Two Maths Question...Urgent

用戶23173

2009-11-23 2:12 pm

1. How many tangent lines to the curve y=x/(x=1) pass through the point (1,2)? At which points do these lines touch the curve?

2. The Normal Line to a curve C at point P is defined as the line that passes through P and is perpendicular to the tangent line to C at P. Find the equation of the normal line to y=1-x^2 at (2, -3). Also sketch the curve and the normal line.

Please use the knowledge of derivative to answer above two questions and in English...

回答 (1)

六呎將軍

2009-11-23 5:32 pm

✔ 最佳答案

(1) y = x/(x + 1) = 1 - 1/(x + 1)

dy/dx = 1/(x + 1)2

Suppose at (h, k), its tangent to the curve passws through (1, 2), then:

dy/dx = 1/(h + 1)2, also slope of line joining (h, k) and (1, 2) is (k - 2)/(h - 1)

Thus,

1/(h + 1)2 = (k - 2)/(h - 1)

(h - 1)/(h + 1)2 = k - 2 = h/(h + 1) - 2 = (-h - 2)/(h + 1)

h - 1 = (-h - 2)(h + 1)

h - 1 = -h2 - 3h - 2

h2 + 4h + 1 = 0

(h + 2)2 - 3 = 0

h = -2 + √3 or -2 - √3

k = (-2 + √3)/(-1 + √3) or (-2 - √3)/(-1 - √3)

k = (-1 + √3)/2 or (1 + √3)/2

(2) y = 1 - x2

dy/dx = -2x

At (2, -3), dy/dx = -4

Hence normal slope = 1/4, with equation:

(y + 3)/(x - 2) = 1/4

x - 2 = 4y + 12

x - 4y - 14 = 0

2009-11-23 10:06:58 補充：
For the sketch in Q2, pls. refer to:
http://i388.photobucket.com/albums/oo325/loyitak1990/Nov09/Crazygraph1.jpg

參考: Myself

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