Advanced Calculus fi..

[Hint: Use induction. For n = 0 the formula becomes the known ruleChapter 6 Infinite Series 437Suppose the formula is true with n replaced by m and prove it is true for m + 1; for thisstep, integrate the remainder term by parts taking u = f(m+')(t), dv = (x - t)m dtlm!.]5. Evaluate to threc decimal places:,0.5 dxa) jd ePxzdx"J,6. Let f (x) = e-'1"' for x # 0 and let f (0) = 0.a) Prove that f (x) is continuous for all x.b) Prove that f '(x) is continuous for x # 0 and thatlim f '(x) = f '(0) = 0,x-toso that f '(x) is continuous for all x. . .. .c) Prove that f (")(x) is continuous for all x and f (")(O) = 0.d) Graph the function f (x).7. Use the remainder formula to estimate the error in the following computations:a) e = 1 + 1 + + + & + 4 [assume that e < 3 is known)1 1c ) l o g ; = ; - ~ + ~8. Let f (x) satisfy the conditions stated in Theorem 41. Let f '(a) = f "(a) = ... =f (")(a) = 0, but f ("+')(a) # 0. Show that f (x) has a maximum, minimum, or horizontalinflection point at x = a according to whether the function f("+')(a)(x - a)"+' has amaximum, minimum, or inflection point for x = a. (This gives another proof of the rulededuced in Section 2.19.)9. (Derivatives and differences) In numerical analysis, one uses approximations for derivatives.For the first derivative of f (x) at x, one chooses a small positive h and usesFor the second derivative, one usesWith the aid of Taylor's formula with remainder (assuming sufficient differentiability),establish the following:a) gl(x,h)- f'(x)haslimitOash + Oand hasafinitelimitash-r 0.b) g2(x, h) - f '(x) has limit 0 as h -+ 0 and 8- has a finite limit as h -r 0.C) g3(x. h) - f "(x) has limit 0 as h -r 0 and has a finite limit as h -r 0.Remark We describe these results by saying that the error in approximating fl(x) bygl(x, h) or ftl(x) by g3(x, h) is of the order h, while the error in approximating f1(x)by gz(x, h) is of the order h2. (The higher the power of h, the better the approximation forsmall h.)0