切線與函數圖形的單調性
Author: 藍邦偉老師

切點的x座標為%v

$ f(x)= $

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$ 切線與函數圖形的單調性$

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c = 1 (slider from floor(x(Q_2)) to floor(x(Corner[2])), step 0.5)
f(x) = -x^(3) + 5 * x^(2) - 8 * x + 4
t = 1 (slider from floor(x(Q_2)) to floor(x(Corner[2])), step 0.5)
r = 0.6 (slider from 0.1 to 3, step 0.1)
List_1 = Sequence(Segment[(-10, k), (10, k)], k, -10, 10)
J = Point(List_1)
R = (1.2829937475160558, 0.20794755623685648)
a_7 = Ray(R, J)
c_a = Circle(R, r)
K = Intersect(c_a, a_7)
O = (-12, -3)
d_a = Circle(O, 3)
A = Point(f)
a_1 = Ray(R, A)
B = Intersect(c_a, a_1)
k_1 = AffineRatio(R, B, A)
a_2 = Ray(O, O + Vector[R, A])
C = Intersect(d_a, a_2)
C' = Dilate(C, k_1, O)
C'' = If(k_1 < 1, C')
loc1 = Locus(C'', A)
D = Point(yAxis)
a_3 = Ray(R, D)
E = Intersect(c_a, a_3)
a_4 = Ray(O, Vector[R, E])
F = Intersect(d_a, a_4)
k_2 = AffineRatio(R, E, D)
F' = Dilate(F, k_2, O)
F'' = If(k_2 < 1, F')
loc2 = Locus(F'', D)
G_1 = Point(xAxis)
a_5 = Ray(R, G_1)
H = Intersect(c_a, a_5)
k_3 = AffineRatio(R, H, G_1)
a_6 = Ray(O, Vector[R, H])
I = Intersect(d_a, a_6)
I' = Dilate(I, k_3, O)
I'' = If(k_3 < 1, I')
loc3 = Locus(I'', G_1)
a_8 = Ray(O, Vector[R, K])
L = Intersect(d_a, a_8)
k_4 = AffineRatio(R, K, J)
L' = Dilate(L, k_4, O)
L'' = If(k_4 < 1, L')
loc4 = Locus(L'', J)
List_2 = Sequence(Segment[(k, -6), (k, 10)], k, -10, 10)
M = Point(List_2)
a_9 = Ray(R, M)
N = Intersect(c_a, a_9)
k_5 = AffineRatio(R, N, M)
a_{10} = Ray(O, Vector[R, M])
P = Intersect(d_a, a_{10})
P' = Dilate(P, k_5, O)
P'' = If(k_5 < 1, P')
loc5 = Locus(P'', M)
c_b = Circle(O, 2.9)
Q_2: 0.3 * Corner[2] + 0.7 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
poly1, a, q_2, q_3, b = Polygon(Corner[1], Q_2, Q_3, Corner[4])
u_{x} = Vector((x(Q_2), 0), (x(Corner[2]), 0))
u_{y} = Vector((0, y(Q_2)), (0, y(Corner[3])))
inputBox1 = Textfield(f)
text1: "$ 縮放倍數：\;" + (LaTeX[4 / r]) + "倍 $"
Q: (t, f(t))
c_1 = Tangent(Q, f)
U = Intersect(d_a, c_b, 1)
V = Intersect(d_a, c_b)
S = Point(c_1)
a_{11} = Ray(R, S)
T = Intersect(c_a, a_{11})
k_6 = AffineRatio(R, T, S)
a_{12} = Ray(O, Vector[R, S])
W = Intersect(d_a, a_{12})
W' = Dilate(W, k_6, O)
W'' = If(k_6 < 1, W')
loc6 = Locus(W'', S)
a_{13} = Ray(R, Q)
Z = Intersect(c_a, a_{13})
k_7 = AffineRatio(R, Z, Q)
a_{14} = Ray(O, Vector[R, Z])
G = Intersect(d_a, a_{14})
G' = Dilate(G, k_7, O)
g = TaylorSeries(f, c, 3)
G'' = If(k_7 < 1, G')
taylor_{form}: "$\bgcolor{yellow}{f(x)= " + (LaTeX[g]) + " }$"
A_1 = (-7.7859734345354985, -3.793914152668443)
d = false
text2: "$ 顯示x=" + (LaTeX[c]) + "附近的泰勒展開式 $"