Numerical Analysis I Math 446/OR 481 (Section DL1) Padmanabhan Seshaiyer Mathematical Sciences George Mason University Email: pseshaiy@gmu.edu Padmanabhan Seshaiyer George Mason University Section 0.5 • • • • • • • Limits Continuity Intermediate Value Theorem Mean Value Theorem Rolle’s Theorem MVT for Integrals Taylors Theorem Padmanabhan Seshaiyer George Mason University Mathematical Preliminaries • Calculus Review – – – – – – – Limits Continuity Intermediate Value Theorem Mean Value Theorem Rolle’s Theorem MVT for Integrals Taylors Theorem Padmanabhan Seshaiyer George Mason University Limit of a function Padmanabhan Seshaiyer George Mason University Limit of a function lim x + 1 x →1 Padmanabhan Seshaiyer George Mason University Limit of a function 1 lim 2 x→0 x Padmanabhan Seshaiyer George Mason University Limit of a function lim f ( x) x →3 2− x f ( x) = sin( x) + 6 if x 3 if x 3 Padmanabhan Seshaiyer George Mason University Continuity of a function Padmanabhan Seshaiyer George Mason University Continuity of a function Padmanabhan Seshaiyer George Mason University Mathematical Preliminaries • Calculus Review – – – – – – – Limits Continuity Intermediate Value Theorem Mean Value Theorem Rolle’s Theorem MVT for Integrals Taylors Theorem Padmanabhan Seshaiyer George Mason University Intermediate Value Theorem (IVT) Padmanabhan Seshaiyer George Mason University Application of IVT • Show that the polynomial has a root in the interval [–1, 2]. Padmanabhan Seshaiyer George Mason University Did you know that there are always two opposite points on the Earth with the same temperature? 13 Padmanabhan Seshaiyer George Mason University Mathematical Preliminaries • Calculus Review – – – – – – – Limits Continuity Intermediate Value Theorem Mean Value Theorem Rolle’s Theorem MVT for Integrals Taylors Theorem Padmanabhan Seshaiyer George Mason University Mean Value Theorem (MVT) If function f(x) is continuous in [a, b] and differentiable on (a, b) then MVT holds. Padmanabhan Seshaiyer George Mason University Rolle’s Theorem Padmanabhan Seshaiyer George Mason University Application of MVT and Rolle’s Theorem • Find a number c satisfying the Mean Value Theorem for f (x) = x2 – 3 on the interval [1, 3]. Padmanabhan Seshaiyer George Mason University Application of MVT and Rolle’s Theorem • For all real numbers a and b, show that: | cos 2a − cos 2b | 2 | a − b | Padmanabhan Seshaiyer George Mason University MVT for Integrals MVT for Integrals (Proof) Padmanabhan Seshaiyer George Mason University • Define a function F(x) on [a, b]: • Applying First fundamental theorem of Calculus: • Applying MVT to F, we have: • From definition of F we have: • Also, we have: • Substituting equations (2) and (3) in equation (1) yields: ……. (1) ……. (2) ……. (3) Padmanabhan Seshaiyer George Mason University Application of MVT for Integrals Padmanabhan Seshaiyer George Mason University Mathematical Preliminaries • Calculus Review – – – – – – – Limits Continuity Intermediate Value Theorem Mean Value Theorem Rolle’s Theorem MVT for Integrals Taylors Theorem Taylor Series Padmanabhan Seshaiyer George Mason University • The Taylor series of a real or complex-valued function f (x) that is infinitely differentiable at a real or complex number x = a is the power series • This can be written compactly as: • Example: Exponential function about x = 0. Padmanabhan Seshaiyer George Mason University Taylor’s Expansion • Determine the Taylor polynomial expansion for f ( x) = e x about a = 0 and plot them. Padmanabhan Seshaiyer George Mason University Taylor’s Approximation • Determine the Taylor polynomial expansion for f ( x) = e x about a = 0 and plot them. Taylor’s Theorem Padmanabhan Seshaiyer George Mason University Taylor’s Theorem Remainder Padmanabhan Seshaiyer George Mason University Padmanabhan Seshaiyer George Mason University Taylor’s Theorem Application Padmanabhan Seshaiyer George Mason University Taylor’s Theorem Padmanabhan Seshaiyer George Mason University Taylor’s Theorem • Determine the third Taylor polynomial for f ( x ) = cos x about a = 0, and use this polynomial to approximate cos(0.01). • Use the polynomial and its remainder terms 0 .1 to approximate: cos x dx 0 Padmanabhan Seshaiyer George Mason University MATLAB routine for exp(x) Padmanabhan Seshaiyer George Mason University MATLAB routine for fact(x)
