Teaching Philosophy: It is extremely important to check ones results.
Ask yourself is the answer reasonable. Ask yourself is their a simple
way to check my result. Often there are easy ways to check, but sometimes
not. Using the TI-89 to check work already done by hand is the killer
application for Calculus 3. Is integral of cos 2x, 2 sin 2x or (1/2)sin 2x?
Use the TI-89 to take the derivative of 2 sin 2x and check. This is
the place in calculus 3 where the TI-89 is the most help. [Well it
does also do nasty integrals, and graphing too.] 

Killer App:
To find the local extremal (local mins and local maxs) for functions
of two variables you will need to two (non-linear) equations in two
unknowns. Many students have little experience with non-linear equations
and will sometimes make silly errors which result in getting either too few
or too many solutions. The TI-89 solves this problem by giving the
correct solutions so you can ``GUIDE'' your algebra and reduce the
number of these kinds of errors.

Examples:
1. Solve x^2-y^2=0 and x^2+y^2=1.

Wrong solution: Rewriting the first equation as x^2=y^2, a student
takes the square root of both sides (incorrectly) getting x=y. Putting
x=y into the second equation, yields 2x^2=1 and the two solutions
(1/sqrt(2),1/sqrt(2)) and (-1/sqrt(2),-1/sqrt(2)).

Corrected solution: Actually there are four solutions. x^2-y^2 =
(x-y)(x+y) = 0 so either x=y [covered above] or x=-y which yields
two more solutions (1/sqrt(2),-1/sqrt(2)) and (-1/sqrt(2),1/sqrt(2)).

TI-89 Solution: 
solve(x^2-y^2=0 and x^2+y^2=1,{x,y})
^^^^^          ^^^^^            ^
alg menu       blanks & alpha   type `,' but it doesn't show on the display


2. Solve x+y=0 and x^2+y^2=1

Wrong solution: The first equation gives y = -x or x = -y. a student
substitutes both equations into the second equation. Which yields
2x^2=1 or x=+/- 1/sqrt(2) and 2y^2=1 or y = +/- 1/sqrt(2). The student
forgets the first equation, and ends up with 4 solutions.
(1/sqrt(2),1/sqrt(2)), (1/sqrt(2),-1/sqrt(2)), (-1/sqrt(2),1/sqrt(2))
and (-1/sqrt(2),-1/sqrt(2)).

Corrected solution: The first equation says x and y have opposites
signs. There was no need to solve 2y^2=1 separately. The two solutions
are (-1/sqrt(2),1/sqrt(2)) and (1/sqrt(2),-1/sqrt(2)). Or if you
would rather x=+/- 1/sqrt(2) and y = -/+ 1/sqrt(2). (Note +/- on
first and -/+ on second.)

TI-89 Solution: 
solve(x+y=0 and x^2+y^2=1,{x,y})
Same keystroke hints.

Wrong solution2: A student, perhaps pressed for time, perhaps wanting
to jump to the interesting part of the problem. Enters the following
into the calculator
solve(x+y=0 and x^2+y=1,{x,y})
Not noticing the typo, the student uses the calculator's incorrect
answers ((1+sqrt(5))/2,-(1+sqrt(5))/2) and ((1-sqrt(5))/2,-(1-sqrt(5))/2).
Making the rest of problem both wrong and much harder to do. It is
hard to give much partial credit for such an answer. It is important
to solve problem by hand. Even more important since some math professors
forbid the use of calculators on tests.