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1 - II sin x, cos x

2 ., ) ) n x n = 0 + x + x + + n x n + n=0 n x c) n = 0 + x c) + x c) + + n x c) n + n=0., 0,,,, n, c. n S n x) = k x k = 0 + x + x + + n x n k=0, {S n x)},,. fx), fx) = k=0 k x k = 0 + x + x + + n x n = lim n S nx)., fx). x. [3.] n x n x = r, x < r x. n=0 x.,. r, r), x > r r. r, r < x < r. x = ±r,. [3.]. n x n r, r = 0, r n=0 r = lim n n n +, r = n n lim n

3 . y = fx) : y = f x) : y = f x) : y = f x) : 3 n fx) n.. f n) x), y n), d n y dx n, d n dx n fx).. n D fx) n, fx) D n,.,.,. R. R. y = e x y = y = = y n) = e x y = sin x y n) = sinx + n π ) y = cos x y n) = cosx + n π ) y = + x) p p, p x > ) y n) = pp )p ) p n + ) + x) p n p, R, y k) = k!, y k+) = y k+) = y k+3 = = 0., y n),, ). y = log + x) x > ) y n) n n )! = ) + x) n, ). log,. + x 3

4 .3, 3) fb) = f) + f c)b ) c b.,,. fx) f) + f )x ) y = fx) x =.,. fx) f) + f )x ) + f ) b ),. f c) x ). [3.3] fx) D n +, D, b fb) = f) + f )b ) + f ) c b. b ) + + f n) ) n!, b ) n + f n+) c) b )n+ n + )! n, b c,., n = 0., R n+ = f n+)c) b )n n + )!. [3.4] fx) 0 D n +, D x fx) = f0) + f 0)x + f 0)! x + + f n) 0) x n + f n+) c) n! n + )! xn+ c 0 x. = 0, b = x., R n+ = f n+)c) n + )! xn 4

5 .R n+, R n+ x n fx), R n+., fx),. fx), x, fx). [3.5] R n+ x) 0 n ) fx) = f0) + f 0)x + f 0)! x + + f n) 0) x n + n!. n.. ) ) 3) e x = + x + x! + x3 3! + + xn + r = ) n! sin x = x x3 3! + x5 xn+ + )n + r = ) 5! n + )! cos x = x! + x4 xn + )n + r = ) 4! n)! p r, p r ) = pp )p ) p r + ) r!, 4) + x) n = + p ) x + p ) x + p n ) x n +. R n+, x = 0 n, R n+. 5

6 .,. [4.] 6) 7) 8) 9) 0) ) cos x = tn x + C sin x = tn x + C x = log x x + + C 0) x + = tn x + C 0) x = sin x + C > 0) x A = log x + x + A + CA 0) [4.] {fx) + gx)dx} = fx)dx + kfx)dx = k fx)dx [4.3] gx)dx ϕx) = t 4.4,. x dx = fϕx))ϕ x)dx = ft)dt [4.4] fx)g x)dx = fx)gx) f x)gx)dx x x + sin x ) + C ) x Adx = x x + A + A log x + x + A + C 6

7 fx) = sin x, g x) =,. sin xdx = x sin x x dx x = x sin x x + C 7

8 .., log ), x3 6x + x + 6 x 8x + 5, ) x 3 6x + x + 6 x 8x + 5 = x + + x 4 x 8x + 5 ) x 4 x 8x + 5 = 3 x 5 x 3 3) x x 5 ) dx = x + x + 3 log x 5 log x 3 + C x 3 4) x x 3 + x + log + C log x 5 3 3,.,. 3x +, x )x + 4). 3x + x )x + 4) = x + x + x + 4 = x + x x x sin x, cos x, tn x = t. 8

9 3 3.,..! 3..,,. fx) [, b] fx) 0, fx) x, x [, b] n, 5) = x 0, x, x,, x n, x n = b. x, x = b n. fx i ), 6) n fx i ) x i=.,,., x 0, n,. { n } 7) S = lim fx i ) x n i=,, fx) 0 ). [5.] fx) [, b]. [, b] n, 5,. 8) b fx)dx = lim n i= n fx i ) x x = b ) n, fx) fx)dx = 0 9

10 ., > b,.,. b fx)dx = b [5.] fx)dx fx) D, D, b, c b fx)dx = c fx)dx + b c fx)dx, ).. fx) [, b], 0 n fx i ) x 0 i=, n, b fx)dx 0., fx) = 0.,,. [5.3] fx), gx) [, b] fx) gx) b fx)dx b gx)dx, fx) = gx). fx) fx), b fx) dx b fx)dx.. [5.4] fx) [, b], b fx)dx = fc)b ) c, b. 0

11 [] 3 :Rodgin

[ ] x f(x) F = f(x) F(x) f(x) f(x) f(x)dx A p.2/29

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