of the function f ( ) =!! is when = 1 and the second derivative, f! ( ) =, of the function f ( ) =!! is when = 1 A local point occurs when = 1! The second derivative, f ( ) =, of the function ( ) =!! f is for all values of The function f ( ) =!! is for all values of and does have a point of inflection Consolidate: If the second derivative of a function is when the first derivative of a function is equal to for a particular -value, then a local point will occur for that particular -value If the second derivative of a function is when the first derivative of a function is equal to for a particular -value, then a local point will occur for that particular -value Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 15 of 19

16 Derivatives of a Cubic Function Investigate: 1 a) Graph each function on the same viewing screen of a graphing calculator f ( ) =!! ( ) =! 6! 9 f " ( ) = 6! 6 f " Use the following WINDOW settings: Xmin = 10 Xma = 10 Xscl = 1 Ymin = 0 Yma = 0 Yscl = 1 Xres = 1 b) Identify each of the functions in part a) as a cubic function, a quadratic function, a linear function, or a constant function f ( ) =!! function ( ) =! 6! 9 f " function ( ) = 6! 6 f " function c) How are the first and second derivative functions related to the original function? d) Write the coordinates of the local maimum point of the function f ( ) =!! e) Write the coordinates of the local minimum point of the function f ( ) =!! f) Write the coordinates of the point of inflection for the function f ( ) =!! Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 16 of 19

17 Derivatives of a Cubic Function (Continued) g) Complete the following chart by circling the correct response for each interval Interval " < <! 1! =! 1! 1 < < 0 = 0 0 < < 1 = 1 1 < <! The first derivative ( ) =! 6! 9 f " Positive The function, f ( ) =!! increasing decreasing local ma local min increasing decreasing local ma local min increasing decreasing local ma local min increasing decreasing The second derivative ( ) = 6! 6 f " The function, f =!! 9 ( ) + 15 pt of inflection pt of inflection pt of inflection Summarize the information in the table in your own words Consolidate: If the second derivative of a function is for a particular -value on the curve and has the opposite sign for points on either side of that particular -value, then a of the function and a of the first derivative of the function will occur for that particular -value Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 17 of 19

18 Derivatives of Rational Functions Investigate: 1 a) Graph each of the following functions on the same viewing screen of a graphing calculator Use the standard viewing window f f "( ) = f ""( ) ( ) =! 1!! 1 (! 1) b) Write the coordinates of the point of inflection for the function = f ( ) =! (! 1) c) Complete the following chart by circling the correct response for each interval Interval! " < <! 1 =! 1! 1 < < 0 = 0 0 < < 1 = 1 1 < <! The first derivative f " ( )! =! 1 (! 1) eist eist eist eist eist eist eist The function, f ( ) =! 1 increasing decreasing local ma local min increasing decreasing local ma local min increasing decreasing local ma local min increasing decreasing The second derivative f "" ( ) = + 6 (! 1) eist not continuous eist eist not continuous eist eist not continuous eist eist The function, f ( ) =! 1 pt of inflection pt of inflection pt of inflection not continuous not continuous not continuous d) Eplain the behaviour of the function f ( ) = at the points where =! 1 and = 1! 1 Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 18 of 19

19 4 Home Activity: Frayer Model Name Date Definition Facts/Characteristics Eamples Second Derivative of a Function Non-eamples Calculus and Vectors: MCV4U: Unit Applying Properties of Derivatives (Draft August 007) Page 19 of 19

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