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DIFFERENTIAL CALCULUS
Reflections 
, November, 1993 -
Journal of the Mathematical Association of NSW
by PAUL WHITE, Australian Catholic University
One of the main obstacles to understanding calculus is the range of advanced mathematical concepts involved. A common solution for teachers to overcome this obstacle has been to reduce calculus to a series of unrelated, manipulative rules. Such a strategy has been reasonably successful for obtaining a satisfactory HSC mark. The HSC 2 Unit paper of 1992 (Question 7(c) to be precise) indicated that such an approach may not suffice into the future. Also, plans for the new syllabuses indicate a less manipulative approach to calculus, with more emphasis on concepts involved. The focus here will be the latter, but we begin with a quick look at what happens now in many classrooms.
Current approach?
The derivative is introduced by looking at a secant through
the points with abscissae x and 
(or x + h) on the curve
, considering how the slope of the secant
changes as the increment decreases and comes quickly to
the conclusion that the slope of the tangent is the
'limiting position' of the secants and so the
slope of the tangent, which is now called the derivative,
is given by
The rule immediately generates the derivative of 
as 2x and is extended to 
. The sequence then moves quickly to finding
all sorts of derivatives with all sorts of complicated
notation.
Questions
Why is the slope of a tangent important?
What is a tangent? What's different about y and
x - why does x get small and not y?
What does limit mean? Why can 0/0 be a a number with limits?
Let's explore.
Example 1
In column 2 are six containers and nine
graphs showing how the height of water increases if the
container is being filled at a constant rate.
(a) Choose the correct graph for each bottle.
(b) For the remaining three graphs, sketch what the bottles would look like.
Example 2
The following is a distancetime graph
of a bus.
(a) What is the average speed of the bus for the first 5 seconds?
(b) What is the actual (instantaneous) speed of the bus after 5 seconds?
What are the graphical interpretations of average and instantaneous rate of change?
Example 3
A class of students working on travel graphs produced the diagrams below. Some of these graphs cannot possibly be correct. Can you select the impossible ones and explain why? For the correct ones, describe what the person moving is doing.
What is the difference between the variable on the horizontal axis and that on the vertical axis?
Example 4
Judy has recorded her height over the years
on each birthday by marking it on a bare patch of wall. This
can be thought of as assigning height to age. (Does it make
sense to assign age to height?) Judy's growth can be
modelled as a function by a table of values where age is in
years and height in centimetres.
| Age | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
|---|---|---|---|---|---|---|---|---|
| Height | 140 | 143 | 147 | 153 | 160 | 166 | 169 | 172 |
Draw a graph representing Judy's height compared to her age. Does it make sense to join the dots? Can you estimate her height at 15 and a half? Can you estimate her height when she was 9? How tall do you expect her to be at 20? Which gives more information, the graph or the table?
What is a function? How are functions usually represented? What does continuous mean?
Differential calculus
Differential calculus is the study of the instantaneous rate of change of continuous functions.
The instantaneous rate of change of a continuous function is called the derivative. Graphically, the slope of the tangent at a given point is the instantaneous rate of change, and so the derivative. Taking the derivative as the dependent variable for each value of the independent variable of a continuous function gives a new function (the gradient or derivative function) which describes how the original function is changing at any given value of the independent variable.
Some reflections on research
The tangent becomes important because it is the instantaneous rate of change, whereas the secant is the average rate of change and the limiting process links the two. Research has shown, however, that the concept of the tangent needs to be more fully developed beyond the limited example of a circle.
Research has also shown that limit theory is too difficult for most students and I would recommend that for 2 Unit students it be addressed only informally. In fact, the whole definition of the derivative can be addressed informally - when was the last time first principle was asked in an HSC or even a 3 Unit paper?
Research has also cast doubts on student ability to handle variables and thus functions. This is a major issue on its own and is not addressed here.
Some reflections on teaching
A possible sequence for teaching introductory calculus along the lines suggested is:
Pre-calculus
In high school, this could be addressed in
Years 9/10 or even extension work in Years 7/8. The basic idea
is to investigate graphs in a non-numeric sense. Such
things as 'which is going faster - why?' or 'which
graph fits which physical situation - why?' are some
suggestions for investigation. Graphs can also be used to
explore the difference between constant and varying rates of
change, as well as increasing and decreasing rates.
Introductory calculus
This could now become a numeric
study of graphs that links the notion of instantaneous rate of
change with the slope of the tangent to a curve. Students can
actually calculate rates of change by drawing tangents to
curves and working out rise and run. (Note that the difference
between instantaneous and average rates of change is vital!)
This can then become the definition of derivative and, also,
some historical background may be appropriate.
Functions
Graphs describe relationships between
changing quantities. The notion of variables as changing
quantities needs to be addressed. The difference between
independent and dependent variables is important. Functions as
sensible relationships between variables can also be
introduced. One way to represent a function is a table, another
is a graph. Which gives more information? Does it make sense to
join the dots?
Derivative
How can instantaneous rates of change be
measured apart from graphical approximations? This is where
algebra is useful because variables describe changing
situations. Hence functions which are described by equations
might now be considered. Using computer packages or
calculators, gradient functions can be found for some well
known functions. For example, calculating the slope of the
tangent to (say) 
for a range of values and then plotting
these slopes against the independent variable will give
the line y=2xas the gradient function.
The formal definition for the derivative can then be
'derived' by seeing how the instantaneous rate of
change (the slope of the tangent) can be approximated by
average rates of change (secants).
Applications
Using change in its graphical
interpretation - the slope of the tangent - as the basis for
calculus allows some of the rules to make sense, even if the
'proof' is less rigorous. For example, graphically (or
with a computer) show that the gradient function of sine is
cosine. Consider the statement: 'The graph of a continuous
function is concave up if the second derivative is
positive.' This statement encapsulates the problem facing
teachers and students of calculus. The rule can be remembered,
correctly applied and yet not be understood. However, with a
re-emphasis towards understanding in the teaching, the position
can be changed. In this case, if the second derivative is seen
as the rate of change of the slope of the tangent, its being
positive means the slope is increasing and hence the curve must
be concave up.
Two examples of generalizations which, when proved
algebraically using limits and 
, are likely to be meaningless and as a
result become rote learnt rules for students, are the
product rule and the chain rule. Using visual descriptions
based on how a change in one variable affects the change
in another, the rules can be seen as generalizations.
The product rule 
can be explained by looking at the change
in the product uv when both u and
v change as the independent variable (probably
good old x) does. The diagram shows the change in
the product:
The shaded part of the diagram represents the pre-change
value of the product. The larger rectangle represents the
post-change value of the product. So, the unshaded part is the
difference. When the difference is compared to the change in
the independent variable (
x), the result is
The product rule emerges after a bit of hand waving.
The chain rule may be demonstrated by turning wheels as shown.
At given time t, wheel A has made m revolutions (say) and wheel B has made n revolutions (say). Suppose wheel A turns at 3 revs/unit time (so dm / dt=3), and that wheel B makes two turns for every one of wheel A (so dn / dm=2). Then obviously wheel B turns at six revs/unit time. So dn / dt =6. By observing that the wheel result works regardless of the numbers used, generalizing gives
Some reflections on resources
A good resource for studying graphs is The Red Box (Swan, 1989). Currently, the set of booklets by Mary Barnes (1992) is the best resource for an informal approach to calculus. The booklets have been developed in cooperation with teachers and educationalists throughout Australia. Computer software for the Macintosh is the program Anugraph. All these are available through the Australian Association of Mathematics Teachers.
References
Barnes, M. (1992). Investigating
Change. Curriculum Corporation, Melbourne.
Swan, M. (1989). The Language of Functions and Graphs. The Shell Centre for Mathematical Education, Nottingham.
For the IBM, David Tall's Graphic Approach to Calculus is the big resource. In Australia it is available through Edsoft.
Some reflections on assessment
Two ideas for slightly different assessment tasks in the Component B style are as follows:
1. Assess student interpretation rates of change from graphs. Independence of response could be introduced by requiring some open contextual description.
Example 1: The following graph shows the amount of petrol in a car during a trip. Looking at the graph, use the petrol consumption information to give a description of a possible trip.
Example 2: The population of the United States in the nineteenth century is given in millions in the following table:
| t | Year | Population |
|---|---|---|
| 0 | 1800 | 5.3 |
| 10 | 1810 | 7.2 |
| 20 | 1820 | 9.6 |
| 30 | 1830 | 12.9 |
| 40 | 1840 | 17.1 |
| 50 | 1850 | 23.3 |
| 60 | 1860 | 31.4 |
| 70 | 1870 | 38.6 |
| 80 | 1880 | 50.2 |
| 90 | 1890 | 62.9 |
| 100 | 1900 | 76.0 |
Using this information, give a description of the population growth in the United States between 1800 and 1900. Are there any predictions you could make (say about 1990)? Are they accurate?
2. Assess student understanding of how rates of change, variables and derivatives are related.
Find dy/dx for 
when x=b where a
and b are constants is a calculus question.
Create a 'word problem' which involves change so
that the solution to the problem is the calculus question
given for some specific values for a and
b. Note that x and y may
represent any variable you wish and you can use different
letters to represent them.
Organization of the tasks could vary according to how the task is presented - standard test questions; open book test; prepared question test (that is, know the question in advance); or assignment, where strikingly similar responses are penalized and / or a brief oral explanation is required by the student.
References
Mack, J. & Stephens, M. (1993) 'Assessing assessment' in The Australian mathematics Teacher, Vol. 49, No. 1.
Mousley, J. (1991). 'Years 7-10: preparing for the VCE' in J. Reilly & S. Wettenhall, Mathematics: Inclusive, Dynamic, Exciting, Active, Stimulating. Mathematics Association of Victoria.
NCTM (1989). Curriculum and Evaluation Standards for School Mathematics.
Ollerton, M. (1991). 'Testing versus assessment' in Mathematics Teaching. MT 135
Shell Centre for Mathematical Education (1984). Problems in Patterns and Number. Joint Matriculation Board.
Shell Centre for Mathematical Education (1985). The Language of Functions and Graphs. Joint Matriculation Board.
Stephens, M. (1988). AAMT Discussion Paper on Assessment and Reporting in Mathematics.
Stephens, M. & Izard (1992). Reshaping Assessment Practices: Assessment in the Mathematical Sciences Under Challenge. ACER, Melbourne.
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