m2 微積分的應用 2題 快.////

1)

dy/dx = 2x + 7
解 2x + 7 = 10 => x = 3/2, y = 39/4
直線方程為 y - 39/4 = 10(x - 3/2) => 4y - 39 = 40x - 60 => 40x - 4y - 21 = 0
得一條切線……

2)

(a)

6x + 2y + 2x(dy/dx) - 2y(dy/dx) = 0 => dy/dx = (3x + y) / (y - x)

(b)

解 (3x + y) / (y - x) = -5/3 => 9x +3y = 5x - 5y => x = -2y
代 3(-2y)^2 + 2(-2y)y - y^2 = 7 => 7y^2 = 7 => y = 1 或 -1
當 y = 1，x = -2，方程為 y - 1 = (-5/3)(x + 2) => 5x + 3y + 7 = 0
當 y = -1，x = 2，方程為 y + 1 = (-5/3)(x - 2) => 5x + 3y - 7 = 0


2012-07-11 10:09:54 補充：
很明顯一係題目錯，一係答案錯，邊本書？我幫你check check﹗﹗﹗

2012-07-11 10:11:27 補充：
我幫你用軟件check過了，答案係得一條切線 40x - 4y - 21 = 0

1) y = x2 + 7x - 3

dy/dx = 2x + 7

y = 10x - 3 的斜率為 10, 所以:

2x + 7 = 10

x = 3/2

y = 9/4 + 21/2 - 3 = 39/4

所以切線方程為:

(y - 39/4)/(x - 3/2) = 10

y - 39/4 = 10x - 15

y = 10x - 21/4

2a) 3x2 + 2xy - y2 = 7

6x + 2y + 2x(dy/dx) - 2y(dy/dx) = 0

3x + y + x(dy/dx) - y(dy/dx) = 0

dy/dx = (3x + y)/(y - x)

b) 5x + 3y = 0 的斜率為 -5/3, 所以:

(3x + y)/(y - x) = -5/3

9x + 3y = 5x - 5y

8y = -4x

y = -x/2

代入 C 原方程:

3x2 + 2x(-x/2) - (-x/2)2 = 7

3x2 - x2 - x2/4 = 7

7x2/4 = 7

x2 = 4

x = 2 或 -2

y = -1 或 1

所以切線方程為:

(y + 1)/(x - 2) = -5/3

3y + 3 = 10 - 5x

5x + 3y - 7 = 0

(y - 1)/(x + 2) = -5/3

3y - 3 = -10 - 5x

5x + 3y + 7 = 0