三角函数的泰勒公式
\u6cf0\u52d2\u516c\u5f0f\u7684\u4e09\u89d2\u51fd\u6570\u662f\u4ec0\u4e48sinx= x- x^3/3! + x^5/5! - x^7/7!+ x^9/9! .... =\u2211(-1)^n x^(2n+1)/(2n+1)!
cosx=1- x^2/2! +x^4/4!\u3002\u3002\u3002=\u2211(-1)^n x^(2n)/(2n)!
\u3000\u6cf0\u52d2\u516c\u5f0f\uff08Taylor's formula\uff09
f(x)=f(0)+f'(0)x+f''(0)/2!?x^2,+f'''(0)/3!?x^3+\u2026\u2026+f(n)(0)/n!?x^n+Rn(x)
\u5176\u4e2dRn(x)=f(n+1\uff09\uff08\u03be\uff09/(n+1)!*(x-x.)^(n+1\uff09\uff0c\u8fd9\u91cc\u03be\u5728x\u548cx\u3002\u4e4b\u95f4\uff0c\u8be5\u4f59\u9879\u79f0\u4e3a\u62c9\u683c\u6717\u65e5\u578b\u7684\u4f59\u9879\u3002
\u3000
\u8bc1\u660e
\u6cf0\u52d2\u516c\u5f0f\u5728x=a\u5904\u5c55\u5f00\u4e3a
f(x)=f(a)+f'(a)(x-a)+(1/2!)f''(a)(x-a)^2+\u2026\u2026+(1/n!)f(n)(a)(x-a)^n+\u2026\u2026
\u8bbe\u5e42\u7ea7\u6570\u4e3af(x)=a0+a1(x-a)+a2(x-a)^2+\u2026\u2026\u2460
\u4ee4x=a\u5219a0=f(a)
\u5c06\u2460\u5f0f\u4e24\u8fb9\u6c42\u4e00\u9636\u5bfc\u6570\uff0c\u5f97
f'(x)=a1+2a2(x-a)+3a3(x-a)^2+\u2026\u2026\u2461
\u4ee4x=a,\u5f97a1=f'(a)
\u5bf9\u2461\u4e24\u8fb9\u6c42\u5bfc\uff0c\u5f97
f"(x)=2!a2+a3(x-a)+\u2026\u2026
\u4ee4x=a,\u5f97a2=f''(a)/2!
\u7ee7\u7eed\u4e0b\u53bb\u53ef\u5f97an=f(n)(a)/n!
\u6240\u4ee5f(x)\u5728x=a\u5904\u7684\u6cf0\u52d2\u516c\u5f0f\u4e3a\uff1a
f(x)=f(a)+f'(a)(x-a)+[f''(a)/2\uff01](x-a)^2+\u2026\u2026+[f(n)(a)/n!](a)(x-a)^n+\u2026\u2026
\u5e94\u7528\uff1a\u7528\u6cf0\u52d2\u516c\u5f0f\u53ef\u628af(x)\u5c55\u5f00\u6210\u5e42\u7ea7\u6570\uff0c\u4ece\u800c\u53ef\u4ee5\u8fdb\u884c\u8fd1\u4f3c\u8ba1\u7b97\uff0c\u4e5f\u53ef\u4ee5\u8ba1\u7b97\u6781\u9650\u503c\uff0c\u7b49\u7b49\u3002
\u53e6\u5916\uff0c\u4e00\u9636\u6cf0\u52d2\u516c\u5f0f\u5c31\u662f\u62c9\u683c\u6717\u65e5\u5fae\u5206\u4e2d\u503c\u5b9a\u7406
f(b)=f(a)+f(\u03b5)(b-a),\u03b5\u4ecb\u4e8ea\u4e0eb\u4e4b\u95f4\u3002
\u3000\u6cf0\u52d2\u516c\u5f0f\u6c42\u5404\u79cd\u4e09\u89d2\u51fd\u6570\uff0c\u5982sin,cosx,tanx,cotx
\u3000\u5c55\u5f00\u4e09\u89d2\u51fd\u6570y=sinx\u548cy=cosx\u3002
\u3000\u3000\u89e3\uff1a\u6839\u636e\u5bfc\u6570\u8868\u5f97\uff1af(x)=sinx,f'(x)=cosx,f''(x)=-sinx,f'''(x)=-cosx,f\u2477\uff08x)=sinx\u2026\u2026
\u3000\u3000\u4e8e\u662f\u5f97\u51fa\u4e86\u5468\u671f\u89c4\u5f8b\u3002\u5206\u522b\u7b97\u51faf(0)=0\uff0cf'(0)=1,f''(x)=0,f'''(0)=-1,f\u2477=0\u2026\u2026
\u3000\u3000\u6700\u540e\u53ef\u5f97\uff1asinx=x-x^3/3!+x^5/5!-x^7/7!+x^9/9!-\u2026\u2026\uff08\u8fd9\u91cc\u5c31\u5199\u6210\u65e0\u7a77\u7ea7\u6570\u7684\u5f62\u5f0f\u4e86\u3002\uff09
\u3000\u3000\u7c7b\u4f3c\u5730\uff0c\u53ef\u4ee5\u5c55\u5f00y=cosx\u3002
\u7ed9\u4f60\u7ed3\u8bba\u5427
sin x = x-x^3/3!+x^5/5!-\u2026\u2026+(-1)^(k-1)*(x^(2k-1))/(2k-1)!+\u2026\u2026\u3002(-\u221e<x<\u221e)
cos x = 1-x^2/2!+x^4/4!-\u2026\u2026+(-1)k*(x^(2k))/(2k)!+\u2026\u2026 (-\u221e<x<\u221e)
tanx=x+x^3/3+2x^5/15+17x^7/315+62x^9/2835++[2^(2n)*(2^(2n)-1)*B(2n-1)*x^(2n-1)]/(2n)!+.(|x|<\u03c0/2).
sinx = x + o(x^2) 是二阶泰勒公式，无穷小量为二阶
泰勒多项式一阶和二阶形式一样，区别在无穷小量上
sin x的二阶泰勒公式和一阶一样。
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