Suppose an approximation of the value of sin :3 is required. in .NET framework

Generating QR Code in .NET framework Suppose an approximation of the value of sin :3 is required.

Suppose an approximation of the value of sin :3 is required.
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1 P4 :3 :3 :3 3 % :2945: 6 The accuracy of this approximation can be determined from examination of the remainder. In particular, (remember j cos j 1)    1 1 243  cos  :3 5  jR4 j   120 105 < :000021 5!
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Thus, the approximation P4 :3 for sin :3 is correct to four decimal places. Additional insight to the process of approximation of functional values results by constructing a graph of P4 x and comparing it to y sin x. (See Fig. 11-2.) x3 P4 x x 6 p The roots of the equation are 0; 6. Examination of the rst and Fig. 11-2 p second derivatives reveals a relative maximum at x 2 and a relative p minimum at x 2. The graph is a local approximation of the sin curve. The reader can show that P6 x produces an even better approximation. (For an example of series approximation of an integral see the example below.)
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SOME IMPORTANT POWER SERIES The following series, convergent to the given function in the indicated intervals, are frequently employed in practice: x3 x5 x7 x2n 1 1. sin x x 1 n 1 1 < x < 1 3! 5! 7! 2n 1 ! x2 x4 x6 x2n 2 2. cos x 1 1 n 1 1 < x < 1 2! 4! 6! 2n 2 ! 2 3 n 1 x x x 3. ex 1<x<1 1 x 2! 3! n 1 ! x2 x3 x4 xn 1<x@1 4. ln j1 xj x 1 n 1 2 3 4 n   3 5 7 2n 1 1 x  x x x x x 5. 1 ln 1<x<1 2  1 x 3 5 7 2n 1 x3 x5 x7 x2n 1 6. tan 1 x 1 @ x @ 1 x 1 n 1 3 5 7 2n 1 p p 1 2 p p 1 . . . p n 1 n x x 7. 1 x p 1 px 2! n! This is the binomial series. (a) If p is a positive integer or zero, the series terminates. (b) If p > 0 but is not an integer, the series converges (absolutely) for 1 @ x @ 1: c If 1 < p < 0, the series converges for 1 < x @ 1: (d) If p @ 1, the series converges for 1 < x < 1.
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For all p the series certainly converges if 1 < x < 1.
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EXAMPLE. Taylor s Theorem applied to the series for ex enables us to estimate the value of the integral ! 1 x2 1 x4 x6 x8 e 10 2 Substituting x for x, we obtain 0 e dx 0 1 x x dx 2! 3! 4! 5! where P4 x 1 x and R4 x e 10 x ; 5! 0<<x 1 4 1 6 1 8 x x x 2! 3! 4!
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ex dx.
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[CHAP. 11
Then 1 1 1 1 P4 x dx 1 % 1:4618 3 5 2! 7 3! 9 4!   1   1 1 10  e   x e    R4 x dx dx < :0021  x10  dx e   5!  5! 11:5
0 0 0 0
Thus, the maximum error is less than .0021 and the value of the integral is accurate to two decimal places.
SPECIAL TOPICS 1. Functions de ned by series are often useful in applications and frequently arise as solutions of di erential equations. For example, the function de ned by ( ) xp x2 x4 1 Jp x p 2 p! 2 2p 2 2 4 2p 2 2p 4
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