Thank you all for the very creative word problems using the concepts behind Newton's Method! We certainly are a very creative crowd, aren't we?
I'm afraid that I wasn't all that creative on my last test on 'Applications of the Derivative' which focused on Optimization and Related Rates. So I wrote a quick and dirty Newton's question:
(1) Let f(x) = 1 - tan(x-pi/2) (a) State the recursive equation describing Newton's Method for Approximating Roots.
(b) Apply this definition to f(x)
(c) Write an equation of the tangent line to f(x) at x0 = pi/2
(d) Use this equation to approximate x1.
(e) Set Mode Fix 9 and use prgmNEWT to create a convergence table with 6 decimal place accuracy starting at x0 given above.
(f) Multiply your best estimate to the first positive root of f(x) by 4/3. State the root exactly.
(g) Make a complete sketch of f(x) and the tangent line at x0. Label x0, x1 and lim n -> inf xn on the x-axis.
Other versions of this question can esily be written simply by changing the first line of the equation (for multiple versions of the test, make up exams, etc.). For example: (1) Let f(x) = 1 - sin^2(x) (1) Let f(x) = cos^2(x) -1 and the like.