曲線C:ρ=a(1+cosθ) 即心臟線,當(dāng)θ=π/2時(shí)對(duì)應(yīng)的點(diǎn)M,求C在點(diǎn)M處的切線方程
θ=π/2時(shí) ρ=a；即M點(diǎn)的極坐標(biāo)為(a,π/2)；M點(diǎn)的直角坐標(biāo)為(0,a)；
將極左邊方程還原成直角坐標(biāo)方程：√(x²+y²)=a[1+x/√(x²+y²)]；
即有x²+y²-ax=a√(x²+y²);
設(shè)F(x,y)=x²+y²-ax-a√(x²+y²);
那么dy/dx=-(∂F/∂x)/(∂F/∂y)=-[2x-a-a/√(x²+y²)]/[2y-a/√(x²+y²)]=-[(2x-a)√(x²+y²)-a]/[2y√(x²+y²)-a]
將x=0,y=a代入得過M點(diǎn)的切線的斜率k=y'(0)=(a+1)/(2a-1)
故過點(diǎn)M的切線方程為y=[(a+1)/(2a-1)]x+a.dy/dx怎么會(huì)用到偏導(dǎo)呢？不能直接對(duì)x隱函數(shù)求導(dǎo)嗎？而且∂F/∂x=2x-a-ax/√(x²+y²)吧，少了個(gè)x?設(shè)F(x，y)=x²+y²-ax-a√(x²+y²)=0【前面我漏寫=0了】dy/dx=-(∂F/∂x)/(∂F/∂y)【是隱函數(shù)的求導(dǎo)公式，與直接求導(dǎo)相比，省去了解出y'的麻煩。你說的對(duì)，確有錯(cuò)！更正如下：】=-[2x-a-ax/√(x²+y²)]/[2y-ay/√(x²+y²)]=-[(2x-a)√(x²+y²)-ax]/[2y√(x²+y²)-ay]將x=0，y=a代入得過M點(diǎn)的切線的斜率k=y'(0)=(a²)/(a²)=1故過點(diǎn)M的切線方程為y=x+a.