How do I calculate the first and second deriatives on the TI-89?

Question

I have a function of f(2) which is a negative y value and f'(2) has a negative tangent but f"(2) is positive. How can I do this on my calculator. If so, How can I use algebra to find the second derivative from the first?

Answer 1

The Second Derivative Test is used to tell whether a critical value of a function is a relative maximum or a relative minimum of the function. Basically, if the graph is concave up at the critical value, then the critical value produces a (relative) minimum. If the graph is concave down at the critical value, the critical value produces a (relative) maximum. It is easy to implement functions on the TI-89 to tell whether a particular value of x produces a relative maximum or minimum. The functions are:

Define ismax(f, x, c) = ((d(f,x)|x=c) = 0) and ((d(f, x, 2)|x=c) < 0)
Define ismin(f, x, c) = ((d(f,x)|x=c) = 0) and ((d(f, x, 2)|x=c) > 0)

(It is easy to get the second function by simply editing the first one. If you are "shaky" on entering user-defined functions into your calculator, click here.) These functions are boolean functions - they return "true" or "false" as their values. You may also find it useful to have a function that just calculates the value of the second derivative at a particular value of x:

Define d2at(f, x, c) = d(f, x, 2)|x=c
This can save you a little typing, but you have to interpret the results for yourself.
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If that doesn't help, try the following website:

http://education.ti.com/educationportal/sites/US/nonProductMulti/pd_onlinecalculus89.html?bid=2