題目 6. Note that z=f(x,y)z=f(x,y)z=f(x,y) where fff is differentiable, x=g(t)x=g(t)x=g(t), and y=h(t)y=h(t)y=h(t). It is known that g(3)=2g(3)=2g(3)=2, g′(3)=5g'(3)=5g′(3)=5, h(3)=7h(3)=7h(3)=7, h′(3)=−4h'(3)=-4h′(3)=−4, fx(2,7)=6f_x(2,7)=6fx(2,7)=6, and fy(2,7)=−8f_y(2,7)=-8fy(2,7)=−8. Determine the value of dz/dtdz/dtdz/dt at t=3t=3t=3. (A) 62 (B) 30 (C) 32 (D) -2 (E) 1。 解答 解法一 思路 展開 對 z=f(x(t),y(t))z=f(x(t),y(t))z=f(x(t),y(t)) 套用二變數鏈鎖規則。 答題過程 展開 dzdt∣t=3=fx(2,7)g′(3)+fy(2,7)h′(3)=6⋅5+(−8)(−4)=62.\left.\frac{\mathrm{d}z}{\mathrm{d}t}\right|_{t=3} =f_x(2,7)g'(3)+f_y(2,7)h'(3) =6\cdot5+(-8)(-4)=62.dtdzt=3=fx(2,7)g′(3)+fy(2,7)h′(3)=6⋅5+(−8)(−4)=62. 答案為 (A)\boxed{\text{(A)}}(A)。
