1549901369-Elements_of_Real_Analysis__Denlinger_

6.5 Taylor's Theorem 331 Thus, !" (0) !"' (0) J C^4 ) (0) T4(x) = f(O)+ f'(O)(x-O)+~(x-0)^2 +- 3 ,-(x-0)^3 +· · ·+- 4 -, -(x-0)^ ...

332 Chapter 6 • Differentiable Functions Example 6.5.6 Find the nth Taylor polynomial of f(x) = ln(l + x) about 0. Solution. We ...

6.5 Taylor's Theorem 333 Because f and Tn(x) have the same value and first n derivatives at a we would expect the graphs of f(x) ...

334 Chapter 6 • Differentiable Functions Definition 6.5.9 Suppose f and its first n derivatives f', f", · · · , f(n) exist in an ...

6.5 Taylor's Theorem 335 Proof. Suppose f is n times differentiable on an open interval containing a and x, where x -:j:. a, and ...

336 Chapter 6 m Differentiable Functions Also, letting t = x in Equation (6), we find n J(kl(x) (x xr+1 G(x) = f(x) + 2::-k-! -( ...

6.5 Taylor's Theorem 337 Since 0 < c < x, we have 1 < ec < ex. Thus, from the above equation, we have x2 x3 x2 x3 1 ...

338 Chapter 6 • Differentiable Functions Theorem 6.5.15 (nth Derivative Test for Maxima/Minima) Suppose that n :'.'.: 2 and f, J ...

6.5 Taylor's Theorem 339 SOME WORD S OF CAUTION Taylor polynomials Tn ( x) about a of a function f are most reliable as approxim ...

340 Chapter 6 • Differentiable Functions Find the sixth Taylor polynomial T 6 (x) for the function f(x) = ln x about Also, writ ...

6.6 *L'Hopital's Rule 341 (d) Show that if n is odd, Rn-i(x) has opposite signs for x to the left of a and to the right of a. Sh ...

342 Chapter 6 • Differentiable Functions We can often evaluate an indeterminate form (1) by algebraically trans- forming ~~:~ in ...

6.6 *L'Hopital's Rule 343 Thus h(a) = h(b), so h satisfies all the hypotheses of Rolle's theorem. Hence, :3 c E (a, b) 3 h' ( c) ...

344 Chapter 6 • Differentiable Functions Theorem 6.6.3 (L 'Hopital 's Rule) Suppose f, g I --t IR., where I is an open interval ...

6.6 *L'Hopital's Rule 345 (d) X-+Q lim f(x) = X-+Q lim g(x) = O; f'(x) ( e) X->e> lim ---;---( g X ) = L (finite, +oo or - ...

346 Chapter 6 1111 Differentiable Functions Case 2: a = xt, and L = +oo. Again, suppose f ,g: I---->~, where f,g, and I satis ...

6.6 *L'Hopital's Rule 347 a change of variables and introduce two new functions. We define functions F and G on (0, ~) by F(u) = ...

348 Chapter 6 11 Differentiable Functions (b) As x-> 1, lnx-> 0 and x -1-> 0. Thus, by Theorem 6.6.4, lnx 1 /x lim --= ...

6.6 *L'Hopital's Rule 349 Proof. First note that this theorem covers 30 cases: a= xt, x 0 , x 0 , +oo, or -oo; L =a real number, ...

350 Chapter 6 • Differentiable Functions Now, as y --+ xci, g(y) --+ +oo, so the left member of this inequality approaches (L - ...