The Tangent Parabola The AMATYC Review, Vol. 3, No., Fall, pp. 5-3. John H. Mathews California State University Fullerton Fullerton, CA 9834 By Russell W. Howell Westmont College Santa Barbara, CA 938 John H. Mathews has been on the faculty at California State University Fullerton since 969. He earned his PhD in mathematics from Michigan State University. He keeps active in the areas of complex analysis, numerical analysis and computer algebra. mathews@fullerton.edu Russell W. Howell has been on the faculty at Westmont College since 978. Prior to that time he was involved in operations research as an officer in the U.S. Army. He completed his PhD from The Ohio State University in 974 (mathematics), and in 986 earned an MSc in computer science from the University of Edinburgh. howell@westmont.edu it is based on two points, f ) and The secant line which is used to approximate y f x, f ) What if we used three points? Then we could determine a polynomial of degree n, which could also be used to approximate y f ( x Thus, we have the concept of the secant parabola with interpolation points, f )), h, f ), and, f ) Figure. The secant parabola approximating y sin( x), at x using h. When the interpolation points are moved closer to the middle point x the secant parabola approaches a limiting position. Figure. The secant parabola approximating y sin( x), at x using h.
The limiting position of the secant parabola is called the tangent parabola. The following figure shows this case when h goes to and all three nodes coincide. Figure 3. The tangent parabola approximating y sin( x), at x where Can you guess what the tangent parabola will turn out to be? We will reveal this pleasant surprise at the end of the article. The secant parabola. A precise discussion of the secant parabola is now presented. Recall that a polynomial of degree n expanded about x x can be written in the form () p( x) a + a ), is to be the interpolating where the coefficients a, a, and a are to be determined. Since p x polynomial for y f ( x), it must pass through the three points, f )), h, f ), and, f ) Using the first point x, f x relation p ) a + a x x ) f ), which implies that a f ( we obtain the Proceeding, we solve for the two coefficients a and a by first making the substitution a f ) in equation () and writing () p( x) f )+ a ). Then make substitutions for the two points h, f ), and, f ) ), respectively, in equation () and obtain two relations and p f )+ a h x h x ) f p ) f )+ a x x ) f Simplification produces the following two equations which will be used to solve for a and a. (3) f ) h a a f, f ) a a f
Subtract the first equation in (3) from the second and then divide by and get a f ( x ) f x h Add the equations in (3), subtract f ( x ) and then divide by and get a f ( x f x ( )+ f x The secant parabola formula., h, f x h ), and the form The secant parabola for y f ( x) which passes through x, f ), f ) involves the variable x and parameters x and h and has (4) p( x, x,h) f )+ f x A numerical experiment. f x h ) f x + f )+ f x x. Formulas for the above graphs of the secant parabola p( x, x, h) for y f ( x) sin( x) are constructed using formula (4 The computations are centered at the point x using the step sizes h.and.. Exploration. Use x and h. in formula (4) and compute the coefficients of x for the secant parabola p( x,,. f(.) f.9. f(.) f (.)+ f.9.89736.783369...8477..4385.78845..5394, and.89736.689497+.783369. which are used with the value f (.).8447 to obtain p( x,,.).8447+.5394( x ).4385( x ) The graphs of y f ( x) sin ( x) and y p( x,,.) are given in Figure. Exploration. In a similar fashion, use x and h. in formula (4) and obtain f(.) f.99..8468384.836598..85866..54933, and and ( x )
f(.) f (.)+ f.99..84464..473 which are used with the value f (.).8447 to obtain.846838446.68949696 +.83659786. p( x,,.).8447+.5493 ( x ).473 ( x ) The graphs of y f ( x) sin ( x) and y p( x,,.) are given in Figure. Finding the limiting numerically. The limit of the secant polynomials is found by evaluating formula (4) using decreasing step sizes h.,.3,.,.3,.,.3,and.. The numerical results are summarized in Table. step size h p ( x, x, h) where x..8447 +.454649 ( x ).3868 x.3.8447 +.5334 ( x ).47589 x..8447 +.5394 ( x ).4385 x.3.8447 +.54( x ).474 x..8447 +.5493( x ).473 x.3.8447 +.543( x ).4735 x..8447 +.543 ( x ).4735 x Table. The secant parabola approximating y sin( x), at x where. Finding the limiting symbolically. The entries in the table show that the coefficients of p( x, x, h) are tending to a limit as. Thus the tangent parabola is (5) P( x, x ) f )+ lim lim f ) f x h f ) f )+ f x x The first limit in (5) is well known, it is lim f ) f f ( x. )+
The second limit in (5) is studied in numerical analysis, and is known to be f verified by applying L hopital s rule using the variable h as follows f ) f )+ f lim lim f ( x ) f ( x f ( x lim f ) f x h 4h, which can be x Therefore, we have shown the limit of the secant parabolas to be (6) P( x, x ) f )+ f x x x + f ) ). Therefore, the tangent parabola in (5) is revealed to be the Taylor polynomial of degree n. For our example with f ( x) sin x, and x, we have P ( x, ) sin( ) + cos() ( x ) sin () ( x ) P x,.84479848 +.543359 x.4735494 x The Lagrange connection. In numerical analysis, the Lagrange interpolation polynomial is constructed, and it can be shown to be equivalent to the formula in (4), however the hand computations are messy. If a computer algebra system, such as Mathematica is used, then it is easy to verify that the two forms are equivalent. First, enter the formula for the Lagrange polynomial ( x x L[ x _ x x f [ x h x x f [ x h + ) ) f [ h + x Then enter formula (4) [ f x h f[ x h f [ x + f [ h + x x x P[ x _ f [ x + f x [ )+ The following Mathematica command will explain L[x ExpandAll[L[x f [ x x f [ x x f [ h + x + x f [ h + x + x f [ h + x + x f [ h + x + x f [ x x + h h h f[ h + x x f [ h + x x f [ h + x x x f [ h + x x f x [ x + f h + x [ x + f h + x [ x h h h
If the command ExpandAll[P[x is issued, the result will be identical. The following Mathematica command can be used to determine if the two symbolic quantities are equivalent and will return either a Boolean expression of true or false. Let s see what happens. ExpandAll[L[xP[x True Therefore, formula (4) is equivalent to Lagrange interpolation, hence the Lagrange form of the remainder applies too. The remainder term. In numerical analysis, the remainder term for a Lagrange interpolation polynomial of degree n, is known to be R( x, x, h) x x ) f ( 3) ( c), 3! where c depends on x and lies somewhere between x h and x. When we take the limit as it is plain to see that we get 3 R( x, x ) x x f ( 3) ( c), 3! which is the remainder term for the Taylor polynomial of degree n. This cinches the fact that the limit of the secant polynomial is the tangent polynomial. Conclusion. The purpose of this article has been to show that the Taylor polynomial is the limiting case of a sequence of interpolating polynomials. The development has been to first show graphical convergence, which is quite rapid. This can be illustrated in the classroom by using graphical calculators or with computer software such as Mathematica or Maple. Then a selected set of interpolating polynomials is tabulated, which is a new twist to the idea of limit, it involves the concept of convergence of a sequence of functions. Finally, the power of calculus is illustrated by discovering that the limiting coefficients are f )and f ( x Then one recognizes that the tangent polynomial is a Taylor polynomial approximation. Moreover, we have motivated the what if exploration by showing what happens to the secant parabola with interpolation points, f )), h, f ), and, f ) when the points collide at the single point, f ) Thus the mystery behind the Taylor polynomial being based on a single point is revealed. It is hoped that teachers reading this article will gain insight to how to use technology in teaching mathematics. Higher degree polynomials have been investigated by the authors in the article cited. References. Russell Howell and John Mathews, Investigation of Tangent Polynomials with a Computer Algebra System, The AMATYC Review, Vol. 4, No., Fall 99, pp. -7.. Kurtis Fink and John Mathews, Numerical Methods Using Matlab, 3 rd edition, Prentice-Hall, Inc, 999.
Numerical Methods Using Matlab, 4 th Edition, 4 John H. Mathews and Kurtis K. Fink ISBN: -3-6548- Prentice-Hall Inc. Upper Saddle River, New Jersey, USA http://vig.prenhall.com/