數學___微分2問 - Ans.fyi 參考答案

數學___微分2問

2009-10-30 5:02 am

1.已知f '(0)=2及f '(1)= -1 .若g(x)=f(sinx)+f(cosx), 求g ' (90度)

2.P(a,b)是曲綫C:y=x^3-2(x^2)+4上的一點.C在P的切綫與X軸交於x=1.

a)證明b=3(a^3)-7(a^2)+4a

b)求a和b的值

更新1:

2b.求a和b的值!!!!

回答 (1)

Prof. Physics

2009-10-30 5:37 am

✔ 最佳答案

1. g(x) = f(sinx) + f(cosx)

g'(x) = cosx f'(sinx) - sinx f'(cosx) (連鎖法則)

g'(90*) = cos90* f'(sin90*) - sin90* f'(cos90*)

= 0 - (1)f'(0)

= -2

2.a. y = x^3 - 2x^2 + 4

dy/dx = 3x^2 - 4x

C在P的切綫的斜率 = 3a^2 - 4a

設該切綫方程為

y = mx + c

m = (3a^2 - 4a)

所以，y = (3a^2 - 4a)x + c

代x = 1，y = 0

0 = (3a^2 - 4a)(1) + c

c = 4a - 3a^2

所以，y = (3a^2 - 4a)x + (4a - 3a^2)

由於切綫通過P(a , b)

代x = a，y = b

b = (3a^2 - 4a)a + (4a - 3a^2)

b = 3a^3 - 7a^2 + 4a

2009-10-30 07:30:26 補充：
b = 3a^3 - 7a^2 + 4a ... (1)
但P(a , b)又合乎C的方程
b = a^3 - 2a^2 + 4 ... (2)
(2) X 3 - (1):
0 = a^2 - 4a + 12
然後就可求得a與b的值

參考: Physics king

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