Math 409-502Harold P. [email protected] mean-value theorem revisitedTheorem. If f is a continuous function on a compact interval [a, b], and if f is differentiable atall interior points of the interval, then there exists an interior point c for which f (b) = f (a) +f0(c)(b − a).Quadratic generalization. Suppose the derivative f0is continuous on [a, b] and the secondderivative f00exists on (a, b). Then there exists an interior point c for whichf (b) = f (a)+ f0(a)(b − a)+12f00(c)(b − a)2.Example: Robinson Crusoe’s approximation for ln(1.1).Take f (x) = ln(x), b = 1.1, a = 1. Then there is a number c between 1 and 1.1 for whichln(1.1) = ln(1)+11(0.1)+12(−1c2)(0.1)2. Therefore ln(1.1) ≈ 0.10, and the exact value is smallerby an amount less than 0.005.(The exact value of ln(1.1) is about 0.09531.)Math 409-502 November 12, 2004 — slide #2Proof of quadratic mean-value theoremLet g be the difference between f and the parabola that intersects the graph of f at a and b andhas the same slope as f at a:g(x) = f (x)−µf (a)+ f0(a)(x − a)+f (b) − f (a) − f0(a)(b − a)(b − a)2(x − a)2¶.Then g(a) = 0 and g(b) = 0. By the original mean-value theorem, there exists a point c1between a and b for which g0(c1) = 0. But g0(a) = 0, so there exists a point c2between a and c1for which g00(c2) = 0. Thenf00(c2) =f (b)− f (a)− f0(a)(b−a)(b−a)2· 2,which simplifies to the required equation.Math 409-502 November 12, 2004 — slide #3Higher-order mean-value theoremTaylor’s theorem. Suppose the function f has at least (n + 1) derivatives on an interval. If xand a are points of the interval, then there is some point c between x and a for whichf (x) = f (a)+ f0(a)(x − a)+12f00(a)(x − a)2+13!f000(a)(x − a)3+ · · · +1n!f(n)(a)(x − a)n+1(n+1)!f(n+1)(c)(x − a)n+1.Math 409-502 November 12, 2004 — slide #4Homework1. Read sections 17.1–17.3, pages 231–236.2. Suppose you were to plot the functions y = cos(x) and y = 1 −x22on the same graph withthe x and y axes scaled in inches (1 inch = 1 radian) using a line thickness of 1 point (where1 inch = 72 points).yx1-1Over what interval of the x-axis would the two curves be indistinguishable? Why?Math 409-502 November 12, 2004 — slide