Category: Calculus

Posted in Algebra, Calculus

8 Different Ways to Think of the Derivative

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In his paper The Transition to Formal Thinking in Mathematics, David Tall presents W.P. Thurston’s seven different ways to think of the derivative:

  1. Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.
  2. Symbolic: the derivative of x^n is nx^n?1, the derivative of sin(x) is cos(x), the derivative of f ? g is f ? ? g ? g? , etc.
  3. Logical: f ?(x) = d if and only if for every ? there is a ? such that when 0 <|?x|< ? , then slope
  4. Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.
  5. Rate: the instantaneous speed of f (t) , when t is time.
  6. Approximation: The derivative of a function is the best linear approximation to the function near a point.
  7. Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power. (Thurston, 1994.)

David Tall argued that the list excluded the global concept of local straightness so he added added the eighth that he believes that other 7 can be built.

8. Embodied: the (changing) slope of the graph itself.

In the same paper, David Tall presents a learning framework for derivative based on his Three Worlds of Mathematics Framework.

understanding derivative

Posted in Algebra, Calculus

Teaching the derivative function without really trying

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New mathematical ideas are usually built on another mathematical idea or ideas. Because of this, the teaching of mathematics if it is to make sense to students, should reflect this ‘building on’ process. Students should be able to see how the new idea is connected to what they already know. Good teaching of mathematics also demand that this new knowledge be useful and connected to the mathematics that students will encounter later.

Here is an example of a lesson that teaches the idea of derivative without really teaching it yet. This means that you can introduce this in Year 9 or 10 in their lesson about graphs of second degree function. The only requirement is that they understand the function of the form f(x) = ax^2. The task requires determining the equation of linear function of the form y = 2ax, which happens to be the derivative of ax^2. Of course you will not introduce the term derivative at this year level. You are just planting the seed for this important concept which students will encounter later.

The lesson uses the applet below. Of course, much of the success of the lesson will still be in questions you will asked after students initial exploration of the applet. You can find my proposed questions for discussion below the applet. [iframe https://math4teaching.com/wp-content/uploads/2012/02/Deriving_function_from_ax_2.html 750 620]

Questions for discussion

  1. You can move point A but not point B. Point B moves with A. What does this imply?
  2. What do you notice about the position of B in relation to the position of A?
  3. What is the path (locus) of point B? Right click it and choose TRACE then move A to verify your conjecture.
  4. What is the same and what is different about the coordinates of points A and B?
  5. To what does the coordinates of B depends on?
  6. What is the equation of the line traced by B?
  7. Refresh the applet then use the slider to change the equation of the graph, say a=3. What is the equation of the line traced by B this time?
  8. What do you think will be the equation of the path of B if the graph is f(x) = ax^2

By the end of this lesson students should have the intuitive notion of derivative and can find its equation given the function f(x) = ax^2.

There are actually 8 ways to think of the derivative. If you want to know more about Calculus, here’s a good reference:

The Calculus Direct: An intuitively Obvious Approach to a Basic Understanding of the Calculus for the Casual Observer

 

Posted in Calculus

Differentiation problem in parametric context with solution

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This is hot off the press-a question taken from the recently concluded 2011 September Preliminary Examinations of a school in Singapore. It deals with applications  of differentiation in the parametric context. Extensive trigonometry is employed here together with the manipulation of surd forms. I have personally worked out everything for your (the student’s) reference.

If you want a real calculus challenge, the problem below should satisfy your appetite. Peace.

QUESTION :

The parametric equations of a curve  are

x = sin2t and y= a cos t

where is a positive constant \frac{-\pi}{2} \le t \le\frac{\pi}{2}

(i) Find the equation of the tangent to the curve at the point P  where t= \frac{\pi}{4}.

(ii) The normal to the curve at the point Q where t = \frac {\pi}{3} intersects the axis at R. Find the coordinates of R and hence show that the area enclosed by the normal at Q , the tangent P and the x-axis is

differentiation

Author

Frederick Koh is a teacher residing in Singapore who specialises in teaching the A level maths curriculum. He has accumulated more than a decade of tutoring experience and loves to share his passion for mathematics on his personal site www.whitegroupmaths.com.

Mr Koh is also the author of the post Working with summation.

I have created a GeoGebra applet to visualize Question 1 above.