Course Profile Advanced Functions and Introductory Calculus (MCB4U), Grade 12, University Preparation, Combined

Unit 2: Underlying Concepts of Calculus

Time: 12 hours

Unit Description

Students study rates of change in the context of mathematical functions and applications from the natural and social sciences. By studying the average and instantaneous rate of change, the idea of the derivative is introduced. Students develop an understanding of the derivative and its connection to the graph of a function by referring to the behaviour of the graphs. This is done using paper-and-pencil methods as well as using graphing technology.

Activity 2.1: Ch-Ch-Ch-Changes

Time: 2.5 hours

Description

Students investigate rates of change problems using tables of values, graphs and equations. In these models, which deal with population growth, displacement-velocity, and temperature gradient, students determine and interpret rates of change. Students also discover a connection between average and instantaneous rates of change using a calculation-based method.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2b - an effective communicator who reads, understands, and uses written materials effectively;

CGE3e - a reflective and creative thinker who adopts a holistic approach to life by integrating learning from various subject areas and experiences.

Strand(s): Underlying Concepts of Calculus

Overall Expectations

CCV.01 - determine and interpret the rates of change of functions drawn from the natural and social sciences;

CCV.02 - demonstrate an understanding of the graphical definition of the derivative of a function.

Specific Expectations

CC1.01 - pose problems and formulate hypotheses regarding rates of change within applications drawn from the natural and social sciences;

CC1.02 - calculate and interpret average rates of change from various models of functions drawn from the natural and social sciences;

CC2.01 - demonstrate an understanding that the slope of a secant on a curve represents the average rate of change of the function over an interval, and that the slope of the tangent to a curve at a point represents the instantaneous rate of change of the function at that point.

Prior Knowledge & Skills

· understanding and calculating rates of change

· finding slopes of straight lines

· plotting data and sketching a curve of best fit

· use of graphing calculator (or similar technology) with temperature probe

Planning Notes

· Students are to do each of the activities individually.

· Students must have access to graph paper.

· The teacher should suggest that students use different coloured pencil or pens on their graphs.

· In Part 1, students analyse a model involving a table of values. If students are to do research for their own data in Part 1, computer time (with Internet access) must be booked.

· In Part 2, students analyse a model involving a defining equation.

· In Part 3, students analyse a model involving a table of values with a curve of best fit. If students are to gather their own data in Part 2, graphing calculators with a temperature probe must be made available otherwise, only graphing calculators are needed.

Teaching/Learning Strategies

A. Teacher Facilitation

· The teacher should facilitate a short class discussion on rates of changes, in order to consolidate the concepts and skills from previous math courses, i.e., calculating a rate of change, connecting slope and rate of change, using examples drawn primarily from the social sciences. The teacher ensures that students understand the meaning of population growth and temperature gradient.

· In the first and third parts of this activity, data is supplied. Alternatively, in Part 1, the teacher could have students research any other type of population growth scenario, e.g., Canada’s population over the last 100 years. In Part 3, the teacher could have groups of students actually perform the suggested experiment and gather their own data, e.g., with a temperature probe. The curve of best fit for this activity could also be found and/or verified using graphing technology, e.g., graphing calculator.

· Excess time should not be spent on the gathering and manipulating of data. The teacher may choose to limit the amount of data and/or the questions involving the manipulation of data. The major emphasis must be on the conceptualization of the underlying concepts of calculus.

· The primary focus in Part 1 is calculation and interpretation of average rates of change and the development of connections with instantaneous rates of change. The focus in Part 2 is the extension of the concepts explored in Part 1 but with the ability to make the intervals as small as necessary. In Part 3 students develop the graphical connections between average and instantaneous rates of change. The tangent to a curve is also introduced.

· In Part 3, the Tangent function on the graphing calculator is used. This function has not been introduced yet, so teachers should familiarize themselves with it and may need to give explicit instructions to the class on how to use it.

· The overall intent of this series of activities is for students to investigate the numerical relationships between average and instantaneous rates of change and tangent lines. These relationships (in particular, the concept of approximating an instantaneous rate of change by using smaller and smaller intervals) are consolidated, formalized, and extended in subsequent activities.

B. Student Activity

Suggestions for teacher facilitation are included throughout this activity in italics. Some solutions are included to aid in the flow of the activity.

Part 1:

World population data from a 1995 UN Census Report and a 2000 US Bureau of Census Report:

Year	Population (millions)	Year	Population (millions)	Year	Population (millions)
1000	310	1910	1750	1965	3345
1250	400	1920	1860	1970	3707
1500	500	1930	2070	1975	4086
1750	790	1940	2300	1980	4454
1800	980	1950	2520	1985	4851
1850	1260	1955	2780	1990	5279
1900	1650	1960	3039	1995	5688
				2000	6083

1. Find the average population growth rate (per year) in the following intervals:

a) 1000-1500 b) 1000-1250 c) 1250-1500

2. Discuss an approach for finding population the growth rate for the year 1250. One approach may be to use the average population growth rates from 1000-1500 to make an educated guess. Another approach may be to compare the average population growth rates for 1000-1250 and 1250-1500 and choose a value between them.

3. For any continuous function, determining the rate of change at a specific data point is referred to as finding an instantaneous rate of change. Estimate an instantaneous population growth rate for the year 1250.

4. For the years 1750, 1950, and 1990, estimate the population growth rates by first calculating the average growth rates before and after each year. Use the average rates to estimate a population growth rate for the given year:

Year	Range before year	Rate of change	Range after year	Rate of change	Estimate of growth rate for the year
1750	1500 – 1750		1750 – 1800
1950	1940 – 1950		1950 – 1960
1990	1985 – 1990		1990 – 1995

5. Discuss the likely accuracy and reliability of your estimates in the previous question. The intent is to have students think about the accuracy of their estimate of the instantaneous rate of change, by considering things such as the number of years in the interval and the differences of each pair of rates. Some students may notice that the ranges before and after 1750 are of different size. A discussion of how that may affect the estimate may be necessary, focusing only on the idea that there are difficulties with using average growth rates. This should provide the impetus for seeking a more efficient and accurate tool, namely the instantaneous rate of change (slope of the tangent) and inevitably, the derivative.

6. Some people would say that our world would not be able to maintain the world’s population in the year 2050. Do you agree or disagree? Explain. You may want to construct a scatter plot as part of your explanation. This question gives students the opportunity to talk about such things as poverty, food shortage, distribution of wealth, family planning, availability of medicine and medical procedures, education, role of governments, natural resources, ethics, etc. [CGE3e]

Part 2:

A ball, which is dropped from the top of a tall building, has a vertical height equation given by:

h(t) = 122.5 ! 4.9t²; where t is time in seconds and h is height above the ground in metres.

1. Some rates of change have special names. In the case of distance, the rate of change of distance with respect to time is called the velocity. Find the average velocity during the intervals:

a) 0-2 seconds b) 2-4 seconds

2. Based on your answers from Question 1 estimate the velocity at 2 seconds. Be sure to use intervals below 2 and above 2. How could you find the instantaneous velocity at 2 seconds more accurately? By using results from Parts 1 and 2, students should use average velocities over 0-2, 1-2, 1.5-2, 1.9-2, as well as 2-2.1, 2-2.5, 2-3, 2-4 etc. to estimate the velocity at 2 seconds. They should realize that they need to make the intervals as small as they can, both above and below the required time, to get an accurate estimate for the instantaneous velocity.

3. What is different between the data for this part and that of Part 1 that makes the estimates for the instantaneous velocity more reliable than the estimates for the instantaneous population growth rate? Students should realize that since the data for this question is generated by an equation, they can make the intervals as small as they wish.

4. Find the velocity at 1.5 seconds.

5. When will the ball hit the ground? Is the velocity 0? Justify your answer using both algebraic and geometric reasoning. What is the velocity when the ball reaches the ground? Use the graph of the given height equation in your explanations. Students can use both the given quadratic height equation and its corresponding parabolic graph to answer these questions.

6. If the ball is thrown upwards from the top of the building (instead of being dropped) sketch a possible height vs. time graph. Discuss how you arrived at this graph. Compare/contrast this graph with the graph obtained in Question 5. Students should be able to note that the curve will have both increasing and decreasing intervals and hence a maximum height. [CGE3e]

7. Over what interval is the velocity positive? Over which interval is the velocity negative? Is the velocity ever 0? If so, when? Be sure to include justification of your answers.

Part 3:

Data for the temperature of a warm cup of hot chocolate over a period of time:

Time (minutes)	Temperature (°C)	Time (minutes)	Temperature (°C)
0	82	24	26.5
5	63.5	25	25
8	56	30	20
11	48	34	16.5
15	40	38	14
18	35	45	12.5

1. Carefully plot the data on graph paper and fit a smooth curve that best represents this data. Be sure to make the scale such that the entire graph paper is used to show the data since some of the questions below will require illustrations on the graph and interpretations that may be difficult to read if the size of the graph is not maximized. The data for this question is exponential.

2. Use a graphing calculator to verify the curve of best fit. Regression will yield y = 78.060(.9570)^x.

For the following question, use estimated points from the graph in Question 1 and verify with the curve of best fit from Question 2.

3. a) Find the average temperature gradient (average rate of change of temperature with respect to time) over the following intervals (use at least one decimal place):

i) 15-30 min. ii) 18-30 min. iii) 24-30 min. iv) 25-30 min.

v) 30-34 min. vi) 30-38 min. vii) 30-45 min.

b) What do the answers in the above question represent graphically? The slope of the secant joining the endpoints of each interval.

c) Lines drawn between data points are called secants. By drawing secants that join each pair of points, illustrate the answers to the above question on the graph in Question 1. Use two different colours, one for the first four pairs and the second for the last three pairs of points. It is very important that the graph in Question 1 is large enough and the scale for the vertical axis is appropriate so these lines are distinguishable (and their slopes are determinable).

4. A Danish mathematician, Thomas Fincke, who wrote about it in Latin in 1583, first used the word “tangent.” Tangent comes from the Latin word tangere, which means to touch.

a) For the equation y = x², use the Tangent function on a graphing calculator to illustrate and find the slope of the tangent line at x = -1, 1, 2, and 5. Use the zoom feature to see how it compares with the shape of the curve. Students should be instructed not to use the tangent button that refers to trigonometry but to use the function from the Draw menu that allows a tangent line to be drawn on the graph.

b) Use the above question and prior statement to discuss what a tangent line is. (Hint: relate it to the shape of a curve and measure of steepness) A formal definition is not necessary at this point. The idea that the tangent is a line touching the curve whose slope a measure of the shape/steepness of the graph close to the specified point will do for now. [CGE3e]

5. Refer back to the original temperature graph. Sketch the tangent line at t = 30 minutes on the graph. Between what values will the slope lie? Estimate the slope of this tangent line. Verify with the tangent feature on the graphing calculator. What does this slope represent? The slope represents the instantaneous rate of change of temperature with respect to time or temperature gradient at time equal 30 minutes. The slope should lie between the slopes of the secants for the intervals 25-30 min. and 30-34 min. (-1<slope<-0.875). The actual value from the regression is -0.92. The students may benefit from a discussion on the closeness of the curve of best fit to the data point at t = 30 to determine if the value from the regression is meaningful. This should remind students that the slope they are finding is from the regression curve, and therefore only an estimate of the true value.

6. Consider the slopes of the secants from Question 3. Look at these values as the size of the time interval gets smaller. Compare these values as the interval gets smaller to the slope of the tangent at
t = 30 min.? What does this mean in terms of rates of change? At this point, students should realize as the interval for time becomes smaller the average rate of change better estimates the instantaneous rate of change. They may also notice that the slope of the tangent is a value between the slopes of the first 4 pairs and the last 3 pairs.

7. Use the process illustrated above in Question 3 to estimate the instantaneous temperature gradient (instantaneous rate of change of temperature with respect to time) at 11 minutes. Verify it with the process used in Question 5.

8. At what time is the (instantaneous) temperature gradient the greatest? Explain. Students may use the process similar to that described in the question above and experiment with different values of t. Students may also begin to realize that temperature gradient is actually a measure of the shape/ steepness of the graph and hence look for the steepest part of the graph.

9. Discuss the relationship between instantaneous and average rate of change in context of temperature gradient. Include graphical interpretations in the discussion. The average rate of change yields the average temperature gradient and represents the slope of the line joining the corresponding data points. The instantaneous rate of change yields the instantaneous temperature gradient and represents the slope of the tangent at that specific data point. As the change in temperature becomes smaller the average temperature gradient becomes a better estimate for the instantaneous temperature gradient. [CGE3e]

C. Follow-up Skills

The teacher should supplement these activities with textbook exercises (include a wide range of paper-and-pencil type questions) that involve various other models from the natural and social sciences.

Accommodations

· Teachers should be aware that some students may need extra time to graph by hand or to manipulate data with the technology.

Assessment & Evaluation of Student Achievement

· Knowledge/Understanding can be formatively assessed using a short quiz on determining and interpreting rates of change after all the activities or after any of the activities, depending on the time students require consolidating skills.

· Application can be formatively assessed using Questions 4 in Part 1, Question 4 in Part 2, and Questions 7 and 8 in Part 3. If technology is used in Part 3, this component can also be assessed for Application.

· Inquiry can be assessed in any of the questions in which the student was asked to find the greatest or least of a particular rate of change, e.g., Question 8 in Part 3.

· Communication can be assessed using any of the questions that ask for a discussion or explanation, e.g., Questions 2 and 5 in Part 1, Questions 6 and 7 in Part 2, Question 9 in Part 3. Criteria that can be used include depth and clarity of explanations, appropriate use of notations, symbols and graphs, proper use of mathematical language.

· Journal writing should be an important theme in this unit, through which students can be asked to formulate and consolidate the underlying concepts of calculus. For this activity, the primary focus is the relationships between average and instantaneous rates of change and their respective graphical interpretations, e.g., the slope of a tangent line, Question 12 in Part 3 would be a good starting point.

Activity 2.2: I Can’t Drive 55

Time: 2.5 hours

Description

Students investigate and compare average and instantaneous rates of change in the context of a motion model (speed-distance-time). Students connect these rates of change graphically with slopes of tangents and slopes of secant lines. The connection between slopes of secant lines and slopes of tangents lines is investigated and will be consolidated and built upon in subsequent activities.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

CGE4a - a self-directed, responsible, life long learner who demonstrates a confident and positive sense of self and respect for the dignity and welfare of others;

CGE4f - a self-directed, responsible, life long learner who applies effective communication, decision making, problem-solving, time, and resource management skills.

Strand(s): Underlying Concepts of Calculus

Overall Expectations

CCV.01 - determine and interpret the rates of change of functions drawn from the natural and social sciences;

CCV.02 - demonstrate an understanding of the graphical definition of the derivative of a function.

Specific Expectations

CC1.03 - estimate and interpret instantaneous rates of change from various models of functions drawn from the natural and social sciences;

CC1.04 - explain the difference between average and instantaneous rates of change within applications and in general;

CC1.05 - make inferences from models of applications and compare the inferences with the original hypothesis regarding rates of change;

Prior Knowledge & Skills

· understanding and calculating rates of change

· finding slopes of straight lines

· plotting data

· relationship between velocity, distance, time

Planning Notes

· If the data is to be gathered, groups of four or five must be assigned, with each group responsible for bringing a bicycle the day of the activity. Each group must have a watch (preferably digital) or the teacher may have to supply stopwatches. Part 2 is to be done individually.

· If students are to use the given data, Part 1 can be done in groups of two or three and Part 2 can be done individually.

· If the graphing is to be done manually, students must have access to graph paper. Teacher should suggest that students use different coloured pencil or pens on their graphs.

· If the analysis is to be done on a spreadsheet, computer time should be booked.

Teaching/Learning Strategies

A. Teacher Facilitation

· Rather than using the given data, the teacher may assign groups of four or five students to gather the data on the school’s track. Markers should be placed around the track at 10-m or 20-m intervals. One student will ride the bicycle, another will be the timer, and the other two or three students (recorders) are positioned evenly around the track to record distances. The timer must be positioned strategically (in the middle of the track) so that the recorders can clearly hear the times.

· Prior to the activity, the teacher should ensure that students are familiar with the relationships between speed-distance-time.

· The intent of Part 1 is for the student to once again relate the average rate of change with average speed (and slope of a secant line) and to connect instantaneous rate of change with instantaneous speed (and slope of a tangent line).

· The intent of Part 2 is to consolidate those concepts from Part 1 and to begin to connect the graphical and numerical values representing an instantaneous rate of change with the idea of being between the left and right average rates of change.

B. Student Activity

Suggestions for teacher facilitation are included throughout this activity in italics. Some solutions are included to aid in the flow of the activity.

An Olympic racing cyclist travels around a 200-m oval velodrome track. The coach gathers distance-time data given in the table below:

Time (s)	Distance (m)	Time (s)	Distance (m)	Time (s)	Distance (m)
0	0	55	292	110	771
5	3	60	367	115	789
10	10	65	428	120	825
15	18	70	444	125	857
20	28	75	469	130	900
25	41	80	491	135	923
30	55	85	518	140	939
35	74	90	565	145	958
40	103	95	636	150	969
45	155	100	701	155	988
50	223	105	758	160	1000

To lessen the tedious task of calculating rates of change, use the table, below, to construct and complete a spreadsheet in order to answer the accompanying questions.

Time (s)	Distance (m)	Rate of Change Using every 4^thData	Rate of Change Using every 2^ndData	Rate of Change Using every Data
0	0
5	3		0.6	0.6
10	10			1.4
15	18	1.2	1.5	1.6
20	28			2
25	41		2.3	2.6
30	55			2.8
35	74	2.8	3.3	3.8
…	…	…	…	…

Part 1

1. Find the average speed over the entire period of time.

2. Find the average speed over the first half of time and over the second half of time. Are these answers the same? In general, should they be? Explain.

3. During what 5-second interval is the average speed the greatest? During what 5-second interval is the average speed the least? Is there any reason for these? Explain. [CGE3c]

4. If the bicycle is equipped with a speedometer, what do you think the speed of the bicycle would be at:

a) 20 s b) 60 s c) 90 s d) 120 s e) 160 s

5. For safety reasons, the coach does not want the cyclist traveling over 55 km/h. Did this cyclist heed instructions? How could this be verified? Be aware of the unit conversions necessary. [CGE4a]

Part 2

Many of the questions below involve graphing and illustrations on the graph. Using the spreadsheet from Part 1, print out 2 copies of a scatter plot each on a separate full page. It is recommended that different coloured pencils or pens be used for clarity.

The first four questions are to be done on the first scatter plot.

1. Connect every 4th data point. Find the slopes. What do they represent? The slopes represent the average speed over each interval.

2. With a different colour connect every 2^nd data point. Find the slopes. What do they represent? [CGE4f]

3. Use another colour to connect every other data point. Find the slopes. What do they represent? [CGE4f]

4. Use your results above to sketch a graph of the distance-time relationship. The intent is to have students begin to make the connection with the slope of the tangent representing the “shape” of the curve.

5. Find the average speed over the following intervals:

a) 35-60 s b) 40-60 s c) 45-60 s d) 50-60 s e) 55-60 s

f) 60-65 s g) 60-70 s h) 60-75 s i) 60-80 s j) 60-85 s

This process could be fairly quick if the spreadsheet is used.

6. Based on the answers in Question 5, between what values can you expect the tangent at t = 60 s to lie? Students should note that the left side secants have average speeds that increase up to 19 m/s as you approach 60 s and the right hand side average speeds increase up to 8.2 m/s as 60 s is approached. This gives a range of possible values for the instantaneous speed at 60 s. The large discrepancy between these values is then graphically depicted in the next question.

7. Illustrate Question 5 on your second scatter plot by drawing secants. Use one colour for the secants on the left side of 60 s and a different colour for secants on the right side of 60 s. Students should be drawing lines that correspond to the slopes in the previous question. Because of the nature of the graph at 60 s, the slopes of the secants on the left and right sides should appear very different making it easier to position the tangent.

8. Draw a tangent line at t = 60 s. Use the graph and values from 5 to estimate the slope of this line. What does it represent? The slope of the tangent line represents the instantaneous speed. [CGE3c]

9. Repeat this process to estimate the slope of the tangent at t = 40 s. What is different about the data here in comparison to the data at t = 60 s? Here the data appears more regular and the change between points around 30 s is not as large.

10. Estimate the time at which the cyclist is travelling at 30 m/s. If you wanted to find a better estimate for the point in time at which the speed is 30 m/s, what changes would you make in the gathering of the data? If data could be gathered every two seconds or even every second, then average speeds would be a better approximation for the speed at an instant. [CGE3c]

11. In some research or industrial settings, the gathering of data may be expensive and/or dangerous. Examples are vehicle crash-testing and “weatherbeater” aircraft that obtain data about hurricanes. As you should have noted above, the more data that is collected, the more accurate the instantaneous rates of change will be. Discuss in your journal some appropriate issues in the collecting of data using the above examples and/or some of your own. [CGE4f]

Extensions

1. Can a speedometer actually measure instantaneous speed?

2. Do radars actually measure instantaneous speed? These two questions give students the opportunity to discuss, from a very practical point of view, that physical apparatuses do not actually measure instantaneous rates of change but rather average rates of change over an extremely small interval.

3. The technique for the gathering of data in this activity may not be the most convenient. What other approaches could have been used? Have students at designated positions and record the times that the cyclist pass by those intervals.

C. Follow-up Skills

1. Draw on a model from the natural and/or social sciences that will generate a graph that can be used to address all the same types of questions in the given activity.

2. Draw on a model from the natural and/or social sciences using an equation that can be used to address all the same types of questions in the given activity.

3. Use graphing software to illustrate slopes of secants and slopes of tangents.

Assessment & Evaluation of Student Achievement

· Learning skills, specifically teamwork, independence, and initiative can be assessed in the group work components of the activities.

· Conferencing can be used to formatively assess Knowledge/Understanding of individuals during the activities.

· In Question 11 in Part 2, Communication skills can be assessed in the journal using criteria such as: degree of clarity of explanations, use of appropriate notations, and correct use of mathematical language.

· Application (using average speed over a small time interval to estimate instantaneous speed) can be formatively assessed using Question 4 in Part 1.

· Question 9 and 10 in Part 2 can be submitted and used to assess Thinking/Inquiry/Problem Solving.

Activity 2.3: Licensed to Derive!

Time: 5 hours

Description

Students explore the characteristics of the graphs of polynomial functions using both graphing techniques and technology. Students develop the concept of a derivative function, and use it to investigate slopes of tangents, increasing and decreasing intervals, critical points, concavity, and points of inflection.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE2c - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

CGE4b - a self-directed, responsible, life long learner who demonstrates flexibility and adaptability;

CGE5a - a collaborative contributor who works effectively as an interdependent team member;

CGE7j - a responsible citizen who contributes to the common goal.

Strand(s): Advanced Functions, Underlying Concepts of Calculus

Overall Expectations

AFV.01 - determine, through investigation, the characteristics of the graphs of polynomial functions of various degrees;

CCV.02 - demonstrate an understanding of the graphical definition of the derivative of a function;

CCV.03 - demonstrate an understanding of the relationship between the derivative of a function and the key features of its graph.

Specific Expectations

AF1.01 - determine, through investigation, using graphing calculators or graphing software, various properties of the graphs of polynomial functions;

CC2.02 - demonstrate an understanding that the slope of the tangent to a curve at a point is the limiting value of the slopes of a sequence of secants;

CC2.03 - demonstrate an understanding that the instantaneous rate of change of a function at a point is the limiting value of a sequence of average rates of change;

CC2.04 - demonstrate an understanding that the derivative of a function at a point is the instantaneous rate of change or the slope of the tangent to the graph of the function at that point;

CC3.01 - describe the key features of a given graph of a function, including intervals of increase and decrease, critical points, points of inflection, and intervals of concavity;

CC3.02 - identify the nature of the rate of change of a given function, and the rate of change of the rate of change, as they relate to the key features of the graph of that function;

CC3.03 - sketch, by hand, the graph of the derivative of a given graph.

Prior Knowledge and Skills

· understanding of a secant line and evaluating its slope

· understanding of a tangent line and its geometric relationship with the curve

· developing and applying of formulas in spreadsheet applications

· understanding of the two different notations for functions that can be used interchangeably
(i.e., y = x² and f(x) = x²)

Planning Notes

· These activities are meant to occur sequentially, but with some minor adjustments, they could become independent activities.

· The teacher should ensure the availability of computers for the use of Zap-a-Graph or other graphing technology and spreadsheet applications. In some instances graphing calculators could be used.

Activity 2.3a: That Function is Very Derivative of Something

Time: 2.5 hours

Description

Students approximate slopes of tangents to a given polynomial function at a series of points along the function using the limiting values of the slopes of secants in the neighbourhood of these points. The slopes of the tangents will then be used to graph the derivative function on the same set of axes.

Planning Notes

· Use Zap-a-Graph or other graphing technology for verification.

· Teachers should put students in groups of two or three.

· The teacher should have at least one acetate sheet and marking pen available per group, as well as several pieces of graph paper.

· Teachers may want to provide a spreadsheet template for the students. This will speed up the calculation process and allow the students to see, very quickly, the direct relation of the limiting value of the slope of the secants to the slope of the tangent. A sample spreadsheet is shown below for the function y = -x³ + 5x² ! 2x ! 8. Note the form of the equation for the indicated cells (the formulas for columns D, E, G, and H are similar) and, of course, the formulas in each column can be copied down as far as necessary.

	A	B	C	D	E	F	G	H	I	J
1	a	f(a)	Slope of Secant Δx = 0.5 to the left of x = a	Slope of Secant Δx = 0.1 to the left of x = a	Slope of Secant Δx = 0.01to the left of x = a	Limiting Value of Slopes of Secants at x = a	Slope of Secant Δx = 0.01 to the right of x = a	Slope of Secant Δx = 0.1 to the right of x = a	Slope of Secant Δx = 0.5 to the right of x = a	Slope of Tangent at x = a
2	-10	1512	-419.75	-405.51	-402.350	-402	-401.650	-398.51	-384.75	-402
3	-8	…	…	…	…	…	…	…	…
…	…	…	…	…	…	…	…	…	…
12	10	…	…	…	…	…	…	…	…

The defining equation for cell B2: -(A2)^3 + 5*(A2)^2 - 2*A2 - 8.

The slope of the secant in cell C2 is calculated using the formula , which can be written using the defining equation: ((-(A2 - 0.5)^3 + 5*(A2 - 0.5)^2 - 2*(A2 - 0.5) - 8) - B2) / (-0.5).

In cell H2, the equation would be:(B2 - (-(A2 + 0.1)^3 + 5*(A2 + 0.1)^2 - 2*(A2 + 0.1) - 8)) / (0.1)

Teaching/Learning Strategies

A. Teacher Facilitation

· Each group should be given a different function to analyse.

· Include functions such as . Simple polynomials such as would also be of interest.

· Other sample functions have been provided. Third-degree functions should be included since they are relatively simple functions, yet show the complexities of inflection points and turning points (maxima and minima). Any polynomial function would be sufficient, however, so long as the characteristics to which the teacher wishes to draw attention occur in a relatively small range
(say .)

· The following sample cubic functions all have integer roots between for :

· Students determine the graph of the derivative by using the limiting value of a sequence of slopes of secants and relating that to the slope of the tangent.

· Each group should record the graphs of their function and its derivative on the same set of axes on an acetate sheet. Teachers may alternatively use chart paper to display these graphs around the room.

· Students should be encouraged to experiment with the scale of their axes so that all of the critical points of the graph are shown.

· This activity exemplifies the power of the spreadsheet to do complex, yet tedious, calculations. To this end, the complexity of some of the spreadsheet formulas will require the teacher to keep careful watch over the class as the activity progresses.

· Most graphing software packages (such as Zap-a-Graph) have utilities that can be used to support the findings of this activity (such as determining the graph of the derivative function). The use of such software packages should be encouraged where appropriate in the MCB4U classroom.

B. Student Activity

Suggestions for teacher facilitation are included throughout this activity in italics. Some solutions are included to aid in the flow of the activity. It may be necessary at this point to remind students of the proper usage of the “y =” and “f(x) =” notations.

1. Set up a spreadsheet with column headings similar to what is shown below:

f(a)

Slope of Secant

Dx = 0.5

to the left of

x = a

Slope of Secant

Dx = 0.1

to the left of

x = a

Slope of Secant

Dx = 0.01

to the left of

x = a

Limiting Value of Slopes of Secants

at x = a

Slope of Secant

Dx = 0.01

to the right of

x = a

Slope of Secant

Dx = 0.1

to the right of

x = a

Slope of Secant

Dx = 0.5

to the right of

x = a

Slope of Tangent

at x = a

2. Using even integers between –10 and 10 for x = a, complete the table of values in the first two columns of the spreadsheet.

3. Graph the given function using the table of values. Be sure to adjust the scales of the vertical and horizontal axes to show all relevant characteristics of the function in the interval . The teacher may want to remind students of the general shape of a cubic function. Graphing software may be used in order to verify the shape of the graph of the given function.

4. For every x-value x = a in the table, calculate the slope of the secant line from the point where x = a to the point where x = a – 0.5, or in other words, Dx = 0.5 to the left of x = a [for example, the points at which x = 6 and x = 5.5.] Record this value in the third column, “Slope of Secant Dx = 0.5 to the left of x = a”. Depending on how knowledgeable the students are with spreadsheets, they may approach this in different ways. Although it may appear complex (see the Planning Notes section), the slope of the secants can be calculated with a single formula, (specifically ). Some students may require the use of intermediate columns to insert the second point
((a ! 0.5), f(a ! 0.5)), and then calculate the slope using the “rise over run” formula .

5. Repeat Question 4, using Dx = 0.1, and record the results in the next column (Column 4). Repeat this question once more, using Dx = 0.01, and record the results in the sixth column, “Slope of Secant Dx = 0.01 to the left of x = a”. [CGE4b]

6. For every x-value in the table, calculate the slope of the secant line from the point where x = a to a point 0.5 units to the right of x = a [for example, between the points at which x = 2 and x = 2.5.] Record this value in Column 9 of the spreadsheet.

7. Repeat Question 6, using Dx = 0.1 and Dx = 0.01, and record the results in Columns 8 and 7, respectively.

8. Complete the spreadsheet by looking across the table, noticing any patterns in the slopes of the secants and entering an appropriate value in Column 6, Limiting Value of Slopes of Secants. Justify this choice. As they look across the columns of the spreadsheet, students will notice the apparent convergence of values on either side of their column. Students should now be reinforcing the idea that the slope of the tangent is the limiting value of the slopes of the secants. If they do not see a convergence, students should be encouraged to insert extra columns for Dx = 0.001 and
Dx =0.0001.

9. Now that there are values in the Limiting Value of the Slopes of Secants column, fill in appropriate values in the Slope of the Tangent column. Students should recall that as the interval for the secant becomes small, the average rate of change approximates the slope of the tangent at the given point. They should be encouraged to use a simple cell reference to automatically transfer the limiting value data to the slope of the tangent column rather than manually copying over the data.

10. On the same set of axes, graph the values of the best estimates of the slopes of the tangent (the last column). That is, plot the set of ordered pairs (x, y) such that the x values are the same as the original function and the y values are the values of the slopes of the tangent at those x values. This graph is called the Derivative of the Function. Describe what the derivative of a function represents. Students should equate the graph of the derivative and the graph of the slopes of the tangents. The range of the derivative function represents the slopes of the tangents for the corresponding domain values. [CGE4b]

11. Conjecture the degree of the derivative function.

12. Using Zap-a-Graph, or some other graphing technology, plot the given function (using Zap-a-Graph, go to the Options menu and choose Derivative to plot the graph of the derivative). Compare this graph to your own. Explain any differences. Note that a function’s derivative is often denoted in the following way: f ' (read “f prime”).

13. Carefully transfer both graphs of the function and the derivative to the acetate sheet provided for display purposes. Place the derivative on the same set of axes as the original function. [CGE2c]

C. Follow-up Skills

1. After each group has completed their respective investigation, they present their results to the rest of the class, drawing attention to Questions 9, 10, and 11. The teacher may wish to facilitate a classroom discussion incorporating some of the results of this investigation. Some important concepts include:

· the connection of the limiting value of the slopes of the secants to the slopes of the tangents;

· the concept of the derivative being the graph of the instantaneous rate of change against the original x-values;

· how the degree of a polynomial function compares to the degree of its derivative.

2. The previous activity involved the calculation of the slope of the tangent at a specific point (for example, x = -4). The teacher may also want to facilitate a discussion regarding the calculation of the slope of the tangent at a general point x = a, using the limiting value of secants. This discussion could then be used as a lead-in to the first-principles definition of the derivative, specifically .

Accommodations

· Although technology is used extensively throughout this activity, it should not be a hurdle for students to reach the conceptual ideas. Thus some students may need some extra support during the activity.

· The groups should be arranged heterogeneously such that students will have appropriate peer support.

Assessment & Evaluation of Student Achievement

Observing and conferencing can be used to assess the students’ Knowledge and Understanding while the activity is in progress. Connecting the limiting value of the slopes of the secant to the tangent and the derivative demonstrates Inquiry/Problem Solving skills. The students’ acetate sheets can be used to assess Communication skills (correct use of mathematical notations, correct use of mathematical language, accuracy of graphs, etc.), and should form the basis of a classroom discussion facilitated by the teacher to confirm and summarize results. The teacher may wish to assign tasks to each member of a group and assess students on the performance of their task. Sample tasks may include “copy producer” (scribe, assessed on written Communication skills), “data engineer” (develops spreadsheet data, assessed on Knowledge and Inquiry skills), “public relations specialist” (representative to explain results, assessed on oral Communication skills), etc. Considering the flexibility and adaptability of the students as they create the spreadsheet data could assess learning skills, specifically initiative. Teamwork skills can be assessed by observing the groups as they complete their investigation. This should focus on the students’ communication skills and time and resource management skills. The students’ organization can be assessed during their presentations using an appropriate oral presentation rubric.

Resources

http://www.ima.umn.edu/~arnold/calculus/tangent/tangent-g.html - a web page that shows an animation of zooming in on the tangent at a point.

http://www.ima.umn.edu/~arnold/calculus/secants/secants2/secants-g.html - a web page that shows an animation of the limiting value of the slope of the secant

http://www.ies.co.jp/math/java/calc/doukan/doukan.html - a java applet that shows the construction of the derivative by “surfing” on the slope of the tangent.

http://www.ies.co.jp/math/java/calc/limsec/limsec.html - a java applet that shows that the limiting value of the slope of the secant is the tangent.

Activity 2.3b: Stay Away from Sweet Graphing Exercises,
They’ll Give You Concavities!

Time: 2.5 hours

Description

Students use the graphs of functions and their respective derivatives to establish the connections between their behaviours. Students define increasing and decreasing functions, and make generalizations about the nature of critical points. Using rates of change of rates of change, students investigate concavity and its relationship to the graphs of a function and its derivative.

Prior Knowledge & Skills

· Students will need to understand that the derivative of a function at a point is the slope of the tangent to the graph of the function at that point.

· Students should be familiar with “prime notation” for denoting derivatives.

Planning Notes

· The teacher prepares an overhead projector and the graph of a polynomial function (preferably of degree four) transferred onto an acetate.

· Students should be placed in groups of two or three. The teacher may wish to keep the same grouping of students as in Activity 2.3a – That Function Is Very Derivative of Something.

· Each group receives a duplicate of each of the acetates completed in the previous activity. If this activity is done independently of the previous activity, the teacher will need to provide each group with the graphs of several polynomial functions and the graphs of their respective derivative functions on a separate set of axes, but on the same sheet. Chart paper may also be used to present the graphs.

· Students will need markers, highlighters, coloured pencils, or some other method of distinguishing various characteristics of graphs. It is recommended that students use yellow and orange highlighters, and black, red, and blue markers.

Teaching/Learning Strategies

A. Teacher Facilitation

· The teacher presents the graph of a polynomial function to the class on an overhead projector. The teacher may wish to make this a polynomial of degree four (e.g., y = x⁴ ! 7x³ + 11x²+ 7x ! 12), as students will be asked to revisit this graph after completing the activity to apply their knowledge in analysing the characteristics of this curve.

· Using a journal, the teacher will ask each student to describe the given curve in such detail as to allow someone else to sketch the curve using only the written description. Students could work in pairs and alternate roles.

· Students should try to avoid a table of values approach, although it is understood that certain points (intercepts, for example) are important to the description.

· At the conclusion of this short writing session, the teacher facilitates a class discussion to consolidate the students’ findings.

· The teacher groups the students and pass out the duplicates. Each group receives all duplicates, but will be assigned a particular function for which they are to be responsible, in case they are asked to present their findings.

· In Unit 1, the students had used the term turning points to represent local maximum and minimum. This activity introduces the more formal name for these, “critical points.” Students should be reminded that in mathematics there may be more than one way to represent the same thing.

· Although they are introduced in this activity, inflection points will not get a very rigorous treatment. The specific conditions and behaviour of these points will be clarified in Unit 6, Curve Sketching. This way, students will have a full compliment of tools to look at first and second derivative behaviour at these points.

B. Student Activity

Suggestions for teacher facilitation are included throughout this activity in italics. Some solutions are included to aid in the flow of the activity.

1. a. A function is said to be increasing if its graph is “going up to the right.” This is obviously not a very formal definition. Refer to the graph of the first function in your handouts. Over what interval(s) is the graph increasing? Trace the increasing portion of the function using a yellow highlighter.

b. Pick two x-values in one of the increasing intervals (call them ), so that . Without referring to the graph, what mathematical property would indicate that the graph is increasing? The y-value of the second point should be greater than the y-value of the first point.

c. Refer now to the graph of the derivative of the first function in your handouts. Use the interval(s) of increase that was (were) determined in Question 1 and trace the graph of the derivative over the same interval using the yellow highlighter. The teacher may point out that students are simply transferring the intervals over to the graph of the derivative, without any regard for what the graph of the derivative looks like in these intervals. Thus, students should not expect the graph of the derivative to be increasing as well. [CGE2c]

d. Over what interval(s) is the function decreasing? Trace the decreasing portion of the function using an orange highlighter.

e. Pick out two x-values in one of the decreasing intervals (call them ), so that . Without referring to the graph, what mathematical property would indicate that the graph is decreasing? The y-value of the second point should be less than the y-value of the first point. [CGE5a]

f. With the orange highlighter, trace the graph of the derivative in the intervals over which the function is decreasing. [CGE2c]

g. It would be important to know the points at which a function stops increasing and begins decreasing. Using a black marker, indicate with a dot the point(s) at which the graph of the function stops increasing or decreasing. On the graph of the derivative, indicate the corresponding point using the black marker. These are the turning points, as introduced in Unit 1. Why might these points be called critical points? Students should be led to the conclusion that critical points are (in polynomial cases at least) sufficient to show the general shape a function, because they show “maximum” and “minimum” points. [CGE2c] The teacher may wish to use this opportunity to formally introduce the terms “local maximum” and “local minimum.”

2. a. Recall that the derivative of a function at a point is the slope of the tangent of the graph of the function at that point. Keeping this in mind, explain increasing and decreasing functions using slopes.

b. Describe the connection between the slope of the graph of the function and the graph of its derivative. When the graph of the function is increasing, the graph of the derivative is positive (above the x-axis). When the graph of the function is decreasing, the graph of the derivative is negative (below the x-axis).

c. Use slopes to explain the significance of critical points. Critical points for polynomial functions occur when the slope of a function has a value of 0.

d. Describe the graph in the neighbourhood of each critical point. What conclusions could be stated regarding the signs of the slopes before and after a critical point in comparison to the shape of the graph near the critical point? Students should conjecture that the sign of the slope will change near a local maximum or minimum. The teacher may need to point out this is not always the case (as in the case of the function y = x³).

3. a. Refer once again to the intervals over which the function is increasing. In what ways is the graph similar over these intervals? In what ways is the graph different over these intervals?

b. Concavity is a term that refers to the curvature of a function. The curvature of a typical bowl placed on a table is said to be concave upward. If the bowl is turned upside down, its curvature is said to be concave downward. Using a red marker, trace the sections of the function that are concave upward. Over what interval(s) is the function concave upward? Trace the graph of the derivative over this (these) interval(s) using the red marker. [CGE2c]

c. Over what interval(s) is the function concave downward? Trace the graph of the function and its derivative in this (these) interval(s) using a blue marker. [CGE2c]

d. Points at which a function’s concavity changes are called points of inflection. Using the black marker, indicate any points of inflection for the original function with a Ä. Indicate as well the corresponding point(s) on the graph of the derivative. [CGE2c]

4. a. Using slopes, describe the difference between an increasing function that is concave upward and an increasing function that is concave downward. Similarly, describe the difference between a decreasing function that is concave upward and a decreasing function that is concave downward. The teacher should encourage students to see that phrases like “increasing at an increasing rate” and “slopes are positive and increasing” are two different ways to describe the same shape.

b. Describe the connection between the concavity of the function and the graph of its derivative. The students’ description should relate the concavity of the function to the slope (or the increasing/decreasing nature) of the graph of the derivative.

c. Concavity can be defined as the “rate of change of the rate of change” of a function. Justify this definition. [CGE5a]

As the teacher circulates the classroom, he/she may wish to pick a few groups to present to the class their results from a particular part of the activity.

Extensions

1. Use the graphs of to support or refute that critical points alone are always enough to determine the shape of the graph? The graphs of and would tend to support the argument that the critical points alone are sufficient for determining the shape of a function. The graph of , however, contains several critical points, all with a value of 0, as well as asymptotes, which are also very important in determining its shape. Through this example, the teacher could introduce students to the notion that values at which a function is undefined can also be thought of as critical values.

2. Consider the vases that would have the profiles shown:

a) If each vase was to be filled with water at a constant rate of a litres/s, discuss how the rate of change of height with respect to time will change for each shape. Sketch a graph of each function. Use this graph to sketch the height function for each vase.

b) Describe any connections between the shape of the vases, the graphs of the height functions, and the rate of change of the height functions.

C. Follow-Up Skills

Many important concepts will need to be reviewed at the conclusion of this activity. These concepts form the foundation upon which the students later solve related rates and optimization problems. In addition, these skills will be applied in postsecondary calculus courses.

· Questions 1b) and 1e) prompt students to define for themselves the concepts of increasing and decreasing functions. The teacher may wish to summarize by using the general definition

Let f be defined on an interval I. Given any two numbers in I such that,

a. f is increasing if and > 0.

b. f is decreasing if and < 0.

· Students are also asked to define increasing and decreasing intervals using slopes (in Question 2), which will be used in the study of optimization problems. The teacher may need to assist the students in formalizing this definition; specifically that function f is increasing if and decreasing if.

· In Question 1g) students describe critical points as points at which a function changes from increasing to decreasing. They refine this description in Question 2c) to propose that critical points are values for which . The teacher should then ask the class whether or not it is possible to produce the graph of a function for which at some point for which the function does not change from increasing to decreasing (like ).

· In Question 4a) students consider the implications of a changing slope on the concavity of a function. It is imperative that students recognize any discussion of “rate of change” or “slope” as a discussion of the derivative. Thus a change of slope implies “a change in the rate of change.” Put in this way, students should be able to more easily make the connection between how a function increases or decreases and its concavity.

D. Supplemental Research

The teacher may wish to assign these questions as a homework exercise, a journal topic, or an assignment to be completed in class, as part of the Follow-Up Skills section.

1. Analyse the polynomial presented at the beginning of the activity, y = x⁴ ! 7x³ + 11x² + 7x ! 12. Include in this analysis a discussion of the intervals of increase and decrease, the slope of the function, critical points, concavity, the slope of the derivative, and points of inflection. This is the polynomial that the teacher is to present on the overhead projector at the beginning of the activity.

2. Discuss characteristics of quadratic functions using the same criteria as Question 1.

3. How does concavity relate to linear functions? Students should make the conclusion that the slope of a linear function does not change, thus .

4. Using increasing/decreasing functions and slopes, analyse the behaviour the lines and . The function neither increases nor decreases. Because of this, its slope has a value of 0. The line x = 3 is not a function. Its slope is undefined, which in the context of increasing and decreasing functions makes perfect sense, since one cannot tell whether the line is increasing or decreasing.

5. Sketch an increasing function that is concave upward. What “real-life” applications could be modelled using such a function? Examples may include population growth models. What applications could be modelled using an increasing, concave downward function? Examples may include projectile motion.

6. Sketch a decreasing function that is concave upward. What “real-life” applications could be modelled using such a function? Examples may include radioactive decay. What applications could be modelled using a decreasing, concave downward function? Examples may include projectile motion.

7. The term concavity is also used in physics and astronomy. What is its meaning in this context?

8. Research the development of “prime notation” for the derivative. Can a derivative be expressed in any other ways?

Assessment & Evaluation of Student Achievement

This activity lends itself to group presentations, which can be assessed using a suitable oral report rubric. The Follow-Up Skills and Supplemental Research sections contain questions requiring students to use their Inquiry, Application, and Communication skills. Criteria for assessment would include hypothesizing and justifying reasoning, and applying their knowledge in an unfamiliar setting. Use of the language of mathematics should also play a prominent role in any assessment of this activity. It is suggested that at this point that Knowledge/Understanding be assessed by a paper-and-pencil task, such as a quiz. Since inflection points are only introduced in this activity, when assessing students’ sketches of derivative functions of given functions, teachers should provide formative, descriptive, feedback only on points of inflection and not penalize students for slight inaccuracies.

Activity 2.4: Summative Assessment

Time: 2 hours

Description

A comprehensive, balanced summative assessment addressing all four Achievement Chart categories should be administered at the end of this unit. Students must be provided with the opportunity to demonstrate their ability to apply the skills and knowledge acquired in this unit. This summative assessment, which has two parts, can be used to determine the degree to which the students have met the expectations of this unit. The first part involves the study of the graphs of functions from a new perspective and is performed in groups. The second part will be a paper-and-pencil task to be completed individually.

Strand(s) & Learning Expectations

Ontario Catholic School Graduate Expectations

CGE4b - a self-directed, responsible, life long learner who demonstrates flexibility and adaptability;

CGE4f - a self-directed, responsible, life long learner who applies effective communication, decision making, problem-solving, time and resource management skills;

CGE5a - a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Advanced Functions, Underlying Concept of Calculus