FDWKCalcSM_ch10

# FDWKCalcSM_ch10 - Section 10.1 421 70. Continued Multiply...

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Section 10.1 421 70. Continued Multiply by x . nn x x x n n () += = 1 2 1 3 1 Replace x by 1 x . x x x x x x n n , + = = > = 1 2 1 1 2 1 1 1 3 2 3 (b) Solve to get for x x x xx = ≈> 2 1 2 769 1 2 3 .. 71. (a) Computing the coefficients, f fx x f ( ) , ( 1 1 2 11 1 4 2 2 = =− + =− ′′ = so x f x + = ′′′ + 1 1 2 1 8 61 3 4 ), ! ( so so = + f f n n ! , ! . 1 3 1 16 1 2 1 In general S o x n n n + ++ + + 1 2 1 4 1 8 1 1 2 2 1 ±± (b) Ratio test for absolute convergence: lim n n n n n x x x →∞ + + + = 1 2 2 1 1 2 1 2 1 i x x <⇒−< < 1 2 3 . The series converges absolutely on( , ) . 13 At the series is which diverges by th x n = 1 1 2 0 ,, e th-term test n . At the series is which diverges x n n = 31 1 2 0 ,( ) , by the th-term test n . The interval of convergence is ( 1, 3). (c) Px 3 2 1 2 1 4 1 8 + P 3 2 05 1 2 05 1 4 8 0 65265 (.) .( . ) . + = 72. (a) Ratio test for absolute convergence: lim lim n n n n n n nx n n x →∞ + + →∞ + = + = 1 2 21 22 2 1 1 i 12 2 x The series converges absolutely on ( 2, 2). The series diverges at both endpoints by the n th-term test, since lim lim ( ) . n →∞ →∞ ≠− 01 0 and The interval of convergence is ( 2, 2). (b) The series converges at 1 and forms an alternating series: −+−+ + − + 1 2 2 4 3 8 4 16 1 2 . n n n The n th-term of this series decreases in absolute value to 0, so the truncation error after 9 terms is less than the absolute value of the10 th term. Thus error << 10 2 001 10 73. (a) x 1 (b) x x 2 2 3 2 (c) x x x 3 23 3 2 2 3 + (d) P 3 07 1 207 3 2 2 3 + Chapter 10 Parametric, Vector, and Polar Functions Section 10.1 Parametric Functions (pp. 531–537) Exploration 1 Investigating Cycloids 1. [0, 20] by [–1, 8] 2. xn a = 2 π for any integer n. 3. at y >− 0 0 and so cos . 4. An arch is produced by one complete turn of the wheel. Thus, they are congruent. 5. The maximum value of y is 2 a and occurs when a =+ for any integer n. 6. The function represented by the cycloid is periodic with period 2 a , and each arch represents one period of the graph. In each arch, the graph is concave down, has an absolute maximum of 2 a at the midpoint, and an absolute minimum of 0 at the two endpoints. Quick Review 10.1 1. tx yt x x =+= −+= + 1 232 132 1 2. t x x x = = −= 3 54 3 54 3 32 3 3 3 3

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422 Section 10.1 3. xt yt tt xy 22 1 1 = = += sin cos cos sin 4. ytt t yx == = sin sin cos 2 5. x y x 2 1 1 = = =+ tan sec sec tan θ 6. x y 2 1 1 + = csc cot cot θθ 7. x y 2 2 1 21 = =− cos cos cos 8. x y 2 2 2 12 = sin cos sin 9. x y 2 2 1 10 = ≤≤ cos sin cos ,( ) θπ 10. x y 2 2 1 = ≤ ≤ cos sin cos ) πθ π Section 10.1 Exercises 1. Yes, y is a function of x. 29 (3, –3) (9, 9) 1 10 10 1 –3 0 x y 2. Yes, y is a function of x. y x = + 2 7 4 (1, 2) (3, 4) 1 5 1 0 x y 4 3. Yes, y is a function of x . 2 1 (1, 0) (2, 1.732) 3 3 1 0 x y 4. No, y is not a function of x. x y 2 2 4 1 (0, 2) (0, –2) 3 1 –3 –1 x y 5. Yes, y is a function of x.
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## This note was uploaded on 09/26/2010 for the course MATH 135 taught by Professor Noone during the Summer '08 term at Rutgers.

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FDWKCalcSM_ch10 - Section 10.1 421 70. Continued Multiply...

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