三次函數的圖形
Author: 藍邦偉　老師

顯示極值點

顯示反曲點

三次函數的圖形

a_1

a_2

a_3

a_4

a=

b=

c=

d=

性質

顯示切線

顯示f'(x)

顯示f''(x)

顯示過程

d = 0
c = 0
b = 0
a = 0
f(x) = a * x^(3) + b * x^(2) + c * x + d
t = -4.32 (slider from x(Corner[1]) to x(Corner[2]))
A, B = Extremum(f)
C: (t, f(t))
L = Tangent(C, f)
m = Slope(L)
p_1(x) = f''(x)
D = TurningPoint(f)
u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
q: p_1(t)
t_1 = false
t_2 = false
inputBox1 = Textfield(a)
inputBox2 = Textfield(b)
inputBox3 = Textfield(c)
inputBox4 = Textfield(d)
t_3 = false
t_4 = false
p_0(x) = f'(x)
t_6 = false
E, F = Root(p_0)
t_A: "" + (LaTeX[A]) + ""
t_B: "$ " + (LaTeX[B]) + " $"
t_{Reflect}: "$ " + (LaTeX[D]) + " $"
t_5 = false
Delta: 4 * b^(2) - 12 * a * c
G = (3.0079008264462734, 3.514142749812173)
t_7 = false
t_f: "$f(x)=" + (LaTeX[If[abs(a) ≠ 1, FractionText[a], If[a ≟ -1, "-", ""]]]) + "x³" + (LaTeX[If[b ≟ 0, "", If[b ≟ 1, "+x^2", If[b ≟ -1, "-x^2", If[b > 0, "+" * abs(b) * "x^2", -abs(b) * "x^2"]]]]]) + "" + (LaTeX[If[c ≟ 0, "", If[c ≟ 1, "+x", If[c ≟ -1, "-x", If[c > 0, "+" * abs(c) * "x", -abs(c) * "x"]]]]]) + "" + (LaTeX[If[d > 0, "+" * d, If[d < 0, "-" * abs(d), ""]]]) + "$
$⇒f'(x)=" + (LaTeX[If[abs(a) ≠ 1 / 3, FractionText[3 * a], If[a ≟ (-1) / 3, "-", ""]]]) + "x^2" + (LaTeX[If[b ≟ 0, "", If[b ≟ 1 / 2, "+x", If[b ≟ (-1) / 2, "-x", If[b > 0, "+" * abs(2 * b) * "x", -abs(2 * b) * "x"]]]]]) + "" + (LaTeX[If[c > 0, "+" * c, If[c < 0, "-" * abs(c), ""]]]) + "=0$
$ " + (LaTeX[If[Delta > 0, "x=" * x(A) * ",x=" * x(B)]]) + "$
$ " + (LaTeX[If[Delta > 0, "f(" * x(A) * ")=" * y(A) * ",\;"]]) + " " + (LaTeX[If[Delta > 0, "f(" * x(B) * ")=" * y(B)]]) + "$
$⇒f''(x)=" + (LaTeX[If[abs(a) ≠ 1 / 6, FractionText[6 * a], If[a ≟ (-1) / 6, "-", ""]]]) + "x" + (LaTeX[If[b > 0, "+" * 2 * b, If[b < 0, "-" * abs(2 * b), ""]]]) + "=0$
$x=" + (LaTeX[x(D)]) + "$
$f(" + (LaTeX[x(D)]) + ")=" + (LaTeX[y(D)]) + "$"
H = (0.8095537190082577, 5.18356423741548)
t_{Property}: "$切線斜率m=f'(t)=" + (LaTeX[m]) + "" + (LaTeX[If[m > 0, ">", "<"]]) + "0," + (LaTeX[If[m > 0, "y=f(x)遞增", "y=f(x)遞減"]]) + "$
$f''(t)=" + (LaTeX[p_1(t)]) + "" + (LaTeX[If[q > 0, ">", "<"]]) + "0,\;y=f(x)開口凹向 " + (LaTeX[If[q > 0, "上", "下"]]) + "$"
Q_2: 0.8 * Corner[1] + 0.2 * Corner[2]
Q_3: (x(Q_2), y(Corner[3]))
poly, e, q_2, q_3, g = Polygon(Corner[1], Q_2, Q_3, Corner[4])