Texas Instruments TI-82 Manual page 32

To test the reasonableness of the conclusion that
positive values of x (since you want the limit as x → ∞). For example, evaluate f (100), f(1000), and f (10,000).
Another way to test the reasonableness of this result is to examine the graph of f (x) =
that extends over large values of x. See, as in Figure 2.79 (where the viewing rectangle extends horizontally from 0
to 100), whether the graph is asymptotic to the horizontal line y = 2 (enter
Figure 2.78: Checking
1.11.2 Numerical Derivatives: The derivative of a function f at x can be defined as the limit of the slopes of secant
+ ∆ −
f x (
'
=
lines, so f x ( ) lim
∆ →
x 0
good approximation to the limit.
The TI-82 has a function nDeriv( in the MATH menu to calculate the symmetric difference
So to find a numerical approximation to f´'(2.5) when f (x) = x
ALPHA X, 2.5 , .001 ) ENTER as shown in Figure 2.80. The format of this command is nDeriv(expression,
variable, value, ∆x). The same derivative is also approximated in Figure 2.80 using ∆x = 0.0001. For most
purposes, ∆x = 0.001 gives a very good approximation to the derivative and is the TI-82's default. So if you do use
∆x = 0.001, just enter nDeriv(expression, variable, value).
Technology Tip: It is sometimes helpful to plot both a function and its derivative together. In Figure 2.82, the
−
5 x 2
function f (x) = and its numerical derivative (actually, an approximation to the derivative given by the
+
2
x 1
32
G T
RAPHING ECHNOLOGY
2 x
lim
x
→∞
x
sin 4 x
lim = 4
x
→
x 0
− ∆
x ) f x ( x )
And for small values of ∆x, the expression
∆
2 x
Figure 2.80: Using nDeriv(
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G : TI-82
UIDE
−
1
= 2, evaluate the function f (x) =
+
1
−
2 x 1
for Y
+
x 1
Figure 2.79: Checking
3
and with ∆x = 0.001, press MATH 8
−
2 x 1
for several large
+
x 1
−
2 x 1
in a viewing rectangle
+
x 1
and 2 for Y
).
1 2
−
2 x 1
lim = 2
+
x 1
→∞
x
+ ∆ − − ∆
f x ( x ) f x ( x )
gives a
∆
2 x
+ ∆ − − ∆
f x ( x ) f x ( x )
∆
2 x
^ 3,
X,T,θ
.