Brain Teaser  Area enclosed by a parabola and a line

09152015, 09:42 PM
Post: #21




RE: Brain Teaser  Area enclosed by a parabola and a line
(09152015 05:08 PM)Gerson W. Barbosa Wrote: I am not familiar with the Prime, but I think it can handle that, can't it?Well, I've tried it first in Xcas, but I couldn't even solve the cubic equation x^3+u*x^2+u^2*x+1/(4*u^3)+u^3=0 for the left intersection point (and I've tried all kinds of solve functions). Then I've tried it also in the HPPrime emulator (in CAS mode), but with the same result: no solution. So I gave up  I know why I stay with my good old Derive CAS! Franz 

09162015, 02:06 AM
(This post was last modified: 09162015 02:41 PM by Gerson W. Barbosa.)
Post: #22




RE: Brain Teaser  Area enclosed by a parabola and a line
(09152015 02:44 PM)CR Haeger Wrote: **or an old student looking at this again after a few decades of rust. In this case Calculus Refresher, by A. A. Klaf might be handy. I bought this book last year in a local bookstore. But don't do like me, who kept it the shelf until today. The equation of the normal line shouldn't be difficult to derive, but it's ready for use on page 70: \[yy_{1}=\frac{1}{\frac{dy}{dx}}(xx_{1})\] For the case y = x^4, we have \(yy_{1}=\frac{1}{4x_{1}^{3}}(xx_{1})\) \((x_{1},y_{1})=(u,u^{4})\) \(yu^{4}=\frac{1}{4u^{3}}(xu)\) \(y=u^{4}\frac{xu}{4u^{3}}\) \(y=\frac{4u^{7}+ux}{4u^{3}}\) Thanks for posting this interesting problem! Gerson. Edited to fix a typo and to add a missing subscript (Thanks, Thomas!). 

09162015, 02:19 AM
Post: #23




RE: Brain Teaser  Area enclosed by a parabola and a line  
09162015, 05:12 AM
Post: #24




RE: Brain Teaser  Area enclosed by a parabola and a line
(09162015 02:06 AM)Gerson W. Barbosa Wrote: The equation of the normal line shouldn't be difficult to derive, but it's ready for use on page 70: There's a typo: \(dx\) and \(dy\) got switched. The correct formula is: \[yy_{1}=\frac{1}{\frac{dy}{dx}}(xx_{1})\] But then this notation can be misleading as the derivative should be evaluated where \(x=x_1\). We could use \(\frac{dy}{dx}\Big_{x=x_1}\) however I prefer to use \(f'(x_1)\). Quote:For the case y = x^4, we have The variable in the derivative should be \(x_1\): \(yy_{1}=\frac{1}{4x_1^{3}}(xx_{1})\) Quote:\((x_{1},y_{1})=(u,u^{4})\) Now we're back on the same road. Quote:Thanks for posting this interesting problem! 1+ Cheers Thomas 

09162015, 03:09 PM
(This post was last modified: 09162015 03:10 PM by Gerson W. Barbosa.)
Post: #25




RE: Brain Teaser  Area enclosed by a parabola and a line
(09162015 05:12 AM)Thomas Klemm Wrote:(09162015 02:06 AM)Gerson W. Barbosa Wrote: The equation of the normal line shouldn't be difficult to derive, but it's ready for use on page 70: Thanks, Thomas, for pointing these out! The example and the illustration in the book make it quite clear, but by looking at the formula alone that notation indeed seems to be misleading. In my handwritten solution above I used a capital X instead of x1, but that might be a bit confusing. I prefer your lowercase u as no subscript is required. I'd revised the formula before posting, but I missed the dy and dx swap. Well, I typed that with the keyboard on my lap, the mouse sliding on top of a pile of books and the monitor far on on the floor, a totally antiergonomic way. (My wife reclaimed the sewingmachine stand I had been using for months, thus disturbing the progress of science ) Cheers, Gerson. 

09162015, 03:55 PM
Post: #26




RE: Brain Teaser  Area enclosed by a parabola and a line
(09162015 03:09 PM)Gerson W. Barbosa Wrote: The example and the illustration in the book make it quite clear, but by looking at the formula alone that notation indeed seems to be misleading. Katie will probably love that title. Cheers Thomas 

09162015, 04:14 PM
Post: #27




RE: Brain Teaser  Area enclosed by a parabola and a line
(09162015 03:55 PM)Thomas Klemm Wrote:(09162015 03:09 PM)Gerson W. Barbosa Wrote: The example and the illustration in the book make it quite clear, but by looking at the formula alone that notation indeed seems to be misleading. The Copyright date is 1944. Perhaps I should have suggested a more modern book, intended for technical persons. Gerson. 

09182015, 05:51 PM
Post: #28




RE: Brain Teaser  Area enclosed by a parabola and a line
Just for the record, here are some numerical results for the cases y=x^4, y=6 and y=8. Notice the minimum areas occur for normal lines passing near y(3/4), y(5/6) and y(7/8), respectively. The values of u for minimum areas should be accurate to 7 or 8 digits.
y=x^4 u: 0.730000000000 Area: 1.232203807010 u: 0.740000000000 Area: 1.229721793524 u: 0.741869418802 Area: 1.229582549556 u: 0.743490000000 Area: 1.229543561785 u: 0.743514882370 Area: 1.229543552881 u: 0.743515349953 Area: 1.229543552884 u: 0.743543197140 Area: 1.229543564411 u: 0.744136760955 Area: 1.229549111764 u: 0.75000000000o Area: 1.230145031369 y=x^6 u: 0.810000000000 Area: 1.120523733964 u: 0.814713174965 Area: 1.119930602299 u: 0.814723174965 Area: 1.119930598819 u: 0.814724747205 Area: 1.119930598753 u: 0.814733174965 Area: 1.119930600634 u: 0.815000000000 Area: 1.119932604732 u: 0.820000000000 Area: 1.120664993006 y=x^8 u: 0.86790000000 Area: 1.24466886334 u: 0.86800000000 Area: 1.24466861630 u: 0.86796579708 Area: 1.24466859303 u: 0.86797579708 Area: 1.24466858824 u: 0.86798579708 Area: 1.24466859303 u: 0.86804996475 Area: 1.24466885164 u: 0.86810000000 Area: 1.24466932686 Gerson. 

09182015, 08:20 PM
Post: #29




RE: Brain Teaser  Area enclosed by a parabola and a line
We start with a slightly different definition of \(h(x)=g(x)f(x)\) but now consider it as a function of two variables \(u\) and \(x\):
\[h(u,x)=\frac{ux}{2u}+u^2x^2\] From this we define the indefinite integral: \[H(u,x)=\int h(u,x)\, dx\] For each \(u\) we find the lower limit \(a(u)\) and the upper limit \(b(u)\) as roots of \(h(u,x)=0\): \[\begin{align*} a(u) &= (u+\frac{1}{2u})\\ b(u) &= u \end{align*}\] We evaluate the antiderivative at these limits, calculate the derivative and set it equal to 0: \[\frac{\mathrm{d} }{\mathrm{d} u}(H(u,b)H(u,a))=0\] To calculate the total differential we use partial differentials: \[\frac{\mathrm{d} }{\mathrm{d} u}H(u,x)=\frac{\partial H}{\partial u}+\frac{\partial H}{\partial x}\frac{\partial x}{\partial u}\] From the definition of \(H(u,x)\) we find: \[\frac{\partial H(u,x)}{\partial x}=h(u,x)\] However this function is 0 when evaluated at the lower or the upper limit. Thus this term vanishes. This leaves only the first term: \[\frac{\partial}{\partial u}H(u,x)=\frac{\partial}{\partial u}\int h(u,x)\,dx \] But we can switch the order of integration and differentiation: \[\begin{align*} \frac{\partial}{\partial u}\int h(u,x)\,dx &=\int \frac{\partial}{\partial u}h(u,x)\,dx \\ &=\int\frac{\partial}{\partial u}(\frac{ux}{2u}+u^2x^2)\,dx \\ &=\int\frac{\partial}{\partial u}(\frac{1}{2}\frac{x}{2u}+u^2x^2)\,dx \\ &=\int\frac{x}{2u^2}+2u\,dx \\ &=\frac{x^2}{4u^2}+2ux \end{align*}\] We evaluate this expression at the upper and lower limit and find the difference. But we can do that for both terms separately: \begin{align*} \frac{b^2a^2}{4u^2} + 2u(ba) &= (ba)\left(\frac{b+a}{4u^2}+2u\right)\\ &= \left(2u+\frac{1}{2u}\right)\left(2u\frac{1}{8u^3}\right) \\ &= \frac{4u^2+1}{2u} \cdot \frac{16u^41}{8u^3} \\ &= \frac{(4u^2+1)(16u^41)}{16u^4} \\ &= \frac{(4u^2+1)(4u^2+1)(4u^21)}{16u^4} \\ &= \frac{(4u^2+1)^2(4u^21)}{16u^4} \\ &= \frac{(4u^2+1)^2(2u+1)(2u1)}{16u^4} = 0 \\ \end{align*} We conclude that \(2u1=0\) which leads to the solution \(u=\tfrac{1}{2}\). Kind regards Thomas 

09212015, 05:53 PM
(This post was last modified: 09212015 09:00 PM by fhub.)
Post: #30




RE: Brain Teaser  Area enclosed by a parabola and a line
(09182015 05:51 PM)Gerson W. Barbosa Wrote: y=x^4Hi Gerson, since your Area result for x^8 looked rather illogical for me (IMO the area should continuously decrease for higher exponents n), I've written a small program (in TurboPascal) for this problem. The left intersection point is of course calculated numerically (no exact value possible for n>4), but the integral is calculated exactly (i.e. no numerical method), and the results really confirm my assumption of continuously decreasing areas: in fact for n>inf the results are: u>1 and A>0 Here's the list for n=2..20: Code:
Code:
Franz 

09212015, 06:40 PM
(This post was last modified: 09212015 06:53 PM by Gerson W. Barbosa.)
Post: #31




RE: Brain Teaser  Area enclosed by a parabola and a line
(09212015 05:53 PM)fhub Wrote: since your Area result for x^8 looked rather illogical for me (IMO the area should continuously decrease for higher exponents n), I've written a small program (in TurboPascal) for this problem. Hello Franz, Thanks again for pointing out yet another wrong result of mine. Thanks also for providing solutions to this plentiful of cases. Since my results for y=x^6 are correct within the accuracy I stated, I surely have made a mistake in the function submitted to the numerical integration and solver for case y=x^8 (I have yet to find it out). Yes, there's no need to use a numerical integrator as the functions are quite easy to integrate [ integral (x^n)dx = x^(n+1)/(n+1) + k, n ≠ 1 ], only the left integration limits being the complicated parts. I've found that Derive 4.11 is installed in my HP200LX and tried some numerical integrations (very slow on the 200LX, BTW). But I haven't explored it much as it is not so intuitive and didn't find a manual. Best regards, Gerson. 

09212015, 09:48 PM
(This post was last modified: 09212015 09:53 PM by Gerson W. Barbosa.)
Post: #32




RE: Brain Teaser  Area enclosed by a parabola and a line
(09212015 05:53 PM)fhub Wrote:(09182015 05:51 PM)Gerson W. Barbosa Wrote: y=x^4 The following might have been closer if my function Y had been Y := u*u*u*u*u*u*u*u  (xu)/(8*u*u*u*u*u*u*u)  x*x*x*x*x*x*x*x rather than Y := u*u*u*u*u*u*u*u  (xu)/(6*u*u*u*u*u*u*u)  x*x*x*x*x*x*x*x. In FreePascal the yellow is brighter and the blue is fainter, but this is not an excuse as I ought to have mistrusted the increase in the area. y=x^8 u: 0.840000000000 Area: 1.038962719953 u: 0.849000000000 Area: 1.034585573337 u: 0.849999511578 Area: 1.034517355546 u: 0.850000000000 Area: 1.034517342489 u: 0.850322146435 Area: 1.034513041971 u: 0.851000000000 Area: 1.034532095727 u: 0.860000000000 Area: 1.038377424828 

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