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FDWKCalcSM_ch10

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

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Section 10.1 421 70. Continued Multiply by x . n n x x x n n ( ) ( ) + = = 1 2 1 3 1 Replace x by 1 x . n n 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 x x = > 2 1 2 769 1 2 3 ( ) . . 71. (a) Computing the coefficients, f f x x f f x ( ) ( ) ( ) , ( ) ( ) ( 1 1 2 1 1 1 4 2 2 = = − + = − ′′ = so x f f x x + ′′ = ′′′ = − + 1 1 2 1 8 6 1 3 4 ) , ( ) ! ( ) ( ) , so so ′′′ = − = + f f n n n n ( ) ! , ! ( ) . ( ) 1 3 1 16 1 2 1 In general S o f x x x 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 1 1 3. The series converges absolutely on ( , ). 1 3 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 = = 3 1 1 2 0 , ( ) , by the th-term test n . The interval of convergence is ( 1, 3). (c) P x x x 3 2 1 2 1 4 1 8 ( ) ( ) = + P 3 2 0 5 1 2 0 5 1 4 0 5 1 8 0 65265 ( . ) . ( . ) . = + = 72. (a) Ratio test for absolute convergence: lim ( ) lim n n n n n n n x n x n n x x x →∞ + + →∞ + = + = 1 2 2 1 2 2 2 1 1 i < ⇒ − < < 1 2 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 n n n n →∞ →∞ 0 1 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 the 10 th term. Thus error < < 10 2 0 01 10 . . 73. (a) P x x 1 1 2 ( ) = − + (b) P x x x 2 2 1 2 3 2 ( ) = − + (c) P x x x x 3 2 3 1 2 3 2 2 3 ( ) = − + + (d) P 3 2 3 0 7 1 2 0 7 3 2 0 7 2 3 0 7 ( . ) ( . ) ( . ) ( . ) = − + + 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. x na = 2 π for any integer n. 3. a t y > 0 1 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 x n a = + ( ) 2 1 π for any integer n. 6. The function represented by the cycloid is periodic with period 2 a π , 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. t x y t x x = = + = + = + 1 2 3 2 1 3 2 1 ( ) 2. t x y t x x = = = = 3 54 3 54 3 3 2 3 3 3 3

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422 Section 10.1 3. x t y t t t x y 2 2 2 2 2 2 2 2 1 1 = = + = + = sin cos cos sin 4. y t t t y x = = = sin sin cos 2 2 2 5. x y x y x 2 2 2 2 2 2 2 1 1 = = = + = + tan sec sec tan θ θ θ 6. x y x y 2 2 2 2 2 2 2 1 1 = = + = = + csc cot cot θ θ θ 7. x y y x 2 2 2 2 2 2 1 2 1 = = = = cos cos cos θ θ θ 8. x y y x 2 2 2 2 2 1 2 1 2 = = = = sin cos sin θ θ θ 9. x y y x 2 2 2 2 2 2 1 1 0 = = = = cos sin cos , ( ) θ θ θ θ π 10. x y y x 2 2 2 2 2 2 1 1 2 = = = = − cos sin cos , ( ) θ θ θ π θ π Section 10.1 Exercises 1. Yes, y is a function of x.
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FDWKCalcSM_ch10 - Section 10.1 421 70 Continued Multiply by...

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