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M138.assn10.W07

# M138.assn10.W07 - MATH 138 Assignment 10(3 pages Winter...

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Unformatted text preview: MATH 138 Assignment 10 (3 pages) Winter 2007 Important: Assignment 10 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 asterisk. Solutions for problems with * will be posted by April ﬁlth. There will be a tutorial for Assignment. 10 on Monday, April 2, from 7-9 pm. in AL 116. Note: 111 this assignment, we use the notation Pl (r) for the 'l‘nylor polynomial of order N. (In your text‘ and in Maple Lab 2, the notation Tn(:1:) is used for order n.) 1. Find the speciﬁed Taylor polynomial I’N(rc) centred at x = a for each of the given functions by evaluating f((i), f’(:r), f”(u.), . . . to determine the coeﬁicients. *a) = mlan', PN(3:), a = 1 *0) ﬂat) = Edi“, 133017), a = C (Use FTC/‘1) *b) = arctanm. 133(1), rt = 1 f) = tanl‘, P3(;r), a = 7r/4 c) = sinar, 111(3):), a = 7r/2 *g) f(3;) = cosx, P4(:r), (L = 7r/3 d) = ln(1 — a3), IDA/(3:), a = 0 h) orf : %/le“tgdt, [73(1), (1.: 0 (Use FTCl) D 2. Find P3 the Taylor polynomial of degree 3 centred at (I) = 0, for each of the following functions by choosing p and :c appropriately in the binomial expansion, and then truncating the series: a) = W b) f(a:) = (1 +1:I;)1/4 *0 f0?) = 4 _ {E *d) flit) = *3. a) Use Taylor’s Remainder Theorem to show that \$2 1.4 COSIL' — — ‘I' b) Find an approximation for cos(0.2), and state an upper bound for the error. 330 <'_—‘ l‘c . _. 720 for WM 736R c) Following the method of Example 6 on page 123 of your Course Notes, use the result from a) to 7r/(3 ﬁnd an approximate value for / cos(1:2]dat', 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. *a) State Taylor’s Remainder Theorem with n = 2 and use it to show that 173 < "‘16 for all a: 2 D. 1 {v1+:c—(1+§;L'—%x2) b) Use the result in a) to approximate each quantity and ﬁnd an upper bound on the error. . i. . 7 1 * '1 1 , . 4W . (i) x/ 1.02 (11) ﬂ (Rewrite as 3 1+ 5) (m) (/1 + §t4rlt (iv) / (/4 + sm Mr. 0 o *5. a) Suppose P2(:r) centred at :1: = a, = 1 for a certain function is 1320:] = 2 + ¢l(:1;— 1) + 3(z1: — ])2. Find the Taylor polynomial T2(:L') centred at a; = 1 for the function g(.'i:) = f(-'L')' MATH 138 — Winter 2007 Assignment #10 Page 2 of 3 b) Find a value of N which guarantees that 2! + N! 2 ,N ex—(1+m+\$—+--- hE—NSlUH4 for mE[—1,1]. 6. Show that if is an odd function, then any Taylor polynomial PN (:23) centred at 1‘ = a = (i will contain only odd powers of b) Prove a similar result for even 7. Use known Maclaurin series to evaluate each limit. Al , 1 —cos 1'5 , 2\/1+.’IJ—2—JJ . Hum—1: ‘ 1—6.1 a) hm———7—[-—) *b) ln’n—————2——— c) liIll—(T- *d) hm ——2—,—T x-«i :1: :u—»0 rc x—di a:—~D 1: sm(:r ) 8. *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 3.1415926, divides by 5, enters 2 . I . . the result in memory (called ‘1" hereafter). Then she calculates :1: <1 — and uses it for the ) value of sin(36°). Explain what she was doing in terms of Taylor polynomials, lind the answer she got, and give an upper bound on the error. b) The electrical potential energy V at a point P due 9 to the charge on a disc of radius a and constant charge density a is V(P) = 2770 («122 + a2 — R), where R is the distance from P to the disc. Use ‘clnavge the substitution (1. = Rzr and the first two terms dens'ﬂgé“ of the Maclaurin series for \/1+ 3:2 to show that for large R, (i.e. for 1: = a/]?, near 0)., WP) e «(Ha/Ii. c) Taylor series approximations can be used to ﬁnd approximate solutions to equations. A nice example of this is the equation sinrc +b(1+ cos2 a; + cos 3:) = 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.) :i) Explain how you know there is a solution near 1: = 0. ii) Expand the equation about 1‘ = 0, using only the linear terms in .’L' (i.e. disregard terms of order 3:2 or higher), and solve for x (in terms of b). ...
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M138.assn10.W07 - MATH 138 Assignment 10(3 pages Winter...

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