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from manimlib.imports import*
import math
class intro(Scene):
def construct(self):
equation=TextMobject("$f(x)=$","${ e }^{ -x^{ 2 } }$")
equation.scale(2)
equation.set_color_by_tex_to_color_map({"${ e }^{ -x^{ 2 } }$":RED})
text=TextMobject("at $a=1$")
text.scale(0.7)
text.shift(DOWN)
shiftText=TextMobject("(Here we shift the origin to the point $x=1$)")
shiftText.scale(0.6)
shiftText.shift(2.4*DOWN)
self.play(Write(equation))
self.wait(0.5)
self.play(FadeIn(text))
self.wait(0.7)
self.play(Write(shiftText))
self.wait(0.7)
self.play(FadeOut(equation),FadeOut(text),FadeOut(shiftText))
def formFormula(coeff_list,variable_list):
coeff_list=[TextMobject("${ a }_{ 0 }$"),TextMobject("${ a }_{ 1 }$"),TextMobject("${ a }_{ 2 }$")]
variable_list=[TextMobject("+"),TextMobject("${ (x-1) }$+"),TextMobject("${ (x-1) }^{ 2 }$")]
coeff_list[0].shift(2.2*UP+1.6*LEFT)
for i in range(0,3):
coeff_list[i].set_color(GOLD_A)
variable_list[i].next_to(coeff_list[i],buff=0.1)
if i!=2:
coeff_list[i+1].next_to(variable_list[i],buff=0.1)
dots=TextMobject("...")
dots.next_to(variable_list[2])
expansion=VGroup(coeff_list[0],coeff_list[1],coeff_list[2],variable_list[0],variable_list[1],variable_list[2],dots)
#expansion.scale(0.7)
return expansion,coeff_list
class graphScene(GraphScene):
CONFIG = {
"x_min": -8,
"x_max": 8,
"y_min": -8,
"y_max": 8,
"graph_origin": ORIGIN,
"function_color": RED,
"axes_color": GREEN,
"x_axis_label": "$x$",
"y_axis_label": "$y$",
"exclude_zero_label": True,
"x_labeled_nums": range(-8, 8, 1),
}
def construct(self):
x_each_unit = self.x_axis_width / (self.x_max - self.x_min)
y_each_unit = self.y_axis_height / (self.y_max - self.y_min)
generalized_eq_coeff=[]
variables_eq=[]
eq,generalized_eq_coeff=formFormula(generalized_eq_coeff,variables_eq)
trText1=TextMobject("let $T_{ n }(x)$:=")
eq.next_to(trText1)
trTextGrup=VGroup(trText1,eq)
trTextGrup.scale(0.5)
trTextGrup.to_corner(UP+RIGHT)
self.play(Write(trTextGrup))
self.setup_axes(animate=True)
fx=TextMobject("${ e }^{ -x^{ 2 } }$")
fx.scale(0.5)
fx.shift(ORIGIN+x_each_unit*7.5*RIGHT+y_each_unit*0.5*UP)
mainfunction=self.get_graph(lambda x:math.exp(-1*pow(x,2)),color=RED,x_min=-8,x_max=8)
self.play(ShowCreation(mainfunction))
self.play(FadeIn(fx))
self.wait(1.4)
coeff=[TextMobject("$e^{-1}$"),TextMobject("$f'(x)$"),TextMobject("$\\frac { f''(x) }{ 2! } $")]
coeff[0].shift(3.33*UP+3.65*RIGHT)
coeff[0].scale(0.45)
coeff[1].shift(3.33*UP+4.13*RIGHT)
coeff[1].scale(0.275)
coeff[2].shift(3.33*UP+5.36*RIGHT)
coeff[2].scale(0.28)
for obj in coeff:
obj.set_color(GOLD_A)
firstApprox=[self.get_graph(lambda x:math.exp(-1),color=BLUE,x_min=-5.5,x_max=5.5)]
secondApprox=[self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1),color=BLUE,x_min=-5.5,x_max=5.5),
self.get_graph(lambda x:math.exp(-1)+3*(x-1)*math.exp(-1),color=BLUE,x_min=-5.5,x_max=5.5),
self.get_graph(lambda x:math.exp(-1)-4*(x-1)*math.exp(-1),color=BLUE,x_min=-5.5,x_max=5.5)]
thirdApprox=[self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1)-2*math.exp(-1)*(x-1)**2,color=BLUE,x_max=5.5,x_min=-5.5),
self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1)-0.1*math.exp(-1)*(x-1)**2,color=BLUE,x_max=5.5,x_min=-5.5),
self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1),color=BLUE,x_max=5.5,x_min=-5.5),
self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1)+0.5*math.exp(-1)*(x-1)**2,color=BLUE,x_max=5.5,x_min=-5.5),
self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1)+2*math.exp(-1)*(x-1)**2,color=BLUE,x_max=5.5,x_min=-5.5)]
firstGraph=self.get_graph(lambda x:math.exp(-1),color=BLUE,x_min=-5.5,x_max=5.5)
secondGraph=self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1),color=BLUE,x_min=-5.5,x_max=5.5)
thirdGraph=self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1)+math.exp(-1)*(x-1)**2,color=BLUE,x_max=5.5,x_min=-5.5)
bottomText1=TextMobject("Apply","$f(1)=T_{n}(1)$")
bottomText2=TextMobject("This gives","$a_{ 0 }=e^{-1}$")
bottomText3=TextMobject("Now it could be of","any slope!")
#show graphs of second approx
bottomText4=TextMobject("Hence","apply","$f'(1)=T_{n}'(1)$")
#final graph
bottomText5=TextMobject("This gives","$a_{ 1 }=-2e^{-1}$")
bottomText6=TextMobject("Since the rate of change of this slope","could vary")
#show third approx graphs
bottomText7=TextMobject("Hence also","apply","$f''(1)=T_{ n }''(1)$")
#final graph
bottomText8=TextMobject("This gives","$a_{ 2 }=e^{-1}$")
bottomText1.set_color_by_tex_to_color_map({"Apply":YELLOW})
bottomText2.set_color_by_tex_to_color_map({"$a_{ 0 }=e^{-1}$":BLUE})
bottomText3.set_color_by_tex_to_color_map({"any slope!":YELLOW})
bottomText4.set_color_by_tex_to_color_map({"apply":YELLOW})
bottomText5.set_color_by_tex_to_color_map({"$a_{ 1 }=-2e^{-1}$":BLUE})
bottomText6.set_color_by_tex_to_color_map({"could vary":YELLOW})
bottomText7.set_color_by_tex_to_color_map({"apply":YELLOW})
bottomText8.set_color_by_tex_to_color_map({"$a_{ 2 }=e^{-1}$":BLUE})
bottomText1.scale(0.4)
bottomText2.scale(0.5)
bottomText3.scale(0.4)
bottomText4.scale(0.4)
bottomText5.scale(0.5)
bottomText6.scale(0.4)
bottomText7.scale(0.4)
bottomText8.scale(0.5)
bottomText1.shift(4.5*RIGHT+2.5*DOWN)
bottomText2.shift(4.5*RIGHT+2.5*DOWN)
bottomText3.shift(4.5*RIGHT+2.5*DOWN)
bottomText4.shift(4.5*RIGHT+2.5*DOWN)
bottomText5.shift(4.5*RIGHT+2.5*DOWN)
bottomText6.shift(4.5*RIGHT+2.5*DOWN)
bottomText7.shift(4.5*RIGHT+2.5*DOWN)
bottomText8.shift(4.5*RIGHT+2.5*DOWN)
self.play(Write(bottomText1))
self.wait(1)
self.play(ShowCreation(firstApprox[0]),ReplacementTransform(bottomText1,bottomText2))
#change coeff in tn(x)
self.play(ReplacementTransform(generalized_eq_coeff[0],coeff[0]))
self.wait(1.5)
self.play(ReplacementTransform(bottomText2,bottomText3))
self.wait(0.5)
self.play(ReplacementTransform(firstApprox[0],secondApprox[1]))
self.wait(0.5)
self.play(ReplacementTransform(secondApprox[1],secondApprox[2]))
# self.wait(0.5)
# self.play(ReplacementTransform(secondApprox[2],secondApprox[0]))
self.wait(1)
self.play(ReplacementTransform(bottomText3,bottomText4),FadeOut(secondApprox[2]))
self.wait(1)
self.play(Write(secondGraph),ReplacementTransform(bottomText4,bottomText5))
#change a1 coeff
self.play(ReplacementTransform(generalized_eq_coeff[1],coeff[1]))
self.wait(1.5)
self.play(ReplacementTransform(bottomText5,bottomText6))
self.play(ReplacementTransform(secondGraph,thirdApprox[0]))
self.wait(0.6)
self.play(ReplacementTransform(thirdApprox[0],thirdApprox[1]))
# self.wait(0.6)
# self.play(ReplacementTransform(thirdApprox[1],thirdApprox[2]))
self.wait(0.6)
self.play(ReplacementTransform(thirdApprox[1],thirdApprox[3]))
self.wait(0.6)
self.play(ReplacementTransform(thirdApprox[3],thirdApprox[4]))
self.wait(1.5)
self.play(ReplacementTransform(bottomText6,bottomText7))
self.wait(1.5)
self.play(ReplacementTransform(bottomText7,bottomText8),ReplacementTransform(thirdApprox[4],thirdGraph))
self.play(ReplacementTransform(generalized_eq_coeff[2],coeff[2]))
self.wait(2)
textFinal=TextMobject("And so on..!")
textFinal.scale(0.7)
textFinal.shift(4.5*RIGHT+2.5*DOWN)
self.play(ReplacementTransform(bottomText8,textFinal))
self.wait(2.5)
finalFormula=TextMobject("Hence","$T_{ n }(x)$","=","$f(1)+f'(1)(x-1)+\\frac { f''(1) }{ 2! }(x-1)^2+..+\\frac { { f }^{ n }(1) }{ n! } { (x-1) }^{ n }$")
finalFormula.scale(0.8)
finalFormula.set_color_by_tex_to_color_map({"$T_{ n }(x)$":GREEN,"$f(1)+f'(1)(x-1)+\\frac { f''(1) }{ 2! }(x-1)^2+..+\\frac { { f }^{ n }(1) }{ n! } { (x-1) }^{ n }$":RED})
self.play(FadeOut(self.axes),FadeOut(textFinal),FadeOut(thirdGraph),FadeOut(trTextGrup),FadeOut(mainfunction),FadeOut(fx),FadeOut(coeff[0]),FadeOut(coeff[1]),FadeOut(coeff[2]))
self.play(Write(finalFormula))
self.wait(2)
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