不好意思，幫忙一下關於微積分，有幾題不會算，謝謝!

Question

我把題目作成圖片放在下面的網址:
http://ima.csim.hk.edu.tw/g2hosts/main.php?g2_view=core.DownloadItem&g2_itemId=16310&g2_serialNumber=2


先謝謝各位大大!

Answer 1

1.z=exsiny∂z/∂x = exsiny∂z/∂y = excosy∂z/∂x∣(s,t)=(0,0) = e0sin0=0∂z/∂y∣(s,t)=(0,0) = e0cos0=1∂z/∂s = (∂z/∂x)(∂x/∂s)+(∂z/∂y)(∂y/∂s)∂z/∂s∣(s,t)=(0,0) = (0)(3)+(1)(4) = 42.f(x,y) = x/y + y/xfx(x,y) = 1/y - y/x²fy(x,y) = -x/y² + 1/x∇f(x,y) = fx(x,y) i + fy(x,y) j = (1/y - y/x²) i + (-x/y² + 1/x) j3.f(x,y) = x²+y²+2x²y+3fx(x,y) = 2x+4xyfy(x,y) = 2y+2x²令2x+4xy=0, 2y+2x²=0得 x(1+2y)=0, x=0 或 1+2y=0, y=-1/2若x=0, 2y+2(0)²=0, y=0若 y=-1/2, 2(-1/2)+2x²=0, 2x²=1, x= ±(1/√2)∴ (0,0), ( 1/√2 ,-1/2), ( -1/√2 ,-1/2) 三個critical point4.設長寬為x,y高為z，須滿足xyz=100並使得f(x,y,z) = xy+2yz+2xz為最小令L= xy+2yz+2xz+λ(xyz-100) 則∂L/∂x = y+2z+λyz∂L/∂y = x+2z+λxz∂L/∂z = 2y+2x+λxy令∂L/∂x=∂L/∂y=∂L/∂z=∂L/∂λ=0解聯立方程式y+2z+λyz = 0...........①x+2z+λxz = 0...........②2y+2x+λxy = 0.........③xyz=100.....................④由① z=-y/(2+λy)代入②x+2(-y/(2+λy))+ λx(-y/(2+λy))=02x+λxy-2y-λxy=02x-2y=0, x=y 代入 ③2y+2y+ λy²=0, 4y+λy²=0, 4+λy=0, λy=-4 代入① 得y+2z-4z=0, y=2zx=y=2z代入④ (2z)(2z)(z)=100, 4z³=100, z³=25, z= 3√(25)故 3√(25)即為所求之高。(註: 3√(25)表示25開立方根，即 251/3 )5.令L = xy-λ(x²+y²-2) 則∂L/∂x = y-2λx∂L/∂y = x-2λy解聯立方程式y-2λx=0........①x-2λy=0........②x²+y²-2=0........③由① y=2λx代入②x-2λ(2λx)=0, x-4λ²x=0, x(1-4λ²)=0x=0不合1- 4λ² =0λ²=1/4λ=±(1/2)若λ=1/2, 由① y-x=0, y=x代入③ 2x²-2=0, x²=1, x=±1得(x,y)=(1,1)或(-1,-1)若λ=-1/2, 由① y+x=0, y=-x代入③ 2x²-2=0, x=±1得 (x,y) = (1,-1) 或 (-1,1)故 x²+y²=2 時 f(x,y)=xy之極小值為(1)(-1)=-1