Wikipedia:Reference desk/Archives/Mathematics/2022 December 15

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

December 15

Approximating y = e^x

I want to approximate y = e^x in the range x = a to x = c with two line segments (a, e^a) (b, e^b) and (b, e^b) (c, e^c) such that the error defined as the area between the original y = e^x and the two segments is minimised. All I really care about is the ratio (b - a) / (c - a).

I'm pretty sure I could do this by finding an expression for the area, taking the derivative, setting it to zero and solving for x to find b, but it feels like the solution shouldn't depend on a or b or even e, so all of that seems like it might be overkill.

Is there an obvious answer that I'm missing?

2A01:E34:EF5E:4640:4210:3448:E6A9:8778 (talk) 22:03, 15 December 2022 (UTC)

If you increase a and c together while keeping (c-a) fixed, say a -> a+d and c -> c+d, that just scales the areas up by a multiplicative factor of e^d. Therefore, the quantities (b-a) and (c-b) only depend on (c-a). We can say b = a + f(c-a) where f is some function. Perhaps that's the "shouldn't depend" that you're intuiting? Given that observation, you can simplify the calculation by setting a=0, then work out the answer just as you suggested. --Amble (talk) 00:13, 16 December 2022 (UTC)

Hi! The only direct thing comes to my mind is Taylor series expansion. However, I am not sure regarding the ratio part of the question. Are you trying to say ration between area under curve in both line segments (?). If so, I would highly recommend to work out on paper and see if this is something you want. I hope this helps! :) Cheers, Nanosci (talk) 00:17, 16 December 2022 (UTC)

The Leibniz integral rule is simple here because the integrands are zero at $x=b$. I get:

${\begin{aligned}0&={\frac {\partial }{\partial b}}\left(\int _{a}^{b}\left({\frac {(b-x)e^{a}+(x-a)e^{b}}{b-a}}-e^{x}\right)\partial x+\int _{b}^{c}\left({\frac {(c-x)e^{b}+(x-b)e^{c}}{c-b}}-e^{x}\right)\partial x\right)\\&=\int _{a}^{b}\left({\frac {\partial }{\partial b}}\left({\frac {(b-x)e^{a}+(x-a)e^{b}}{b-a}}-e^{x}\right)\right)\partial x+\int _{b}^{c}\left({\frac {\partial }{\partial b}}\left({\frac {(c-x)e^{b}+(x-b)e^{c}}{c-b}}-e^{x}\right)\right)\partial x\\&={\frac {e^{a}+e^{b}(c-a)-e^{c}}{2}}\end{aligned}}$

so

$b=\ln \left({\frac {e^{c}-e^{a}}{c-a}}\right)$

catslash (talk) 01:10, 16 December 2022 (UTC)

Try $a=0$, $c=1$, then $b=\ln(e-1)\simeq 0.541325$ which seems plausible. catslash (talk) 01:30, 16 December 2022 (UTC)

On further consideration, for any function $f(x)$, the area is stationary when

$f'(b)={\frac {f(c)-f(a)}{c-a}}$

that is, the point at which the slope of the function is the average slope over the interval. catslash (talk) 02:32, 16 December 2022 (UTC)

I think this is correct. Here is another way of obtaining the result. By scaling we can choose $a=0,c=1.$ The piecewise linear function $f_{b}$ on the interval $[0,1]$ fixed by the three points $(0,1),(b,e^{b}),(1,e)$ never assumes values exceeding those of $y=e^{x},$ so to minimize the difference in areas we can simply seek to maximize the area between $f_{b}$ and the x-axis, formed by two trapezoids glued back-to-back:

$I_{b}=\int _{0}^{1}f_{b}(x)dx={\frac {1}{2}}b(1+e^{b})+{\frac {1}{2}}(1-b)(e^{b}+e).$

Solving ${\frac {d}{db}}I_{b}=0$ results in $b=\log(e-1).$  --Lambiam 07:21, 16 December 2022 (UTC)

Thankyou all. So as I suspected a and b are not relevant, but e actually is. I had the feeling that the base would end up canceling out and the ratio would end up being a common constant such as 0.5 or the golden ratio or something. 08:34, 16 December 2022 (UTC) — Preceding unsigned comment added by 2A01:E34:EF5E:4640:6743:DBBA:BC3:542 (talk)

Knowing the answer from having ground through the integrals, hindsight makes it obvious geometrically:

• The area between the line segments and the curve is the area under the line segments minus the area under the curve.
• The area under the curve is fixed, so the problem reduces to minimizing the area under the line segments (subject to the middle point being on the curve)
• That is equivalent to maximizing the area of a triangle whose (upward facing) 'base' is a single straight line between the end points and whose 'apex' is the middle point (b, f(b)) of the line segment pair.
• The area of a triangle is half the base times the height, so the area is stationary (and possibly maximal) when the apex is moving parallel to the base.
• So the solution lies at a point where the slope of the curve is equal to the average slope over the interval.

catslash (talk) 19:23, 16 December 2022 (UTC)