Math problem where graphing calculator may slow you down  part II.

09122014, 12:43 PM
(This post was last modified: 09122014 12:43 PM by CR Haeger.)
Post: #1




Math problem where graphing calculator may slow you down  part II.
Hello  here is a problem that should be fairly easily solvable by hand and with a trig table:
Assume f(x) = max(sin(x), cos(x)) using radians. If the area under the curve f(x) = 4.0 and the lower limit of x = 0.0, what is the upper limit of integration? What do your machines and/or brains come up with? Note  I got this to work on the HP Prime using the AREA() function in Function APP, but had problems using solve() or fsolve() properly. NumSolv on a TI36x had to think about this a looong time. 

09122014, 04:33 PM
(This post was last modified: 09122014 04:54 PM by Marcus von Cube.)
Post: #2




RE: Math problem where graphing calculator may slow you down  part II.
(09122014 12:43 PM)CR Haeger Wrote: If the area under the curve f(x) = 4.0Is the area positive if the function is negative? Or are we just talking about the definite integral? EDIT: If the area is always positive, the answer 2π. I split the interval at π/4, π, π+π/4, π+π/2, 2π. All the square roots cancel each other. Marcus von Cube Wehrheim, Germany http://www.mvcsys.de http://wp34s.sf.net http://mvcsys.de/doc/basiccompare.html 

09122014, 05:22 PM
Post: #3




RE: Math problem where graphing calculator may slow you down  part II.
(09122014 04:33 PM)Marcus von Cube Wrote:(09122014 12:43 PM)CR Haeger Wrote: If the area under the curve f(x) = 4.0Is the area positive if the function is negative? Or are we just talking about the definite integral? Hi Marcus, I should have stated that its a definite integral where area is negative when f(x) is negative. Your solution and abs(f(x)) are interesting too  thanks! 

09132014, 04:54 AM
Post: #4




RE: Math problem where graphing calculator may slow you down  part II.
My solution by hand: \(2\pi+\arccos(3\sqrt{2}4)\)
This is about: 7.6089 I used the HP15C with this program to calculate the function: Code: LBL A Cheers Thomas 

09132014, 09:12 PM
Post: #5




RE: Math problem where graphing calculator may slow you down  part II.
By hand and a bit of brain....
f(x) is periodic with a 2pi period. from 0 to pi/4 f(x)=cos(x) from pi/4 to 5pi/4 f(x)=sin(x) from 5pi/4 to 2pi f(x)=cos(x) Integrating is straightforward and gives 4/sqrt(2) over 0 to 2pi going up to 9pi/4 gives 5/sqrt(2). Then the value searched for is between 9pi/4 and 9pi/4+pi, in other words on the second interval of the definition of f(x). This equivalent to search for X with: sum from pi/4 to X of sin(x) dx=45/sqrt(2). Integrating analytically and solving lead to: X=acos(6/sqrt(2)4) The solution is then 9pi/4+Xpi/4~7.6089 Nothing more necessary than any standard scientific calculator... BTW, I'm cheating slightly, I checked the value of the integral with my own python software ;) which uses a HP like algorithm.... 

09142014, 02:18 PM
Post: #6




RE: Math problem where graphing calculator may slow you down  part II.
(09132014 09:12 PM)Bunuel66 Wrote: Integrating is straightforward and gives 4/sqrt(2) over 0 to 2pi We should make sure that the integral is always < 4. Thus we calculate it up to the first zero of the function \(f(x)\) which occurs at \(x=\pi\): \[ \int_{0}^{\pi}\max(\sin(x),\cos(x))dx=1+\sqrt{2}\approx 2.41421 \] Cheers Thomas 

09142014, 07:22 PM
Post: #7




RE: Math problem where graphing calculator may slow you down  part II.
Actually I made the check for the first maximum, I wanted just not to be too long with the explanations, just giving the general idea.
Regards 

11142014, 10:01 PM
(This post was last modified: 11142014 10:40 PM by Gilles.)
Post: #8




RE: Math problem where graphing calculator may slow you down  part II.
I used the numeric solver of the 50G in FIX 4
FIX 6 takes more time and returns X: 7.608894 This help to get the exact answer with the 50G CAS : 'X=2*PI+ACOS(3*SQRT(2)4)' 

11162014, 04:06 PM
(This post was last modified: 11162014 04:21 PM by CR Haeger.)
Post: #9




RE: Math problem where graphing calculator may slow you down  part II.
Thanks Giles,
Has anyone tried solving this using the HP Prime CAS, Home or Function APP? It seems to me there are only a couple of ways to find a numeric solution. I have not found any exact solution using the device either. Screenshot with CAS settings Exact, Complex and Use i unchecked. Graphically, it worked out pretty well, once I figured out AREA() syntax. Note that using integral from template did not work for me. 

11162014, 08:40 PM
(This post was last modified: 11162014 08:45 PM by Gilles.)
Post: #10




RE: Math problem where graphing calculator may slow you down  part II.
Hi CR Haeger, I tried with the Prime both in CAS and with the SOLVE APPS without success.
I agree with you about the plotter of the Prime : just fabulous ! I like very much to zoom instantly with 2 fingers 

11282014, 06:17 PM
Post: #11




RE: Math problem where graphing calculator may slow you down  part II.
I'm amazed at the TI 36 Pro. It took 30+ minutes (in numsolve), but it came up with the result!


11292014, 12:49 AM
Post: #12




RE: Math problem where graphing calculator may slow you down  part II.
I sped things up to under 15 minutes by doing a couple of experimental integrations to get an idea of where the integral would be ~4, then a quick table to zoom in on an appropriate guess, and then enter a good guess into numsolve.


11292014, 03:31 AM
Post: #13




RE: Math problem where graphing calculator may slow you down  part II.
...which leads me to...how to accomplish this on the WP34S? How do I solve for an integral ?


11292014, 09:30 AM
Post: #14




RE: Math problem where graphing calculator may slow you down  part II.  
11292014, 01:52 PM
(This post was last modified: 11292014 01:54 PM by CR Haeger.)
Post: #15




RE: Math problem where graphing calculator may slow you down  part II.
(11292014 12:49 AM)lrdheat Wrote: I sped things up to under 15 minutes by doing a couple of experimental integrations to get an idea of where the integral would be ~4, then a quick table to zoom in on an appropriate guess, and then enter a good guess into numsolve. Try turning off pretty print mode and setting the integrals resolution to say 0.1. May speed things up. I agree  the TI36x is pretty straightforward and capable for these type if solver problems. 

11302014, 01:33 AM
Post: #16




RE: Math problem where graphing calculator may slow you down  part II.
Thanks Thomas,
It works...couldn't remember how to have an integral in solve. Rather slow on this sort of problem! 

11302014, 12:16 PM
Post: #17




RE: Math problem where graphing calculator may slow you down  part II.
(11302014 01:33 AM)lrdheat Wrote: Rather slow on this sort of problem! For this specific problem the derivative is trivial: \[ \begin{align} F(x)&=\int_{0}^{x}f(t)dt4 \\ F'(x)&=f(x) \\ \end{align} \] Thus we can use Newton's method: \[ \begin{align} x'&=x\frac{F(x)}{F'(x)} \\ &=x\frac{\int_{0}^{x}f(t)dt4}{f(x)} \\ &=x+\frac{\int_{x}^{0}f(t)dt+4}{f(x)} \\ \end{align} \] Instead of starting the integration from \(0\) over and over again we can reuse the result of \(F(x)\) from the previous loop: \[ F(x')=F(x)+\int_{x}^{x'}f(t)dt \] This value is saved in register 01. Code: LBL'FX' Code: LBL'NWT' This will speed up the calculation. Cheers Thomas 

11302014, 07:04 PM
Post: #18




RE: Math problem where graphing calculator may slow you down  part II.
Excellent, clear, concise as usual.


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