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MathA9 - MATH 138 Assignment 9(3 pages Winter 2006...

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Unformatted text preview: MATH 138 Assignment 9 (3 pages) Winter 2006 Important: Assignment 9 covers the last five lectures on Taylor series, Taylor polynomials, and Taylor's Remainder Theorem. While it is not for submission, 10-15% of the final exam will be based on this material. Thus it is essential that you do as many of the problems as possible. Some have been suggested with an *. There will be a tutorial for Assignment 9 on Wednesday, March 29, from 3:30 — 5:30 in RCH 101. Solutions will be posted by April 511’. Note: In this assignment, we use the notation PN(z) for the Taylor polynomial of order N. (In your text, and in Maple Lab 2, the notation Tn(:r) is used for order n.) ' 1. Find the specified Taylor polynOmial PMs) centred at z = a for each of the given functions by evaluating f(a), f'{$}, f”(a.), . . . to determine the coefficients. *a) = :rlnas, PN(:r), n =1 *e) f(m) = xii-grit, P3(m), a. = e (Use FTCl) ’b) fir) = arctanz', P3(z), a 21 f) f(r) = tans, Paw), a: 1r/4 6) fix) = sin cc, 194(2), :1 = n/2 'g) flat) = 0055:, P4(a:), a = 1r/3 d) f(.r) = ln(1 — 11:), PN(cr), a = 0 h} erf = %/me“2dt, P3(a:}, a = 0 (Use FTCI) o 2. Find Pam), the Taylor polynomial of degree 3 centred at :r: = 0, for each of the following functions by choosing p and to appropriately in the binomial expansion, and then truncating the series: a) flw) = x/‘ml 1+ w b) fix) = (1 +3)“ * = id = c) for) m ) to) {2 + w 3. a) Use Taylor’s Remainder Theorem to show that . I3 I :r I5 — — — < —--w smzr (a: 6) _ 120 for all :rEIR b) Find an approximation for sin(.3), and state an upper bound for the error. e) F0110wing the method of Example 6 on page 133 of your Course Notes, use the result from a) to 1r/6 find an approximate value for sin(x2)d:r, and derive an upper bound on the error. Compare 0 this upper bound with the error bound predicted by the corollary of Alternating Series Test. 4. 51) Use Taylor’s Remainder Theorem to show that 9:2 _¢_ _ _ e (1 n+2) 1 a b) Use the result from a) to find an approximate value for / KG) tilt and find an upper bound On 0 3 5% forall 120. the error. HINT: Follow the same procedure as suggested in #3 above. MATH 138 - Winter 2006 Assignment #9 Page 2 of 3 5. *a) State Taylor‘s Remainder Theorem with n = 2 and use it to show that 3 (.1— forsll 1'20. |\/1+2—(1+l$—lx2) _16 2 8 b) Use the result in a) to approximate each quantity and find an upper bound on the error. 1 7r (i) «1.02 (ii) fl (Rewriteasg 1+4—19) ‘(iii) Afll+ét4dt (iv) [0 x/4+siutdt *6. Suppose P2(a:) centred at cc = a = 1 for a certain function f(:c) is P2(z) = 2 + 4(x — 1) + 3(a: —1)2. 1 Find the Taylor polynomial T2(I) centred at 2: = 1 for the function g(:c) = 7. a) Find a constant M such that ‘m— (1+lz) 1 < 2 —_ . 2 _Mm for $6[ 2,0] b) Find P3(a:) for the function f(x) 2 ln(1 + x}, centred at :r = a = 0. If we wish to have error |f(z) — P3(m)| 5 2.5 x1[}5 for x e [—d s m g all, what is the largest possible value for at? (You will need Maple to solve the equation for d.) *c) Find a value of N which guarantees that m 22 EN —4 e — 1+z+§+m+fi $10 for :I:E[—1,1]. 8. a) Show that if f (x) is an odd function, then any Taylor polynomial PN(:1:) centred at :r = a. = 0 will contain only odd powers of 9:. b) Prove a similar result for even f (x). 9. Taylor’s Remainder Theorem can be used to evaluate limits. For example, we know that e” = 1+$+k332, E where k = ET for some c between 0 and :r (TRT fer a = 0, n = 1). Thus . em—l . 1+o+kz2—1 11m = 11m ——-—— zdfl :1: r—n'.) E = lim {1 + In) z—fl = 1. Use a similar method to find each limit. _ —:I:2 2 _ _ t _ a) m 1_e b) m __n_\/1+w2$ fie) Maw m—ml « cos x :z—~o 32 :40 :53 MATH 138 - Winter 2006 Assignment #9 Page 3 of 3 10. 'a) You are about to calculate sin(36°) when the batteries die on your calculator. You roommate has an old calculator, but it has no ‘sin’ key. Unperturbed, she enters 3.14159226, divides by 5, enters the result in memory (called ‘2? hereafter). Then she calculates a: (1 — and uses it for the value of sin(36°). Explain what she was doing in terms of Taylor polynomials, find the answer she got, and give an upper bound on the error. b} The electrical potential energy V at a point P due to the charge on a. disc of radius a and constant charge density 0' is VLF) = 2710(VR2 + o.2 — R), where R is the distance from p to the disc. Use the substitution o = Ra: and the first two terms of the Maclaurin series for v1 + 22 to show that for large R, (i.e. for a: —-» o/R near 0), V(P) Rs info/R. c) Taylor series approximations can be used to find approximate solutions to equations. A nice example of this is the equation sina: + b(1+ c0529: + cosm) = 0, where b is a very small positive constant. (This equation arose in Einstein’s calculations pre- dicting the bending of light by the gravitational field of the sun.) 1) Explain how you know there is a solution near a: z 0. ii) Expand the equation about a: = 0, using only the linear terms in at" (i.e. disregard terms of order m2 or higher)1 and solve for m (in terms of b). ...
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