8.4.2 Table of variations of a function
The table of variations of a function consists of
-
The first row, for the variable, which gives the endpoints of
subintervals of the domain, as well as any critical points and
inflection points.
- The second row, for the derivative, which gives the values of the
derivative at the values in the first row (or limits as the variable
approaches one of the values) and between them the sign (+ or −)
of the derivative in the corresponding subinterval.
- The third row, for the function, which gives the values of the
function at the values in the first row, and between them whether
the function is increasing or decreasing in the corresponding
subinterval.
- The fourth row, for the second derivative, which
gives the values of the second derivative at the values in the
first row, and between them whether the second derivative is
positive or negative (and hence whether the graph is concave up or
concave down) in the subinterval.
The tabvar command finds the
table of variations of a function.
-
tabvar takes one mandatory argument and one
optional argument.
-
expr, an expression of a single variable.
- Optionally, x, the variable (by default, x=x).
- tabvar(expr ⟨,x ⟩) returns the
table of variations of the function f(x)=expr and
draws the graph on the DispG screen, accessible
with the menu
Cfg ▸ Show ▸ DispG.
Examples
plotfunc(x^2-x-2,x=((-3.393824) .. 4.574184))
Inside Xcas you can see the function with Cfg>Show>DispG.
|
| |
| ⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣ | | x | −∞ | | | | +∞ |
| y′=2 x−1 | −∞ | − | 0 | + | +∞ |
| y=x2−x−2 | +∞ | ↓ | | ↑ | +∞ |
| y′′ | 2 | +(⋃) | 2 | +(⋃) | 2
|
| ⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦ |
|
| | | | | | | | | | |
|
plotfunc((2*t-1)/(t-1),t=((-2.893824) .. 5.074184))
Inside Xcas you can see the function with Cfg>Show>DispG.
|
| |
| ⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣ | | t | −∞ | | 1 | 1 | | +∞ |
| 0 | − | || | || | − | 0 |
| 2 | ↓ | −∞ | +∞ | ↓ | 2 |
| y′′ | 0 | −(⋂) | || | || | +
(⋃) | 0
|
| ⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦ |
|
| | | | | | | | | | |
|
Note that in the second example the value 1 appears twice in the first row, so
that both one-sided limits of y can be displayed at the
vertical asymptote t=1. The values of 2 for y at −∞
and ∞ indicate a horizontal asymptote of y=2.