Two calculator exam problems

05272014, 01:47 PM
Post: #1




Two calculator exam problems
I would greatly appreciate seeing solutions to the following problem:
A. Let f(x) = 1 +x(2  x(3  x(4  x(1 + x)))). Find the solution to f(f(x)) = 1/2 that is nearest to x = 0. And, especially if they would provide further insight, solutions to this: B. Let f(x) = sin (1/x) and define g(x) = f(f(f(f(x)))). Find the maximum value of g(x) for x in the range 102 to 103.2. Many thanks in advance. 

05272014, 02:48 PM
Post: #2




RE: Two calculator exam problems
Tailor series ?


05272014, 03:24 PM
(This post was last modified: 05272014 03:30 PM by CR Haeger.)
Post: #3




RE: Two calculator exam problems
(05272014 01:47 PM)Peter Murphy Wrote: I would greatly appreciate seeing solutions to the following problem:Using HP Prime A. Defining f(x):=1 +x(2  x(3  x(4  x(1 + x)))), then g(x):=f(f(x)) then fsolve(g(x)=1/2, x, 1..1) yields five roots between 1 to +1 with +0.21399.... closest to zero. Defining Function.F1:=g(x)1/2 lets you view this interesting curve. B. Similarly. define f(x):=sin(1/x) then g(x):=f(f(f(f(x))). Defining Function.F2:=g(x) lets you graphically explore g(x) in the range 102103.2 (assumed degrees?). Although 102 appears highest, 103.2 is max at 0.9459...  there seems to be a local max of ~1.00 at ~103.25 degrees.. Best, Carl 

05282014, 02:11 PM
Post: #4




RE: Two calculator exam problems
Carl, the Prime seems ideally suited for such problems.
Can you (or anyone) show me how to set up and solve them on a 50g (in RPN mode) or a 48G? I've puzzled over this for a good while. The concept of the userdefined function (UDF) for the 48/50 calculators seems promising, but what I've done hasn't produced the structure that you exhibit in your Prime examples. I now find that David Hayden solved such a problem (problem 13) back in 2009: [url:http://www.hpmuseum.org/cgisys/cgiwrap/hpmuseum/archv019.cgi?read=159915] but he's taking for granted what I cannot see at all, namely how one sets up such nested functions in the first place. 

05282014, 05:02 PM
Post: #5




RE: Two calculator exam problems
Peter  Im not very expert at the HP50G so I hope others will provide you (and I) some tips on how to setup these problems.
I will say that the HP Prime has been very nice to work with when in comes to symbolic/numerical solving and function graphing analysis tasks. 

05282014, 06:30 PM
Post: #6




RE: Two calculator exam problems
(05282014 02:11 PM)Peter Murphy Wrote: but he's taking for granted what I cannot see at all, namely how one sets up such nested functions in the first place. Code: 'f(x)=SIN(1/x)' ENTER You can have a look at how these function are defined. Just push them on the stack. For instance f(x): Code: \<< \> x 'SIN(1/x)' \>> Should you prefer RPL you could use this instead: Code: \<< \> x \<< INV SIN \>> \>> HTH Thomas 

05282014, 06:54 PM
(This post was last modified: 05282014 06:57 PM by dizzy.)
Post: #7




RE: Two calculator exam problems
The first one is easy:
Code: \<< Which yields: Quote:{X=0.213992196216 X=... } Similarly: Code: \<< You could graph this or use try to optimize it somehow. 

05282014, 09:29 PM
Post: #8




RE: Two calculator exam problems
Thanks Thomas and Dizzy for exactly what I was hoping to see.
Now to sit down and work out your solutions; I'll report back tomorrow. Peter 

05292014, 02:32 PM
Post: #9




RE: Two calculator exam problems
Thanks again Thomas and Dizzy. Your suggestions helped me break through my mental barrier on this stuff.
I still must explore Dizzy's solution to problem A, but I rendered f(x) as the corresponding coefficient array [ 1 1 4 3 2 1] 'F' STO. Then this programmed expression << F F * PEVAL PEVAL .5  >> STEQ allowed me to plot g(x) and find x = 0.21399. Now for problem B. I am grateful for the existence of this Forum and for the generosity of its Users. 

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