Re: [微分] 問題請教...
※ 引述《victor7935 (victor)》之銘言：
: 1.
: Find A and B given that the function
: f(x) = { x^3 , x 小於等於1
: { Ax +B , x> 1
應該有說函數是連續且可微分的吧?
-
f'(x->1 ) = 3
+
f'(x->1 ) = A
A = 3
-
f(x->1 ) = 1
+
f(x->1 ) = A + B = 3 + B
B = -2
: 2.Find conditions on a,b,c,d which guarantee that the graph
: of the cubic p(x)=ax^3+bx^2+cx+d
cubic 三次方
: (1.) exactly two horizontal tangents.
a =/= 0 因為cubic
p' = 3ax^2 + 2bx + c
因為有兩條水平線
表示有兩個"相異"x使得p'(x) = 0
所以判別式要 > 0
4b^2 - 12ac > 0
=> b^2 > 3ac
: 2 2
: 3.Express d y/d x
: (a) 4tany = x^3
題目的意思是要你用x的函數來表達y''
4(secy)^2 * y' = 3x^2
8(secy)(secy)(tany)y'y' + 4(secy)^2 y'' = 6x
8(secy)^2 (tany) 9x^4
=> ----------------------- + 4(secy)^2 y'' = 6x
16 (secy)^4
9 x^4 tany
=> --------------------- + 4 [1 + (tany)^2] y'' = 6x
4 [1 + (tany)^2]
9 x^4 x^3
=> -------------------- + 4 [1 + x^6 /16]y'' = 6x
16 [1 + x^6 /16]
移項後就等式左邊y'' 右邊都是x的函數
即為所求
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