Texas Instruments TI-89 Tip List page 420

Some numbers not effected when I change certain mode settings-why?
Check to see what type of numbers you are using. The "Display Digits" and "Exponential Formats" do
not effect integers and rational numbers.
Summation function (symbolic) algorithm?
Iterated addition is used for modest numeric limits. Other sums are determined from formulas given in
references such as "Standard Mathematical Tables and Formulae", CRC Press, together with tests for
telescoping sums. Limits or special formulas are used for infinite summation limits.
solve() function - why does it not find all the solutions?
The TI-92's solve() function attempts to solve an equation analytically to get an exact representation of
the solution(s). When it cannot do so, it uses numerical search techniques to attempt to find
approximate solutions. These search techniques cannot guarantee to find any or all solutions. When
multiple solutions are found, they cannot be guaranteed to be consecutive ones. Search techniques
are highly sensitive to the nature of the equation and the form in which it is given, the properties of the
individual functions, the fixed precision limitations of the floating point arithmetic, and any
boundaries placed upon the search.
In the case of sin(x)=cos(x) the solve() function can determine the exact solutions and can represent
them as an infinite family using the "arbitrary integer" notation (@n1). However, in the case of
cos(x)=-2e^(-2x), the equation is transcendental. The number of solutions is infinite, and the
solutions do become closer and closer to the roots of cos(x) as x moves toward infinity. However, the
solutions are not periodic and, in general, can only be approximated.
A very useful technique when using the solver is to graph the left and right sides of the equation and
look for intersections on the graph. Or, graph the difference between the left and right sides and look
for crossings of the x-axis. Investigating the equation graphically often suggests reasonable
bounds that can be placed on the solve() function using the with (|) operator. Providing reasonable
bounds often greatly improves the success of the numerical search techniques.
Solve ignores With ( | ) constraints with trig functions - why?
If the TI-92 is able to find an exact solution to an equation, and this solution is an infinite family of
solutions, the restrictions after the With operator are ignored. The answer returned will contain the
arbitrary integer notation and will not return specific answers within the given range.
If the TI-92 is unable to find an exact solution, then it resolves to using a numeric search for the
solutions (except in Exact mode). During a numeric search, the TI-92 will apply the constraints to the
solutions returned. If nsolve is used, the solution returned will be within the given range. However,
nsolve will only return one solution.
If the TI-92 is able to find a finite many exact answers, then the constraints will also be applied.
Examples:
solve(sin(x/2)=cos(x/2),x) | x>=0 and x<2Pi returns ((4@n -3)*pi) / 2 (radian/ auto mode)
nsolve(sin(x/2)=cos(x/2),x) returns -42.4115 (radian/auto mode)
nsolve(sin(x/2)=cos(x/2),x) | x>=0 and x<2Pi returns 1.5708 (radian/auto mode)
solve(sin(2x)=cos(x),x) returns a list of several values (radian/auto mode)
solve(sin(2x)=cos(x),x) | x>=0 and x<2Pi returns list of answers in given range (radian/auto mode)
B - 9