2-1導數的概念

2-1 導數的概念

2-1導數的概念


(甲) 切線與瞬時變化率

(1) P點的切線為 y + 3 = m(x-2) , m 為 f(x)在 P點上的斜率 , m = f'(x) | x = 2 .
f'(x) = 3x^2 - 6x , 所以 m = 12-12 = 0 , 所以切線 y = -3 . 但法線斜率為 m' , m*m' = -1 , m' = ?
應該無.
(2) L 與 f(x) 有交點的話, 即求L與f(x)的聯立方程式解集合.
      y = - 3 .
      y = x^3 -3x^2 + 1
      - 3  = x^3 -3x^2 + 1
       x^3 -3x^2  + 4 = 0 , but how to find its roots  ?

(1) r = 2 , V(2) = 4/3*8*π = 32/3*π , r = 4 , V(4) = 4/3*64*π = 256/3*π , V(4) - V(2) = 112/3*π

(2) 瞬時變化率 = 為體積相對時間的變化率 , 即 V'(r) = 4πr^2 . r = 2 , V'(2) =16π . 

f(x) = x^1/2 , f'(x) = 1/2 x^(-1/2) . m = 1/2 4^-1/2 = 1/4

(1) t = 2 , f(t) = 2*4 + 3*2 = 8 + 6 = 14
     t = 4 , f(t) = 2*16+3*4 = 32 + 12 = 44 , f(4)- f(2) / 4 -2 = 30 /2 = 15
(2) 瞬時速度 = f(t)的微分 , f'(t) = 4t + 3 , f'(2) = 11