Download Differentiation with the TI-89

Differentiation with the TI-89
To find the derivative of some function f (x) using the TI-89, go to F3: Calc . Then select
1: d( differentiate . The command line should then be completed in the following form:
d(f(x), x)
Example
q 1: To find the slope of the tangent line to the graph of function
f (x) = 3 (x2 − 1)/(x2 + 1) at the point where x = 2:
d(((x∧ 2 - 1)/(x∧ 2 + 1)) ∧ (1/3), x) enter
ans(1) | x = 2 enter
(The first command will display the derivative as an expression in x. We want to evaluate that
expression at x = 2. Hitting the 2nd key followed by the (-) key gives ans; that is, the last
result obtained. The final display will be 8 ∗ 52/3 ∗ 31/3 /225, or .149944.)
Example 2: Find the absolute maximum and/or absolute minimum value of the function
f (x) = 5x6 + 18x4 + 10x2 − 50x − 10 (if they exist).
5x∧ 6 + 18x∧ 4 + 10x∧ 2 - 50x - 10 sto enter
d(f, x) enter
ans(1) sto f1 enter
solve(f1 = 0, x) enter
(At this point we’re finding critical numbers of function f . The only value returned is .736527.)
d(f1, x) enter
ans(1) sto f2 enter
f2 | x = .736527 enter
(The value returned is 181.315, the value of the second derivative at the critical number. Since
there is only one critical number, and f 00 is positive at that number, the absolute minimum of
f is guaranteed to exist at that critical number. In this particular case, the terms of the second
derivative have all positive coefficients and all exponents are even, so it’s clear that the second
derivative will be positive.)
f | x = .736527 enter
(The value displayed is -35.3065, the absolute minimum value of f . There is no absolute
maximum.)
q
Exercise: Let f (x) = (sin(x))2 /(x2 + 1). Use the TI-89 to find:
a. the slope of the tangent line to the graph at x = π/2
b. the critical numbers of f between x = 0 and x = π.
c. the absolute maximum value of the function on the interval [0, π].
[Answer: a. -.243284 b. 1.33535 c. .5668]