微積分高手20點(能寫多少就多少)

第一題f(x)=sinx at a=0 g(x)=e^x at a=2 h(x)=ln(1+x^2) at a=0
(1) Write Rn(x) of order n for the functions in Taylor's formula.Ans:sin(x)=x-x^3/3!+x^5/5!-x^7/7!+-.....Rn=(-1)^(n+1)*x^(2n-1)/(2n-1)!
ln(1+x)=x-x^2/2+x^3/3-x^4/4+-.....ln(1+x^2)=x^2-x^4/2+x^6/3-x^8/4+-.....Rn=(-1)^(n+1)*x^2n/n!
e^x=(1+x+x^2/2!+x^3/3!+.....)*e^2Rn=x^2*e^2/n!
(2) Prove that the Taylor series of f(x) converges for all x∈-∞~+∞ Ans:By ratio test:limit(n->oo)[A(n+1)/An]=limit(n->oo)[x^(2n+1)/(2n+1)!]/[x^(2n-1)/(2n-1)!]=limit(n->oo){x^2/[2n*(2n+1)]}=0<1=> converge
第二題Given a parametric curve x=cos^3(t), y=sin^3(t), 0<=t<=2π.(1)Find the length of the curveAns:(dL)^2=(dx)^2+(dy)^2=[-3*cos^2(t)*sin(t)dt]^2+[3*sin^2(t)*cos(t)dt]^2=9[cos(t)*sin(t)]^2*[sin^2(t)+cos^2(t)]*(dt)^2dL=3[cos(t)*sin(t)]*dt.......(1)
L對x.y軸都對稱.擁有4段等長曲線
L=4*{3∫[cos(t)*sin(t)]dt};;;t=0~pi/2=4*(3/4)∫sin(2t)]*d(2t)=-3*cos(2t);;;t=0~pi/2=-3*(-1-0)=3.........ans
(2)Find the area of the surface generated by revolving the curve about the x-axisAns:S=4*segments=4∫2*pi*y*dL=8*pi∫sin^3(t)*3[cos(t)*sin(t)]*dt.......by Eq.(1)=24*pi∫sin^4(t)*d[sin(t)];;;t=0~pi/2=(24*pi/5)*sin^5(t)=24*pi/5.....ans


2014-04-02 16:13:05 補充：
第三題

(1) Find the minimum speed of the particle during the period t∈[0,3].

Ans:

v(t)=dr(t)/dt={-2sin(t),3cos(t),1}

V=|v|

=√[4sin^2(t)+9cos^2(t)+1]

=√[5+4cos^2(t)]

V^2=5+4cos^2(t)

兩邊微分:

2V*V'=0

=-8cos(t)*sin(t)

=-4sin(2t)

=> 2t=180

t=90(deg)=pi/2=1.57=0~3


V(90)=√[5+4cos^2(90)]=√5

2014-04-02 16:13:37 補充：
(2) Find the unit tangent vector of the particle at t=π/2

Ans:

T(t)=v(t)/V(t)

T(90)=v(90)/V(90)

={-2sin(t),3cos(t),1}/√5

={-2,0,1}/√5

=(-2i+k)/√5.....ans

2014-04-02 16:18:44 補充：
第四題

Let r(t)=(t+2)i-(t+1)j+t*k, 0<=t<=3, be a vector-valued function.

(1) Find the unit tangent vector.

v(t)=r'(t)=i-j+k

V=|v|=√3

T=v/V=(i-j+k)/√3

2014-04-02 16:20:15 補充：
(2) Find the curvature of the curve at t=1


k=|dT/dt|/V=0.....ans

2014-04-02 16:28:30 補充：
(3) r(t)=(t+2)i-(t+1)j+t*k

z=t, y=-t-1=-z-1, x=t+2=z+2

Add for x,y: x+y=1

f(x,y,z)=x+y-1

N=▽f/|▽f|=(i+j)/√2

2014-04-02 16:28:49 補充：
超過字限

2014-04-02 16:29:16 補充：
不能再貼

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