麻烦图像结合讲解.有简便方法更好.

Question

麻烦图像结合讲解.有简便方法更好.
 

Answer 1

答案C
不知道是否有简便做法.
f(x) = x³ - x
f'(x) = 3x² - 1
P₁(x₁, x₁³ - x₁), f'(x₁) = 3x₁² - 1
过P₁的切线: y - (x₁³ - x₁) = (3x₁² - 1)(x - x₁), y = (3x₁² - 1)x - 2x₁³
与f(x) = x³ - x联立, 整理得: (x - x₁)²(x + 2x₁) = 0
P₂(-2x₁, -8x₁³ + 2x₁)
过P₁的切线: y - (-8x₁³ + 2x₁) = [3(-2x₁)² - 1](x + 2x₁)
y = (12x₁² + 1)x + 16x₁³
与f(x) = x³ - x联立, 整理得: (x + 2x₁)²(x - 4x₁) = 0
P₃(4x₁, 64x₁³ - 4x₁)
S₁ = ∫[(3x₁² - 1)x - 2x₁³ - (x³ - x)]dx x₁ -> -2x₁
= ∫(-x³ + 3x₁²x - 2x₁³)dx x₁ -> -2x₁
= -x⁴/4 + 3x₁²x²/2 -2x₁³x x₁ -> -2x₁
= 27x₁⁴/4
S₂ = ∫[(12x₁² + 1)x + 16x₁³ - (x³ - x)]dx -2x₁ -> 4x₁
= ∫(-x³ + 12x₁²x + 16x₁³)dx -2x₁ -> 4x₁
= -x⁴/4 + 6x₁²x² - 16x₁³x -2x₁ -> 4x₁
= 108x₁⁴
S₁/S₂ = (27x₁⁴/4)/(108x₁⁴) = 1/16