HP 39gs Master Manual page 83

Gradient at a point as the limit of the slope of a chord
The true gradient at a point is available in a number of ways. For example, via the
view or via the δ differentiation operator. For students first being introduced to calculus a common task
PLOT
is to investigate the slope of the chord joining two points as the length of the chord tends towards zero.
⎛ f
ie.
lim ⎜
⎝
h→0
This can be effectively introduced via the Function aplet.
Begin by entering the function being studied into
To examine the gradient at x=3, store 3 into memory A in the
view as shown right.
Return to the view, un-
SYMB
expression:
F2(X)=(F1(A+X)-F1(A))/X
This is the basic differentiation formula quoted above with
role of h and being the point of evaluation, in this case with
A
Change to the
NUM SETUP
Your Own". By entering successively smaller values for
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
view, enter a new value into
HOME
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
at http://www.hphomeview.com.
( ) ( )
+ − ⎞
x h f x
⎟
⎠
h
the function
in
F2(X)
view and change the
and then return to the
A
as shown.
F1(X)
HOME
and enter the
F1(X)
.
taking the
X
.
A = 3
to "Build
NumType
you can now
X
view.
NUM
83
Slope tool in the