A Generalization of the Parabolic Chord PropertyAuthor(s): John MasonReviewed work(s):Source: The College Mathematics Journal, Vol. 42, No. 4 (September 2011), pp. 326-328Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/10.4169/college.math.j.42.4.326 .Accessed: 24/09/2012 02:27

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References

1. J. Hass, M. D. Weir, and G. B. Thomas, University Calculus, Pearson Addison-Wesley, Boston, 2007.2. C. H. Edwards and D. E. Penny, Calculus, Early Transcendentals, 7th edition, Prentice Hall, Upper Saddle

River NJ, 2008.3. C. R. McConnell and S. L. Brue, Microeconomics: Principles, Problems, and Policies, 15th edition, McGraw

Hill, Boston, 2002.

A Generalization of the Parabolic Chord PropertyJohn Mason (j.h.mason@open.ac.uk) Oxford, England

It is well known that the tangents at either end of a chord of a parabola meet in apoint aligned vertically with the midpoint of the chord. In other words, the point ofintersection of the tangents at the two ends of a chord on a parabola and the midpointof that chord lie on a line parallel to the axis of the parabola. In this JOURNAL, Sten-lund [3] showed that this midpoint property characterizes quadratic polynomials; thenKrasopoulos [1] extended the property to Rn; and Xu [4] showed that the propertycharacterizes quadratics in Rn . The property has a natural generalization.

Theorem. Given two distinct points on a polynomial of degree d, the Taylor polyno-mials of degree d − 1 at those points meet in d − 1 points whose mean is verticallyaligned with the midpoint of the chord joining the two points.

Proof. Without loss of generality, translate the origin to one end of the chord. Letthe degree d polynomial be

p(x) =d∑

k=1

ck x k,

where cd 6= 0, then the Taylor polynomial of degree d − 1 at the origin is simply

T0(x) =d−1∑k=1

ck x k .

Let (t, p(t)) be another point on the graph of the polynomial. Then the Taylor poly-nomial at t is

T1(x) =d−1∑k=0

p(k)(t)

k!(x − t)k .

The two “tangential functions” T0 and T1 are both of degree d − 1, and so, in gen-eral, they meet in d − 1 points (some of which may be complex) whose first coordi-nates are the roots of D(x) = T1(x)− T2(x). The sum of the roots of D is, apart fromsign, the coefficient of xd−2 divided by the coefficient of xd−1.

One way to find the terms of T of degree k, avoiding derivatives, is to collect p(x +t) in powers of x , delete the powers of x greater than k, and then substitute x − t for

http://dx.doi.org/10.4169/college.math.j.42.4.326

326 „ THE MATHEMATICAL ASSOCIATION OF AMERICA

x (see McAndrew [2]). Following this procedure, and concentrating on the terms thatgive rise to powers of x of degree d − 2 or greater leads to

p(x + t) = cd(x + t)d + cd−1(x + t)d−1+ cd−2(x + t)d−2

+ · · ·

= cd xd+ (dtcd + cd−1) xd−1

+

(d(d − 1)

2t2cd + (d − 1)tcd−1 + cd−2

)xd−2+ · · · ,

which makes

T1(x) = (dtcd + cd−1) (x − t)d−1

+

(d(d − 1)

2t2cd + (d − 1)tcd−1 + cd−2

)(x − t)d−2

+ · · · ,

in turn making

D(x) = dtcd xd−1−

(d(d − 1)

2t2cd xd−2

+ · · ·

),

where by assumption dtcd is not zero.Consequently, the sum of the d − 1 roots is (d − 1)t/2, and so the mean of the first

coordinates of the intersection points is t/2, i.e., the first coordinate of the midpoint ofthe chord. Thus the midpoint of the chord and the mean of the intersection points arevertically aligned.

Conversely The mean-of-the-intersection-points property essentially characterizespolynomials of degree d amongst functions with at least d − 1 continuous derivatives.Let p(x) be a function with at least d − 1 continuous derivatives on an interval con-taining the origin. If p has the property that the mean of the points of intersection ofthe Taylor series of degree d − 1 at the origin with the Taylor series of degree d − 1at any other point in the interval is always vertically aligned with the midpoint of thecorresponding chord, then p is a polynomial of degree d .

For the proof, let the Taylor polynomials of degree d − 1 be T1 and T2, as before,and let

D(x) =d−1∑k=0

(p(k)(t)

k!(x − t)k − ck x k

)be the polynomial whose roots give the first coordinates of the intersection pointsof the two Taylor series. The hypothesis requires that the mean of the roots be t/2,and so the sum of the roots must be t

2 (d − 1). Imposing this condition and using thebinomial theorem to pick out the relevant coefficients of xd−1 and xd−2, gives

p(d−2)(t)

(d − 2)!− cd−2 +

pd−1(t)

(d − 1)!− t (d − 1) =

−t (d − 1)

2

(p(d−1)(t)

(d − 1)!− cd−1

)which simplifies to

p(d−2)(t)

(d − 2)!−

p(d−1)(t)

(d − 1)!

t (d − 1)

2= cd−2 +

t (d − 1)

2cd−1.

VOL. 42, NO. 4, SEPTEMBER 2011 THE COLLEGE MATHEMATICS JOURNAL 327

Chasing coefficients is one route to the result, but there is a quicker way. Let

g(t) =p(d−2)(t)

(d − 2)!.

Then the imposed condition is transformed into

g(t)− (1/2)tg′(t) = cd−2 + cd−1t (d − 1)/2.

This differential equation in g forces g to be a quadratic, and hence p(t) must be ofdegree d .

Summary. The well known property of quadratic functions, that the tangents at either endof a chord of a parabola meet in a point aligned vertically with the midpoint of the chord isextended to polynomials of degree d. Given two distinct points on a polynomial of degree d,the Taylor polynomials of degree d − 1 at those points meet in d − 1 points whose mean isvertically aligned with the midpoint of the chord joining the two points. This characterizespolynomials of degree d.

Acknowledgment. The differential equation approach is due to the Classroom Capsule editor.

References

1. P. T. Krasopoulos, Tangent planes of a quadratic function, College Math. J. 34 (2003) 205–206; available athttp://dx.doi.org/10.2307/3595802

2. A. McAndrew. An elementary, limit-free calculus for polynomials. Math. Gazette, 94, no. 529 (2010) 67–83.3. M. Stenlund, On the tangent lines of a parabola, College Math. J. 32 (2001) 194–196; available at http:

//dx.doi.org/10.2307/2687469

4. C. Xu, A Characterization of a quadratic function in Rn . College Math. J. 41 (2001) 212–213.

Intriguing Limit

Limits depending on the expression (1+ α

x )x are a never ending story. What is

this one:

limx→∞

(e(−1)n

n

(. . .(

e14

(e−

13

(e

12

(e−1(

1+1

x

)x )x )x )x )x. . .)x

︸ ︷︷ ︸(n+1)−times

= ?

—Roman Wituła and Damian Słota,Silesian University of Technology

328 „ THE MATHEMATICAL ASSOCIATION OF AMERICA