Sets 6, . Exercise 20.2 Let f Xo. Show that h f (f f) makes sense and compute h. What is the regu...
6. Let f [a, b R be a thrice differentiable function and xo E [a, b]. Show that da 6. Let f [a, b R be a thrice differentiable function and xo E [a, b]. Show that da
More generally, let f(x) be smooth with f"(xo) formula 0. Show that the truncation error in the with hi = h and ho = h/2 must decrease linearly, and not faster, as h → 0. More generally, let f(x) be smooth with f"(xo) formula 0. Show that the truncation error in the with hi = h and ho = h/2 must decrease linearly, and not faster, as h → 0.
Exercise 4.6-2: Find the optimal value for h that wll minimize the error for the formula f' (xo) = f(x0+h) _ f(xo) _ h f"(e) in the presence of roundoff error, using the approach of Section 4.6 a) Consider estimating f(1) where f(x)using the above formula. What is the optimal value for h for estimating f'(1), assuming that the roundoff error is bounded by E-10-16 (which is the machine epsilon 2-53 in the 64-bit floating point representation). b) Use Julia...
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C) (5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
3. By hand, compute Di (20) and DI (Xo) for f(x) = x4 – x3 at Xo = 1, with h=1/2, where Dfim _ f (20 + h) – f(x0) Ôf (m)_ f(xo + h) – f(xo – h) 2h In each case, what is the resulting error?
Let f : [a, b] → R and xo e (a,b). Assume that f is continuous on [a,b] \{x0} and lim x approaches too x0 f(x) = L (L is finite) exists. Show that f is Riemann integrable. 1. (20 pts) Let f : [a, b] R and to € (a,b). Assume that f is continuous on [a, b]\{ro} and limz-ro f (x) = L (L is finite) exists. Show that f is Riemann integrable. Hint: We split it into...
Exercise 1.3. Let {A,:y er be a family of sets indexed by ſ. a) Show that 4, C U4, Wyel. b) Show that n A, CA,, Vyer. Yel mer
Let f [a, b [a, b] such Exercise 28: R be continuous. Show that there exists x E : that 1 f(x)= f. 6-a Hint: Intermediate value theorem Let f [a, b [a, b] such Exercise 28: R be continuous. Show that there exists x E : that 1 f(x)= f. 6-a Hint: Intermediate value theorem
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a < x < b.) (c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a
I. Let f : R2 → R be defined by f(x)l cos (122) 211 Compute the second order Taylor polynomial of f near the point xo - 0. A Road Map to Glory (On your way to glory, please keep in mind that f is class C) a) Fill in the blanks: The second order Taylor's polynomial at h E R2 is given by T2 (h) = 2! b) Compute the numbers, vectors and matrices that went into the blanks...