4.3 Lagrange Approximation


 Della Jane Doyle
 5 years ago
 Views:
Transcription
1 206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average of known function values at neighboring points. Linear interpolation uses a line segment that passes through two points. The slope between (x 0, y 0 ) and (x 1, y 1 ) is m = (y 1 y 0 )/(x 1 x 0 ), and the pointslope formula for the line y = m(x x 0 ) + y 0 can be rearranged as (1) y = P(x) = y 0 + (y 1 y 0 ) x x 0 x 1 x 0.
2 SEC. 4.3 LAGRANGE APPROXIMATION 207 When formula (1) is expanded, the result is a polynomial of degree 1. Evaluation of P(x) at x 0 and x 1 produces y 0 and y 1, respectively: (2) P(x 0 ) = y 0 + (y 1 y 0 )(0) = y 0, P(x 1 ) = y 0 + (y 1 y 0 )(1) = y 1. The French mathematician Joseph Louis Lagrange used a slightly different method to find this polynomial. He noticed that it could be written as x x 1 x x 0 (3) y = P 1 (x) = y 0 + y 1. x 0 x 1 x 1 x 0 Each term on the right side of (3) involves a linear factor; hence the sum is a polynomial of degree 1. The quotients in (3) are denoted by (4) L 1,0 (x) = x x 1 and L 1,1 (x) = x x 0. x 0 x 1 x 1 x 0 Computation reveals that L 1,0 (x 0 ) = 1, L 1,0 (x 1 ) = 0, L 1,1 (x 0 ) = 0, and L 1,1 (x 1 ) = 1 so that the polynomial P 1 (x) in (3) also passes through the two given points: (5) P 1 (x 0 ) = y 0 + y 1 (0) = y 0 and P 1 (x 1 ) = y 0 (0) + y 1 = y 1. The terms L 1,0 (x) and L 1,1 (x) in (4) are called Lagrange coefficient polynomials based on the nodes x 0 and x 1. Using this notation, (3) can be written in summation form (6) P 1 (x) = 1 y k L 1,k (x). k=0 Suppose that the ordinates y k are computed with the formula y k = f (x k ).IfP 1 (x) is used to approximate f (x) over the interval [x 0, x 1 ], we call the process interpolation. If x < x 0 (or x 1 < x), then using P 1 (x) is called extrapolation. The next example illustrates these concepts. Example 4.6. Consider the graph y = f (x) = cos(x) over [0.0, 1.2]. (a) Use the nodes x 0 = 0.0 and x 1 = 1.2 to construct a linear interpolation polynomial P 1 (x). (b) Use the nodes x 0 = 0.2 and x 1 = 1.0 to construct a linear approximating polynomial Q 1 (x). (a) Using (3) with the abscissas x 0 = 0.0 and x 1 = 1.2 and the ordinates y 0 = cos(0.0) = and y 1 = cos(1.2) = produces P 1 (x) = x 1.2 x = (x 1.2) (x 0.0).
3 208 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION y y y = f(x) y = f(x) 0.6 y = P 1 (x) 0.6 y = Q 1 (x) x x (a) Figure 4.11 (a) The linear approximation y = P 1 (x) where the nodes x 0 = 0.0 and x 1 = 1.2 are the endpoints of the interval [a, b]. (b) The linear approximation y = Q 1 (x) where the nodes x 0 = 0.2 andx 1 = 1.0 lie inside the interval [a, b]. (b) (b) When the nodes x 0 = 0.2 and x 1 = 1.0 with y 0 = cos(0.2) = and y 1 = cos(1.0) = are used, the result is Q 1 (x) = x 1.0 x = (x 1.0) (x 0.2). Figure 4.11(a) and (b) show the graph of y = cos(x) and compare it with y = P 1 (x) and y = Q 1 (x), respectively. Numerical computations are given in Table 4.6 and reveal that Q 1 (x) has less error at the points x k that satisfy 0.1 x k 1.1. The largest tabulated error, f (0.6) P 1 (0.6) = , is reduced to f (0.6) Q 1 (0.6) = by using Q 1 (x). The generalization of (6) is the construction of a polynomial P N (x) of degree at most N that passes through the N + 1 points (x 0, y 0 ), (x 1, y 1 ),...,(x N, y N ) and has the form (7) P N (x) = N y k L N,k (x), where L N,k is the Lagrange coefficient polynomial based on these nodes: (8) L N,k (x) = (x x 0) (x x k 1 )(x x k+1 ) (x x N ) (x k x 0 ) (x k x k 1 )(x k x k+1 ) (x k x N ). k=0 It is understood that the terms (x x k ) and (x k x k ) do not appear on the right side of
4 SEC. 4.3 LAGRANGE APPROXIMATION 209 Table 4.6 Comparison of f (x) = cos(x) and the Linear Approximations P 1 (x) and Q 1 (x) x k f (x k ) = cos(x k ) P 1 (x k ) f (x k ) P 1 (x k ) Q 1 (x k ) f (x k ) Q 1 (x k ) equation (8). It is appropriate to introduce the product notation for (8), and we write (9) L N,k (x) = Nj=0 j =k Nj=0 j =k (x x j ) (x k x j ). Here the notation in (9) indicates that in the numerator the product of the linear factors (x x j ) is to be formed, but the factor (x x k ) is to be left out (or skipped). A similar construction occurs in the denominator. A straightforward calculation shows that for each fixed k, the Lagrange coefficient polynomial L N,k (x) has the property (10) L N,k (x j ) = 1 when j = k and L N,k (x j ) = 0 when j = k. Then direct substitution of these values into (7) is used to show that the polynomial curve y = P N (x) goes through (x j, y j ): (11) P N (x j ) = y 0 L N,0 (x j ) + +y j L N, j (x j ) + +y N L N,N (x j ) = y 0 (0) + +y j (1) + +y N (0) = y j. To show that P N (x) is unique, we invoke the fundamental theorem of algebra, which states that a nonzero polynomial T (x) of degree N has at most N roots. In other words, if T (x) is zero at N + 1 distinct abscissas, it is identically zero. Suppose that P N (x) is not unique and that there exists another polynomial Q N (x) of degree N that also passes through the N +1 points. Form the difference polynomial T (x) =
5 210 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION y 1.0 y x y = f(x) y = P 2 (x) 0.5 x y = f(x) y = P 3 (x) (a) Figure 4.12 (a) The quadratic approximation polynomial y = P 2 (x) based on the nodes x 0 = 0.0, x 1 = 0.6, and x 2 = 1.2. (b) The cubic approximation polynomial y = P 3 (x) based on the nodes x 0 = 0.0, x 1 = 0.4, x 2 = 0.8, and x 3 = 1.2. (b) P N (x) Q N (x). Observe that the polynomial T (x) has degree N and that T (x j ) = P N (x j ) Q N (x j ) = y j y j = 0, for j = 0, 1,..., N. Therefore, T (x) 0 and it follows that Q N (x) = P N (x). When (7) is expanded, the result is similar to (3). The Lagrange quadratic interpolating polynomial through the three points (x 0, y 0 ), (x 1, y 1 ), and (x 2, y 2 ) is (x x (12) P 2 (x) = y 1 )(x x 2 ) 0 (x 0 x 1 )(x 0 x 2 ) + y (x x 0 )(x x 2 ) 1 (x 1 x 0 )(x 1 x 2 ) + y (x x 0 )(x x 1 ) 2 (x 2 x 0 )(x 2 x 1 ). The Lagrange cubic interpolating polynomial through the four points (x 0, y 0 ), (x 1, y 1 ), (x 2, y 2 ), and (x 3, y 3 ) is (13) (x x P 3 (x) = y 1 )(x x 2 )(x x 3 ) 0 (x 0 x 1 )(x 0 x 2 )(x 0 x 3 ) + y (x x 0 )(x x 2 )(x x 3 ) 1 (x 1 x 0 )(x 1 x 2 )(x 1 x 3 ) (x x + y 0 )(x x 1 )(x x 3 ) 2 (x 2 x 0 )(x 2 x 1 )(x 2 x 3 ) + y (x x 0 )(x x 1 )(x x 2 ) 3 (x 3 x 0 )(x 3 x 1 )(x 3 x 2 ). Example 4.7. Consider y = f (x) = cos(x) over [0.0, 1.2]. (a) Use the three nodes x 0 = 0.0, x 1 = 0.6, and x 2 = 1.2 to construct a quadratic interpolation polynomial P 2 (x). (b) Use the four nodes x 0 = 0.0, x 1 = 0.4, x 2 = 0.8, and x 3 = 1.2 to construct a cubic interpolation polynomial P 3 (x). (a) Using x 0 = 0.0, x 1 = 0.6, x 2 = 1.2 and y 0 = cos(0.0) = 1, y 1 = cos(0.6) =
6 SEC. 4.3 LAGRANGE APPROXIMATION , and y 2 = cos(1.2) = in equation (12) produces (x 0.6)(x 1.2) (x 0.0)(x 1.2) P 2 (x) = ( )( ) ( )( ) (x 0.0)(x 0.6) ( )( ) = (x 0.6)(x 1.2) (x 0.0)(x 1.2) (x 0.0)(x 0.6). (b) Using x 0 = 0.0, x 1 = 0.4, x 2 = 0.8, x 3 = 1.2 and y 0 = cos(0.0) = 1.0, y 1 = cos(0.4) = , y 2 = cos(0.8) = , and y 3 = cos(1.2) = in equation (13) produces (x 0.4)(x 0.8)(x 1.2) P 3 (x) = ( )( )( ) (x 0.0)(x 0.8)(x 1.2) ( )( )( ) (x 0.0)(x 0.4)(x 1.2) ( )( )( ) (x 0.0)(x 0.4)(x 0.8) ( )( )( ) = (x 0.4)(x 0.8)(x 1.2) (x 0.0)(x 0.8)(x 1.2) (x 0.0)(x 0.4)(x 1.2) (x 0.0)(x 0.4)(x 0.8). The graphs of y = cos(x) and the polynomials y = P 2 (x) and y = P 3 (x) are shown in Figure 4.12(a) and (b), respectively. Error Terms and Error Bounds It is important to understand the nature of the error term when the Lagrange polynomial is used to approximate a continuous function f (x). It is similar to the error term for the Taylor polynomial, except that the factor (x x 0 ) N+1 is replaced with the product (x x 0 )(x x 1 ) (x x N ). This is expected because interpolation is exact at each of the N + 1 nodes x k, where we have E N (x k ) = f (x k ) P N (x k ) = y k y k = 0for k = 0, 1, 2,...,N. Theorem 4.3 (Lagrange Polynomial Approximation). Assume that f C N+1 [a, b] and that x 0, x 1,...,x N [a, b] are N + 1 nodes. If x [a, b], then (14) f (x) = P N (x) + E N (x),
7 212 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION where P N (x) is a polynomial that can be used to approximate f (x): (15) f (x) P N (x) = N k=0 f (x k )L N,k (x). The error term E N (x) has the form (16) E N (x) = (x x 0)(x x 1 ) (x x N ) f (N+1) (c) (N + 1)! for some value c = c(x) that lies in the interval [a, b]. Proof. As an example of the general method, we establish (16) when N = 1. The general case is discussed in the exercises. Start by defining the special function g(t) as follows: (17) g(t) = f (t) P 1 (t) E 1 (x) (t x 0)(t x 1 ) (x x 0 )(x x 1 ). Notice that x, x 0, and x 1 are constants with respect to the variable t and that g(t) evaluates to be zero at these three values; that is, g(x) = f (x) P 1 (x) E 1 (x) (x x 0)(x x 1 ) (x x 0 )(x x 1 ) = f (x) P 1(x) E 1 (x) = 0, g(x 0 ) = f (x 0 ) P 1 (x 0 ) E 1 (x) (x 0 x 0 )(x 0 x 1 ) = f (x 0 ) P 1 (x 0 ) = 0, (x x 0 )(x x 1 ) g(x 1 ) = f (x 1 ) P 1 (x 1 ) E 1 (x) (x 1 x 0 )(x 1 x 1 ) = f (x 1 ) P 1 (x 1 ) = 0. (x x 0 )(x x 1 ) Suppose that x lies in the open interval (x 0, x 1 ). Applying Rolle s theorem to g(t) on the interval [x 0, x] produces a value d 0, with x 0 < d 0 < x, such that (18) g (d 0 ) = 0. A second application of Rolle s theorem to g(t) on [x, x 1 ] will produce a value d 1, with x < d 1 < x 1, such that (19) g (d 1 ) = 0. Equations (18) and (19) show that the function g (t) is zero at t = d 0 and t = d 1. A third use of Rolle s theorem, but this time applied to g (t) over [d 0, d 1 ], produces a value c for which (20) g (2) (c) = 0.
8 SEC. 4.3 LAGRANGE APPROXIMATION 213 Now go back to (17) and compute the derivatives g (t) and g (t): (21) g (t) = f (t) P 1 (t) E 1(x) (t x 0) + (t x 1 ) (x x 0 )(x x 1 ), (22) g (t) = f 2 (t) 0 E 1 (x) (x x 0 )(x x 1 ). In (22) we have used the fact the P 1 (t) is a polynomial of degree N = 1; hence its second derivative is P 1 (t) 0. Evaluation of (22) at the point t = c and using (20) yields (23) 0 = f 2 (c) E 1 (x) (x x 0 )(x x 1 ). Solving (23) for E 1 (x) results in the desired form (16) for the remainder: (24) E 1 (x) = (x x 0)(x x 1 ) f (2) (c), 2! and the proof is complete. The next result addresses the special case when the nodes for the Lagrange polynomial are equally spaced x k = x 0 + hk, fork = 0, 1,..., N, and the polynomial P N (x) is used only for interpolation inside the interval [x 0, x N ]. Theorem 4.4 (Error Bounds for Lagrange Interpolation, Equally Spaced Nodes). Assume that f (x) is defined on [a, b], which contains equally spaced nodes x k = x 0 + hk. Additionally, assume that f (x) and the derivatives of f (x), up to the order N + 1, are continuous and bounded on the special subintervals [x 0, x 1 ], [x 0, x 2 ], and [x 0, x 3 ], respectively; that is, (25) f (N+1) (x) M N+1 for x 0 x x N, for N = 1, 2, 3. The error terms (16) corresponding to the cases N = 1, 2, and 3 have the following useful bounds on their magnitude: (26) E 1 (x) h2 M 2 8 valid for x [x 0, x 1 ], (27) E 2 (x) h3 M valid for x [x 0, x 2 ], (28) E 3 (x) h4 M 4 24 valid for x [x 0, x 3 ]. Proof. We establish (26) and leave the others for the reader. Using the change of variables x x 0 = t and x x 1 = t h, the error term E 1 (x) can be written as (29) E 1 (x) = E 1 (x 0 + t) = (t2 ht) f (2) (c) 2! for 0 t h.
9 214 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION The bound for the derivative for this case is (30) f (2) (c) M 2 for x 0 c x 1. Now determine a bound for the expression (t 2 ht) in the numerator of (29); call this term (t) = t 2 ht. Since (t) = 2t h, there is one critical point t = h/2 that is the solution to (t) = 0. The extreme values of (t) over [0, h] occur either at an end point (0) = 0, (h) = 0 or at the critical point (h/2) = h 2 /4. Since the latter value is the largest, we have established the bound (31) (t) = t 2 ht h2 4 = h2 4 for 0 t h. Using (30) and (31) to estimate the magnitude of the product in the numerator in (29) results in (32) E 1 (x) = (t) f (2) (c) 2! h2 M 2, 8 and formula (26) is established. Comparison of Accuracy and O(h N+1 ) The significance of Theorem 4.4 is to understand a simple relationship between the size of the error terms for linear, quadratic, and cubic interpolation. In each case the error bound E N (x) depends on h in two ways. First, h N+1 is explicitly present so that E N (x) is proportional to h N+1. Second, the values M N+1 generally depend on h and tend to f (N+1) (x 0 ) as h goes to zero. Therefore, as h goes to zero, E N (x) converges to zero with the same rapidity that h N+1 converges to zero. The notation O(h N+1 ) is used when discussing this behavior. For example, the error bound (26) can be expressed as E 1 (x) =O(h 2 ) valid for x [x 0, x 1 ]. The notation O(h 2 ) stands in place of h 2 M 2 /8 in relation (26) and is meant to convey the idea that the bound for the error term is approximately a multiple of h 2 ; that is, E 1 (x) Ch 2 O(h 2 ). As a consequence, if the derivatives of f (x) are uniformly bounded on the interval [a, b] and h < 1, then choosing N large will make h N+1 small, and the higherdegree approximating polynomial will have less error.
10 SEC. 4.3 LAGRANGE APPROXIMATION 215 y y y = E 2 (x) y = E 3 (x) x x (a) (b) Figure 4.13 (a) The error function E 2 (x) = cos(x) P 2 (x). (b) The error function E 3 (x) = cos(x) P 3 (x). Example 4.8. Consider y = f (x) = cos(x) over [0.0, 1.2]. Use formulas (26) through (28) and determine the error bounds for the Lagrange polynomials P 1 (x), P 2 (x), and P 3 (x) that were constructed in Examples 4.6 and 4.7. First, determine the bounds M 2, M 3, and M 4 for the derivatives f (2) (x), f (3) (x), and f (4) (x), respectively, taken over the interval [0.0, 1.2]: f (2) (x) = cos(x) cos(0.0) = = M 2, f (3) (x) = sin(x) sin(1.2) = = M 3, f (4) (x) = cos(x) cos(0.0) = = M 4. For P 1 (x) the spacing of the nodes is h = 1.2, and its error bound is (33) E 1 (x) h2 M 2 (1.2)2 ( ) = For P 2 (x) the spacing of the nodes is h = 0.6, and its error bound is (34) E 2 (x) h3 M (0.6)3 ( ) 9 = For P 3 (x) the spacing of the nodes is h = 0.4, and its error bound is (35) E 3 (x) h4 M 4 24 (0.4)4 ( ) 24 = From Example 4.6 we saw that E 1 (0.6) = cos(0.6) P 1 (0.6) = , so the bound in (33) is reasonable. The graphs of the error functions E 2 (x) = cos(x) P 2 (x) and E 3 (x) = cos(x) P 3 (x) are shown in Figure 4.13(a) and (b),
11 216 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Table 4.7 Comparison of f (x) = cos(x) and the Quadratic and Cubic Polynomial Approximations P 2 (x) and P 3 (x) x k f (x k ) = cos(x k ) P 2 (x k ) E 2 (x k ) P 3 (x k ) E 3 (x k ) respectively, and numerical computations are given in Table 4.7. Using values in the table, we find that E 2 (1.0) = cos(1.0) P 2 (1.0) = and E 3 (0.2) = cos(0.2) P 3 (0.2) = , which is in reasonable agreement with the bounds and given in (34) and (35), respectively.
12 Numerical Methods Using Matlab, 4 th Edition, 2004 John H. Mathews and Kurtis K. Fink ISBN: PrenticeHall Inc. Upper Saddle River, New Jersey, USA
4.5 Chebyshev Polynomials
230 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION 4.5 Chebyshev Polynomials We now turn our attention to polynomial interpolation for f (x) over [ 1, 1] based on the nodes 1 x 0 < x 1 < < x N 1. Both
More informationPiecewise Cubic Splines
280 CHAP. 5 CURVE FITTING Piecewise Cubic Splines The fitting of a polynomial curve to a set of data points has applications in CAD (computerassisted design), CAM (computerassisted manufacturing), and
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n1 x n1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
More informationNovember 16, 2015. Interpolation, Extrapolation & Polynomial Approximation
Interpolation, Extrapolation & Polynomial Approximation November 16, 2015 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationINTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).
INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x
More informationThe Method of Partial Fractions Math 121 Calculus II Spring 2015
Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationZeros of Polynomial Functions
Zeros of Polynomial Functions Objectives: 1.Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions 2.Find rational zeros of polynomial functions 3.Find conjugate
More informationLagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.
Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method
More informationAn important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate.
Chapter 10 Series and Approximations An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate 1 0 e x2 dx, you could set
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationApproximating functions by Taylor Polynomials.
Chapter 4 Approximating functions by Taylor Polynomials. 4.1 Linear Approximations We have already seen how to approximate a function using its tangent line. This was the key idea in Euler s method. If
More information3.2 The Factor Theorem and The Remainder Theorem
3. The Factor Theorem and The Remainder Theorem 57 3. The Factor Theorem and The Remainder Theorem Suppose we wish to find the zeros of f(x) = x 3 + 4x 5x 4. Setting f(x) = 0 results in the polynomial
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
More information5 Numerical Differentiation
D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives
More information3.3 Real Zeros of Polynomials
3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section
More informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
More informationApplication. Outline. 31 Polynomial Functions 32 Finding Rational Zeros of. Polynomial. 33 Approximating Real Zeros of.
Polynomial and Rational Functions Outline 31 Polynomial Functions 32 Finding Rational Zeros of Polynomials 33 Approximating Real Zeros of Polynomials 34 Rational Functions Chapter 3 Group Activity:
More informationLectures 56: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5: Taylor Series Weeks 5 Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More information6.4 Logarithmic Equations and Inequalities
6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More information7. Some irreducible polynomials
7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationInteger roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationCubic Functions: Global Analysis
Chapter 14 Cubic Functions: Global Analysis The Essential Question, 231 Concavitysign, 232 Slopesign, 234 Extremum, 235 Heightsign, 236 0Concavity Location, 237 0Slope Location, 239 Extremum Location,
More information1 Cubic Hermite Spline Interpolation
cs412: introduction to numerical analysis 10/26/10 Lecture 13: Cubic Hermite Spline Interpolation II Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore 1 Cubic Hermite
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationSMT 2014 Algebra Test Solutions February 15, 2014
1. Alice and Bob are painting a house. If Alice and Bob do not take any breaks, they will finish painting the house in 20 hours. If, however, Bob stops painting once the house is halffinished, then the
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationSequences and Series
Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite
More informationAlgebra 1 Course Title
Algebra 1 Course Title Course wide 1. What patterns and methods are being used? Course wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
More informationMarch 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationContinuity. DEFINITION 1: A function f is continuous at a number a if. lim
Continuity DEFINITION : A function f is continuous at a number a if f(x) = f(a) REMARK: It follows from the definition that f is continuous at a if and only if. f(a) is defined. 2. f(x) and +f(x) exist.
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
More information8 Polynomials Worksheet
8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions  Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph
More informationEquations, Inequalities & Partial Fractions
Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationSeparable First Order Differential Equations
Separable First Order Differential Equations Form of Separable Equations which take the form = gx hy or These are differential equations = gxĥy, where gx is a continuous function of x and hy is a continuously
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationModern Algebra Lecture Notes: Rings and fields set 4 (Revision 2)
Modern Algebra Lecture Notes: Rings and fields set 4 (Revision 2) Kevin Broughan University of Waikato, Hamilton, New Zealand May 13, 2010 Remainder and Factor Theorem 15 Definition of factor If f (x)
More informationFactoring Cubic Polynomials
Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More information7.6 Approximation Errors and Simpson's Rule
WileyPLUS: Home Help Contact us Logout HughesHallett, Calculus: Single and Multivariable, 4/e Calculus I, II, and Vector Calculus Reading content Integration 7.1. Integration by Substitution 7.2. Integration
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationPolynomial and Synthetic Division. Long Division of Polynomials. Example 1. 6x 2 7x 2 x 2) 19x 2 16x 4 6x3 12x 2 7x 2 16x 7x 2 14x. 2x 4.
_.qd /7/5 9: AM Page 5 Section.. Polynomial and Synthetic Division 5 Polynomial and Synthetic Division What you should learn Use long division to divide polynomials by other polynomials. Use synthetic
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationI. Pointwise convergence
MATH 40  NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationis the degree of the polynomial and is the leading coefficient.
Property: T. HrubikVulanovic email: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 HigherDegree Polynomial Functions... 1 Section 6.1 HigherDegree Polynomial Functions...
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More information63. Graph y 1 2 x and y 2 THE FACTOR THEOREM. The Factor Theorem. Consider the polynomial function. P(x) x 2 2x 15.
9.4 (927) 517 Gear ratio d) For a fixed wheel size and chain ring, does the gear ratio increase or decrease as the number of teeth on the cog increases? decreases 100 80 60 40 20 27in. wheel, 44 teeth
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More information3.6 The Real Zeros of a Polynomial Function
SECTION 3.6 The Real Zeros of a Polynomial Function 219 3.6 The Real Zeros of a Polynomial Function PREPARING FOR THIS SECTION Before getting started, review the following: Classification of Numbers (Appendix,
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationMATH 132: CALCULUS II SYLLABUS
MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early
More informationSection 4.4. Using the Fundamental Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 4.4 Using the Fundamental Theorem As we saw in Section 4.3, using the Fundamental Theorem of Integral Calculus reduces the problem of evaluating a
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More information3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationx a x 2 (1 + x 2 ) n.
Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationAnalyzing Functions Intervals of Increase & Decrease Lesson 76
(A) Lesson Objectives a. Understand what is meant by the terms increasing/decreasing as it relates to functions b. Use graphic and algebraic methods to determine intervals of increase/decrease c. Apply
More informationSlope and Rate of Change
Chapter 1 Slope and Rate of Change Chapter Summary and Goal This chapter will start with a discussion of slopes and the tangent line. This will rapidly lead to heuristic developments of limits and the
More informationSolving Quadratic & Higher Degree Inequalities
Ch. 8 Solving Quadratic & Higher Degree Inequalities We solve quadratic and higher degree inequalities very much like we solve quadratic and higher degree equations. One method we often use to solve quadratic
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationThe degree of a polynomial function is equal to the highest exponent found on the independent variables.
DETAILED SOLUTIONS AND CONCEPTS  POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE
More information1.6 A LIBRARY OF PARENT FUNCTIONS. Copyright Cengage Learning. All rights reserved.
1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationComputational Geometry Lab: FEM BASIS FUNCTIONS FOR A TETRAHEDRON
Computational Geometry Lab: FEM BASIS FUNCTIONS FOR A TETRAHEDRON John Burkardt Information Technology Department Virginia Tech http://people.sc.fsu.edu/ jburkardt/presentations/cg lab fem basis tetrahedron.pdf
More informationCorollary. (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality
Corollary For equidistant knots, i.e., u i = a + i (ba)/n, we obtain with (f є C n+1 [a,b]). Proof: This follows directly from the preceding theorem using the inequality 120202: ESM4A  Numerical Methods
More information9.10 calculus 09 10 blank.notebook February 26, 2010
Thm. Short form: A power series representing a function is its Taylor series. 1 From the previous section, Find some terms using the theorem. 2 Know this and the Maclaurin series for sin, cos, and e x.
More information2.5 ZEROS OF POLYNOMIAL FUNCTIONS. Copyright Cengage Learning. All rights reserved.
2.5 ZEROS OF POLYNOMIAL FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions.
More informationx(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3
CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract 
More information1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).
.7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationPartial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:
Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than
More informationNumerical Solution of Differential
Chapter 13 Numerical Solution of Differential Equations We have considered numerical solution procedures for two kinds of equations: In chapter 10 the unknown was a real number; in chapter 6 the unknown
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More informationReal Roots of Univariate Polynomials with Real Coefficients
Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials
More information