有關微分法一問,請各位大大幫忙

石頭速度 ds/dt = d(5t2)/dt = 10t

t = 5, 時, ds/dt = 50 m/s

設 x 為 石頭與小船的距離, 則:

x = √[(200 - s)2 + 1002]

= √[(200 - 5t2)2 + 1002]

= √(40000 - 2000t2 + 25t4 + 10000)

= 5√(2000 - 80t2 + t4)

dx/dx = {5/[2√(2000 - 80t2 + t4)]} d(2000 - 80t2 + t4)/dt

= 5(-160t + 4t3)/[2√(2000 - 80t2 + t4)]

= 5(2t3 - 80t)/√(2000 - 80t2 + t4)

t = 5 時:

dx/dt = 5(250 - 400)/√(2000 - 2000 + 625)

= - 30 m/s

s = 5t^2
ds/dt = 10t = speed
so when t = 5, speed = 10(5) = 50m/s.
Distance between the bottom of the cliff and the stone = 200 - 5t^2
By Pythagoras theorem, distance between the boat and the stone,
x^2 = 100^2 + (200 - 5t^2)^2
2x(dx/dt) = 2(200 - 5t^2)(- 10t)
dx/dt = - 10t(200 - 5t^2)/x
For t = 5,
x^2 = 100^2 + (200 - 125)^2 = 10000 + 5625 = 15625
x = 125
so dx/dt = - 50(200 - 125)/125 = - 50(75)/125 = 30 m/s. decreasing.

s=(5t^2)
=(5x5^2)/100
=5x25/100
=125100
=1.25