In Problems 1-8, determine the first three nonzero termsin the Taylor polynomial approximationsinitial value problem.for the given5. x" + tx = 0;x(0) = 1,x' (0) = 0

Answered: In Problems 1-8, determine the first…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let x = a and x = b be the brackets for a root-finding problem for a scalar function f(x). If 8 = b-a is the size of the original bracket, the size of the interval containing the root after four iterations of the bisection method is (a) 8 (unchanged). (b) 8/4 (c) 6/8 (d) 8/16 (e) not knowable with the given information.
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Thus far in the class we've only seen one kind of so-called "linear" difference equation models: the Malthusian ones of the form N(t+1)= XN (t) This is called linear because N(t+1) is a linear function of N(t) - in other words, if we view N(t+1) as "y" and N(t) as "r" then this is an equation of the form "y = mx" since À is a constant. Another more general type of "linear" difference equation is: N(t+1) = AN(t) + a where a is some constant. This is again a linear model because N(t+1) is a linear function of N(t), this time of the form "y=mx+b". This is sometimes called a "linear difference equation with a constant term". The next few problems have to do with linear difference equations with constant terms. 4. Consider the difference equation N(t + 1) = 0.5N(t) + 40 where we are given that N(0) = 100. a) Check that N(t) = 100* (0.5) is not a solution to this difference equation. b) Check that N(t) = 100* (0.5) +40 is not a solution to this difference equation. c) Check that N(t) = 20 *…
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5. In many applications the error function erf(r) = du is important. Compute and graph the fourth order Taylor polynomial of er f(x) about a = 0.
Suppose f(0) = 4, f'(0) = 2, f''(0) = 2, and f(3) (0) = 6. Find the third-order Taylor polynomial for f centered at 0 and use it to approximate f(0.1). The third-order polynomial is p3(x) = 4 + 2x + x² + x² (Simplify your answer.) The value of f(0.1) is approximately P3 (0.1) = (Simplify your answer.) C +
9. Write out P7, the Taylor polynomial of degree n = 7 approximating g near x = 0, given that %3D g(x) = ) (-1)²+13ì 2i-1 (i – 1)! i=1
Suppose f(x) is a function whose 4th order Taylor polynomial at the base point 0 is T₁(x) = x² + √²x³ + x² + x + 1 18 Determine the value of f(4) (1). Of(4) (0) = 24 ƒ(4) (0) = 2/1/2 1 48 O f(4) (0) = = f(4) (0) = = 1 12 None of the listed answers is correct.
Problem 1.10. Find the Taylor polynomial of degree N = 5 for f(x) = In (1+x) expanded about a = 0 and find the upper bound for the error when you approximate In (1.2). %3D
Problem 38 Let T2(z) be the Taylor polynomial of f(r) = V centered at a = 4. Use the Error Bound to estimate the accuracy of the approximation (i.e.to find the maximum value of the error) of VI=T;(x) for 3.9 <1<4

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In Problems 1-8, determine the first three in the Taylor polynomial approximations for the given initial value problem.