# The Hermite Function; The Horner Function; The Variable Vx - HP 49g+ User Manual

Graphing calculator.

## The HERMITE function

The function HERMITE [HERMI] uses as argument an integer number, k, and
returns the Hermite polynomial of k-th degree. A Hermite polynomial, He
is defined as
=
, 1
He
He
0
An alternate definition of the Hermite polynomials is
*
=
, 1
H
H
0
n
n
where d
/dx
= n-th derivative with respect to x. This is the definition used in
the calculator.
Examples: The Hermite polynomials of orders 3 and 5 are given by:
HERMITE(3) = '8*X^3-12*X'
And
HERMITE(5) = '32*x^5-160*X^3+120*X'.

### The HORNER function

The function HORNER produces the Horner division, or synthetic division, of a
polynomial P(X) by the factor (X-a).
polynomial P(X) and the number a.
polynomial Q(X) that results from dividing P(X) by (X-a), the value of a, and the
value of P(a), in that order.
example, HORNER('X^3+2*X^2-3*X+1',2) = {'X^2+4*X+5', 2, 11}. We
could, therefore, write X
example: HORNER('X^6-1',-5)=
6
15624}
i.e.,
X
-1 = (X

### The variable VX

A variable called VX exists in the calculator's {HOME CASDIR} directory that
takes, by default, the value of 'X'.
independent variable for algebraic and calculus applications. Avoid using
the variable VX in your programs or equations, so as to not get it confused
with the CAS' VX. If you need to refer to the x-component of velocity, for
example, you can use vx or Vx.
variable see Appendix C.
n
d
2
n
x
2 /
(
)
=
(
) 1
(
x
e
n
n
dx
n
d
2
n
x
( *
)
=
(
) 1
x
e
n
n
dx
The input to the function are the
The function returns the quotient
In other words, P(X) = Q(X)(X-a)+P(a).
3
2
2
+2X
-3X+1 = (X
+4X+5)(X-2)+11.
{'X^5-5*X^4+25*X^3-125*X^2+625*X-3125',-5,
5
4
3
2
-5*X
+25X
-125X
+625X-3125)(X+5)+15624.
This is the name of the preferred
For additional information on the CAS
2
x
2 /
),
=
1
2 ,
,...
e
n
2
x
(
),
=
1
2 ,
,...
e
n
,
A second
Page 5-20
(x)
k
For  