數學微積分考卷題

dL/dt = 2
L(t) = 2t + C , C為常數

dW/dt = 1
W(t) = t + K , K為常數

設當 t = T 時,長度為5,寬度為3.
L(T) = 2T + C = 5 .....(1式)
W(T) = T + K = 3 .....(2式)
(1式) - 2*(2式)得:
C - 2K = -1
C = 2K-1

A(t)
= L(t)W(t)
= (2t+2K-1)(t+K)
= 2t^2 + 2Kt + (2K-1)t + K(2K-1)
= 2t^2 +(4K-1)t + K(2K-1)

dA/dt
= 4t + (4K-1)
= 4(t+K) - 1

dA/dt ｜ t = T
= 4(T+K) - 1
= 4*3 - 1 ← 利用(2式)
= 11
Ans: 當長度為5單位,寬度3單位時,面積以每秒11平方單位增加.

A = XY

dX/dt = 2
dY/dt = 1
問: dA/dt = ?