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1 files changed, 239 insertions, 0 deletions

diff --git a/FSF-2020/calculus/series-and-transformations/Taylor Series/gifs/video2_TaylorExpansionGeneralForm.py b/FSF-2020/calculus/series-and-transformations/Taylor Series/gifs/video2_TaylorExpansionGeneralForm.py
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+++ b/FSF-2020/calculus/series-and-transformations/Taylor Series/gifs/video2_TaylorExpansionGeneralForm.py
@@ -0,0 +1,239 @@
+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("about $x=1$")
+        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,MovingCameraScene):
+    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 setup(self):
+        GraphScene.setup(self)
+        MovingCameraScene.setup(self)
+    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)   
+
+        equation=TextMobject("$f(x)=$","${ e }^{ -x^{ 2 } }$")
+        equation.scale(0.55)
+        equation.set_color_by_tex_to_color_map({"${ e }^{ -x^{ 2 } }$":RED})
+        text=TextMobject("about $x=1$")
+        text.scale(0.55)
+        equation.shift(3.39*UP+5*LEFT)
+        text.shift(3*UP+5*LEFT)
+
+        self.add(equation)
+        self.add(text)
+
+
+        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,scalee=1)
+        
+        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! } $")]
+        coeff[0].shift(4.1*y_each_unit*UP+5.15*RIGHT*x_each_unit)
+        coeff[0].scale(0.3)
+        coeff[1].shift(4*y_each_unit*UP+5.7*RIGHT*x_each_unit)
+        coeff[1].scale(0.2)
+        coeff[2].shift(4*y_each_unit*UP+7.3*RIGHT*x_each_unit)
+        coeff[2].scale(0.18)
+
+        for obj in coeff:
+            obj.set_color(GOLD_A)
+
+        firstApprox=[self.get_graph(lambda x:math.exp(-1),color=BLUE,x_min=-3,x_max=4)]
+        secondApprox=[self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1),color=BLUE,x_min=-3,x_max=4),
+                    self.get_graph(lambda x:math.exp(-1)+3*(x-1)*math.exp(-1),color=BLUE,x_min=-3,x_max=4),
+                    self.get_graph(lambda x:math.exp(-1)-4*(x-1)*math.exp(-1),color=BLUE,x_min=-3,x_max=4)]
+        thirdApprox=[self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1)-2*math.exp(-1)*(x-1)**2,color=BLUE,x_max=4,x_min=-3),
+                    self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1)-0.1*math.exp(-1)*(x-1)**2,color=BLUE,x_max=4,x_min=-3),
+                    self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1),color=BLUE,x_max=4,x_min=-3),
+                    self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1)+0.5*math.exp(-1)*(x-1)**2,color=BLUE,x_max=4,x_min=-3),
+                    self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1)+2*math.exp(-1)*(x-1)**2,color=BLUE,x_max=4,x_min=-3)]
+                    
+        firstGraph=self.get_graph(lambda x:math.exp(-1),color=BLUE,x_min=-3,x_max=4)
+        secondGraph=self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1),color=BLUE,x_min=-3,x_max=4)
+        thirdGraph=self.get_graph(lambda x:math.exp(-1)-2*(x-1)*math.exp(-1)+math.exp(-1)*(x-1)**2,color=BLUE,x_max=4,x_min=-3)
+
+        bottomText1=TextMobject("Apply","$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)
+        bottomText2.shift(5*RIGHT*x_each_unit+2.5*DOWN*y_each_unit)
+        bottomText3.shift(5*RIGHT*x_each_unit+2.5*DOWN*y_each_unit)
+        bottomText4.shift(5*RIGHT*x_each_unit+2.5*DOWN*y_each_unit)
+        bottomText5.shift(5*RIGHT*x_each_unit+2.5*DOWN*y_each_unit)
+        bottomText6.shift(5.7*RIGHT*x_each_unit+2.5*DOWN*y_each_unit)
+        bottomText7.shift(5.7*RIGHT*x_each_unit+2.5*DOWN*y_each_unit)
+        bottomText8.shift(5.7*RIGHT*x_each_unit+2.5*DOWN*y_each_unit)
+
+        bottomText2.scale(0.7)
+        bottomText3.scale(0.7)
+        bottomText4.scale(0.7)
+        bottomText5.scale(0.7)
+        bottomText6.scale(0.7)
+        bottomText7.scale(0.7)
+        bottomText8.scale(0.7)
+
+        self.play(Write(bottomText1))
+        self.wait(0.8)
+        self.camera_frame.save_state()
+        self.play(self.camera_frame.set_width, 8,
+                self.camera_frame.move_to, x_each_unit*UP+x_each_unit*2*RIGHT,
+                ApplyMethod(trTextGrup.move_to,4*y_each_unit*UP+6.1*RIGHT*x_each_unit),
+                ApplyMethod(bottomText1.move_to,5.4*RIGHT*x_each_unit+2.5*DOWN*y_each_unit),
+                ApplyMethod(equation.shift,1.39*DOWN+2*RIGHT+RIGHT*x_each_unit*2),
+                ApplyMethod(text.shift,1.39*DOWN+2*RIGHT+RIGHT*x_each_unit*2),)
+        self.play(ApplyMethod(text.scale,0.5),ApplyMethod(equation.scale,0.5),ApplyMethod(bottomText1.scale,0.6),ApplyMethod(trTextGrup.scale,0.7))
+        self.play(ApplyMethod(text.shift,0.25*UP))
+        self.wait(0.6)
+
+        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(0.35)
+        textFinal.shift(5.7*RIGHT*x_each_unit+2.5*DOWN*y_each_unit)
+        self.play(ReplacementTransform(bottomText8,textFinal))
+        self.wait(1)
+        self.play(FadeOut(equation),FadeOut(text))
+        self.play(self.camera_frame.set_width, 15,
+                    self.camera_frame.move_to, 0)
+
+        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)
\ No newline at end of file