asked in 教育與參考考試 · 1 decade ago

微積分計算問題

<1>suppose that a differentiable function y=f(x) is implicitly defined by

the equation x^y=y^x .

find dy/dx, dx/dy

<2>Estimate∫sin(t^2) dt correct to within an error 0.001

範圍是0~1

<3>∫sec^5x dx

Update:

dx/dy =x ( x - lnx ) /y( y - x lny )

ln x前面是不是少ㄍy

1 Answer

Rating
  • 1 decade ago
    Favorite Answer

    1.xy = yx, =>ln xy = ln yx so that y lnx = x lny . Differentiate on both sides with respect to x , we get y'.( lnx )+ y.( 1 / x )= 1.ln y+ x.(y'/ y). And hence dy/dx = y' =y( y - x lny )/x ( x - lnx ), where dx/dy =1/(dy/dx) ∴dx/dy =x ( x - lnx ) /y( y - x lny ) . 2.Estimate∫10 sin( t2 )dt correct to within an error 0.001. (See my blog! )3.∫sec5 xdx We use the reduced form to find the answer. Let In=∫secn xdx=1/(n-1).secn-2 x.tanx+(n-2/n-1).∫secn-2 xdx , thus ∫sec5 xdx =(1/4 ).sec3 x.tanx+(3/4).I 3 =(1/4 ).sec3 x.tanx+(3/4).((1/2)secx.tanx + (1/2)∫secxdx )==(1/4 ).sec3 x.tanx+(3/8).secx.tanx +(3/8).ln|secx+tanx|+c

    2008-02-02 06:49:49 補充:

    dx/dy =x ( x -y lnx ) /y( y - x lny )

    漏打了!!

    Source(s): myself
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