8.+Tangents+and+Derivatives+-+Teachers

Limits and Discontinuity Textbook Analysis

Limits and Discontinuity : The Facts
The BOS syllabus does //not// require formal definitions of limits, continuity and differentiability to be covered - so it is sad that our text books then choose to attempt to do so and make a total mess of it.

It would be best to take the syllabus advice and stay with an intuitive but mathematically correct feel for limits. If you do want to go there, this approach might work:
 * Limits**
 * When we say a function f(x) has a limit L as x->a, we are looking at what happens to the values of f(x) when our x value is in a tight range around x=a. Every value of f(x) in that range of x will be in the near the limit value L.
 * To be more precise, if the limit exists, then for //any// interval around the limit L, let's call it L-e < L < L+e, we can //always// find a small range x-d < x < x+d around a where every value of f(x) will be inside that interval L-e < L < L+e.
 * //The actual value of the function (if defined) at f(a) is irrelevant// to the value of the limit at f(a). Substituting a value to find a limit is equally meaningless - even though it works for regular polynomial functions students are used to.
 * Pender's limit definition is the closest to being accurate (7I p.268) but misses the key idea of getting as close as we want to the f(x) value by choosing our x value to be in a range around x=a.
 * Another interesting function to look at is the Dirichilet function f(x) = 1, x is rational, and = 0, x is irrational. A very surprising function - it's a nice recap of piecewise functions and of rational/irrational numbers. I ask my students to draw it to see what they come up with before discussing it.


 * Continuity and Discontinuity**
 * **It is meaningless to discuss a function in places it is not defined.** As such, it is meaningless to describe the continuity or otherwise at asymptotes - the function is just not defined there. It is therefore incorrect to say the function 1/x is discontinuous at x=0 : it isn't defined there.
 * Pender is the only text to accurately define continuity at a point (7I p.270) although he does several times fall into statements about discontinuity at asymptotes.
 * **"Smooth curves"** : Some very interesting 'smooth' curves don't have limits at certain points. Check out the function sin(1/x) near x=0 (not to be confused with (sin x)/x). Probably best not to show this function to students at this stage though.


 * Differentiability**
 * Only Pender goes here (7K). Good explanation for checking differentiability of piecewise function and looking at points on a function where the tangent is vertical. Note this is not in the syllabus.

**Textbook Analysis**
//Personal opinions of Mr Zuber - these comments do not necessarily reflect the views of the faculty.//

Grove's presentation is totally counter to the advice of experts on teaching calculus (Barnes, Little).
 * Grove "Maths in Focus Extension 1 Prelim, 2nd Edition"**
 * Section 8.1 has a promising start in the text, but the exercises require students to draw accurate gradient functions before any real exploration has occurred.
 * Of more concern, Section 8.2 immediately dives into differentiability - prior to a good grounding in the ideas of differentiation. //Differentiability is not in the syllabus// - and introducing an entire exercise on the concept at this stage is a conceptual overload when we should be creating a clear space to focus on the essentials of this topic.
 * 8.3 is mechanical calculation of limits exercise, and at this stage has no obvious (to the student) connection to the topic. Only serves to add to the confusion created in 8.2
 * 8.4 introduces dy/dx notation into mix - before students have practised working with f'(x). Even Pender - who assumes stronger students - waits until much more work using f'(x) has been done.
 * 8.5 has a section aptly titled "the Basic Rules" - this topic is reduced to a set of rules - a theme that continues throughout the rest of the chapter.
 * 8.11 "Angle between two curves" - just for kicks, add in this extension content that is not even part of the BOS Topic 8 syllabus. Is this really the place for this? Again - even Pender doesn't cover this content at this stage.

Another presentation totally counter to the advice of experts. An extremely dense sequence, heavy on limits and formulaic approaches.
 * Fitzpatrick "New Senior Mathematics Two Unit Course"**
 * The introduction promises a good start with a discussion on velocity and the limiting process.
 * 14.1 dives into limits, with many limit theorems introduced (without proof).
 * No exploration of gradient function suggested - straight into the limits definition, then differentiation from first principles.
 * 14.4 is aptly titled "Rules for differentiation" - a purely rules-based approach and no proofs.
 * 14.5 on differentiability has nice graphs.
 * 14.6 "Further rules for differentiation" continues the rules-based approach. In just one section covers *all* the rules. At least there is an sketch of a proof of the product and chain rules.

A careful, well paced introduction to differentiation. Possibly too heavy in places for some students.
 * Pender "Cambridge Mathematics 3 Unit Year 11"**
 * 7A: Chapter opens with a quality geometric exploration. Perhaps a little heavy getting students to calculate derivatives of semi-circles at this stage! Good explanation of the difficulties with using the word 'tangent' as compared to the use of tangents of cicles.
 * 7B: Clear explicit statement of derivative of the sum of functions and of the multiple of a function.
 * 7D: Pender wisely holds off using the Leibnitz notation until this section and is explicit about what it is and is not.
 * 7E: Composite functions are carefully explained as a 'chain of functions'. Differentiation of parametric and inverse functions - may be pushing it for some students at this stage.
 * 7H: Good early work on rates of change, providing stronger intuitive feel of the topic. Placed in the context of the chain rule.
 * 7I/7J : Saves the limits, continuity and differentiability discussion for the //end// of the topic - in contrast to most other text books.
 * 7K : Extension section on Implicit Differentation. Again - may be pushing it for some students at this stage.


 * Jones & Couchman "3 Unit Mathematics Books 1"**
 * A good opening sequence developing intuitive understanding of gradient of a curve at a point.
 * Use of delta x notation helpful in understanding limiting values and in working through examples of differentiation by first principles.
 * Excellent presentation of differentiation by first principles for sqrt(x) and 1/x.
 * Last section of the chapter goes into Topic 14: Application of calculus.

Resources
Mary Barnes Reflections article 1995: @http://hsc.csu.edu.au/maths/teacher_resources/2384/prof_reading/journals/barnes/M_Barnes_Nov_95.html //"This will help to combat the meaningless symbol manipulation which is all that many students experience in traditional calculus courses" //

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