四次函數的圖形
Author: 藍邦偉　老師

顯示極值點

顯示反曲點

四次函數的圖形

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)

e=

e

e = 3
a = 1
b = 0
c = -4
d = 0
f(x) = a * x^(4) + b * x^(3) + c * x^(2) + d * x + e
t = -4.32 (slider from x(Corner[1]) to x(Corner[3]), step 0.01)
A, B, H = Extremum(f)
C: (t, f(t))
L = Tangent(C, f)
m = Slope(L)
p_1(x) = f''(x)
D, G = 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_5 = false
text14: "" + (LaTeX[A]) + ""
text15: "$ " + (LaTeX[B]) + " $"
text17: "$ " + (LaTeX[D]) + " $"
t_6 = false
inputBox5 = Textfield(e)
text20: "" + G
text21: "" + H
E, F = Root(p_1)
I = (5.164129917355363, 8.881916303531181)
t_f: "$f(x)=" + (LaTeX[If[abs(a) ≠ 1, FractionText[a], If[a ≟ -1, "-", ""]]]) + "\;x⁴" + (LaTeX[If[b ≟ 0, "", If[b ≟ 1, "+x^3", If[b ≟ -1, "-x^3", If[b > 0, "+" * abs(b) * "x^3", -abs(b) * "x^3"]]]]]) + "" + (LaTeX[If[c ≟ 0, "", If[c ≟ 1, "+x^2", If[c ≟ -1, "-x^2", If[c > 0, "+" * abs(c) * "x^2", -abs(c) * "x^2"]]]]]) + "" + (LaTeX[If[d ≟ 0, "", If[d ≟ 1, "+x", If[d ≟ -1, "-x", If[d > 0, "+" * abs(d) * "x", -abs(d) * "x"]]]]]) + "" + (LaTeX[If[e > 0, "+" * e, If[e < 0, "-" * abs(e), ""]]]) + "$
$⇒f'(x)=" + (LaTeX[If[abs(a) ≠ 1 / 4, FractionText[4 * a], If[a ≟ (-1) / 4, "-", ""]]]) + "\;x^3" + (LaTeX[If[b ≟ 0, "", If[b ≟ 1 / 3, "+x^2", If[b ≟ (-1) / 3, "-x^2", If[b > 0, "+" * abs(3 * b) * "x^2", -abs(3 * b) * "x^2"]]]]]) + "" + (LaTeX[If[c ≟ 0, "", If[c ≟ 1 / 2, "+x", If[c ≟ (-1) / 2, "-x", If[c > 0, "+" * abs(2 * c) * "x", -abs(2 * c) * "x"]]]]]) + "" + (LaTeX[If[d > 0, "+" * d, If[d < 0, "-" * abs(d), ""]]]) + "=0$
$⇒x=" + (LaTeX[x(A)]) + "," + (LaTeX[x(B)]) + "," + (LaTeX[x(H)]) + "$
$⇒f(" + (LaTeX[x(A)]) + ")=" + (LaTeX[y(A)]) + "$
$⇒f(" + (LaTeX[x(B)]) + ")=" + (LaTeX[y(B)]) + "$
$⇒f(" + (LaTeX[x(H)]) + ")=" + (LaTeX[y(H)]) + "$"
t_{f2}: "$⇒f''(x)=" + (LaTeX[If[abs(a) ≠ 1 / 12, FractionText[a], If[a ≟ (-1) / 12, "-", ""]]]) + "\;x²" + (LaTeX[If[b ≟ 0, "", If[b ≟ 1 / 6, "+x", If[b ≟ (-1) / 6, "-x", If[b > 0, "+" * abs(6 * b) * "x", -abs(6 * b) * "x"]]]]]) + "" + (LaTeX[If[c > 0, "+" * 2 * c, If[c < 0, "-" * abs(2 * c), ""]]]) + "=0$
$⇒x=" + (LaTeX[x(D)]) + "," + (LaTeX[x(G)]) + "$
$⇒f(" + (LaTeX[x(D)]) + ")=" + (LaTeX[y(D)]) + "$
$⇒f(" + (LaTeX[x(G)]) + ")=" + (LaTeX[y(G)]) + "$"
J = (5.776389917355362, 2.91903630353118)
t_{property}: "$切線斜率m=f'(t)=" + (LaTeX[m]) + "" + (LaTeX[If[m > 0, ">", "<"]]) + "0," + (LaTeX[If[m > 0, "y=f(x)遞增", "y=f(x)遞減"]]) + "$
$f''(t)=" + (LaTeX[p_1(t)]) + "" + (LaTeX[If[q > 0, ">", "<"]]) + "0,y=f(x)開口凹向" + (LaTeX[If[q > 0, "上", "下"]]) + "$"
K = (-7, 10)
Q_2: 0.2 * Corner[2] + 0.8 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
poly, g, q_2, q_3, h = Polygon(Corner[1], Q_2, Q_3, Corner[4])