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;