HP 39gs_40gs_Mastering The Graphing Calculator_English_E_F2224-90010

Gradient at a point as the limit of the slope of a chord

PLOT view or via the δ differentiation operator. For students first being introduced to calculus a common task

is to investigate the slope of the chord joining two points as the length of the chord tends towards zero.

The true gradient at a point is available in a number of ways. For example, via the

Slope tool in the

⎛

(

xh fx

)

⎞

)

−

(

ie.

lim

⎜ ⎟

h→0

⎝

⎠

This can be effectively introduced via the Function aplet.

Begin by entering the function being studied into

F1(X) as shown.

To examine the gradient at x=3, store 3 into memory A in the

HOME

view as shown right.

Return to the

SYMB view, un- the function F1(X) and enter the

expression:

F2(X)=(F1(A+X)-F1(A))/X in F2(X).

This is the basic differentiation formula quoted above with

X taking the

role of h and

A being the point of evaluation, in this case with A = 3.

Change to the

NUM SETUP view and change the NumType to “Build

Your Own”. By entering successively smaller values for

X you can now

investigate the limit as h tends towards zero.

In this case it is clear that the limit for x=3 is the value 6.

To investigate the gradient at a different point simply change back to the

HOME view, enter a new value into A and then return to the NUM view.

The disadvantage of the previous method is that

it is not very visual. An alternative is to use an

aplet downloaded from the web. An aplet that

will automate the process and provide a visual display of the

chord diminishing in length can be found on the author’s website

http://www.hphomeview.com