add topic-name folder

1 files changed, 198 insertions, 0 deletions

diff --git a/FSF-2020/calculus/series-and-transformations/Taylor Series/script1.py b/FSF-2020/calculus/series-and-transformations/Taylor Series/script1.py
new file mode 100644
index 0000000..e83eff8
--- /dev/null
+++ b/FSF-2020/calculus/series-and-transformations/Taylor Series/script1.py
@@ -0,0 +1,198 @@
+from manimlib.imports import*
+import math
+
+def formFormula(coeff_list,variable_list):
+    coeff_list=[TextMobject("${ a }_{ 0 }$"),TextMobject("${ a }_{ 1 }$"),TextMobject("${ a }_{ 2 }$")]
+    variable_list=[TextMobject("+"),TextMobject("${ x }$+"),TextMobject("${ x }^{ 2 }$")]
+    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 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("let $a=0$")
+        text.scale(0.7)
+        text.shift(DOWN)
+
+        self.play(Write(equation))
+        self.wait(0.5)
+        self.play(FadeIn(text))
+        self.wait(0.7)
+        self.play(FadeOut(equation),FadeOut(text))
+
+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("$1$"),TextMobject("$f'(x)$"),TextMobject("$\\frac { f''(x) }{ 2! } $")]
+        coeff[0].shift(3.39*UP+4.88*RIGHT)
+        coeff[0].scale(0.5)
+        coeff[1].shift(3.39*UP+5.3*RIGHT)
+        coeff[1].scale(0.275)
+        coeff[2].shift(3.39*UP+5.98*RIGHT)
+        coeff[2].scale(0.28)
+
+        for obj in coeff:
+            obj.set_color(GOLD_A)
+
+        firstApprox=[self.get_graph(lambda x:1,color=BLUE)]
+        secondApprox=[self.get_graph(lambda x:1,color=BLUE),
+                    self.get_graph(lambda x:x+1,color=BLUE),
+                    self.get_graph(lambda x:-x+1,color=BLUE)]
+        thirdApprox=[self.get_graph(lambda x:1-2*math.pow(x,2),color=BLUE),
+                    self.get_graph(lambda x:1-0.1*math.pow(x,2),color=BLUE),
+                    self.get_graph(lambda x:1,color=BLUE),
+                    self.get_graph(lambda x:1+0.1*math.pow(x,2),color=BLUE),
+                    self.get_graph(lambda x:1+math.pow(x,2),color=BLUE)]
+                    
+        firstGraph=self.get_graph(lambda x:1,color=BLUE)
+        secondGraph=self.get_graph(lambda x:1-math.pow(x,2),color=BLUE)
+
+        bottomText1=TextMobject("The polynomial should","satisfy","the function at $x=0$")
+        bottomText2=TextMobject("This gives","$a_{ 0 }=1$")
+        bottomText3=TextMobject("Now it could be of","any slope!")
+        #show graphs of second approx
+        bottomText4=TextMobject("Hence the","slopes","should","even match")
+        #final graph
+        bottomText5=TextMobject("This gives","$a_{ 1 }=0$")
+        bottomText6=TextMobject("Since the rate of change of this slope","could vary")
+        #show third approx graphs
+        bottomText7=TextMobject("Hence the","rate of change of these slopes","should also be","same!")
+        #final graph
+        bottomText8=TextMobject("This gives","$a_{ 2 }=-1$")
+
+        bottomText1.set_color_by_tex_to_color_map({"satisfy":YELLOW})
+        bottomText2.set_color_by_tex_to_color_map({"$a_{ 0 }=1$":BLUE})
+        bottomText3.set_color_by_tex_to_color_map({"any slope!":YELLOW})
+        bottomText4.set_color_by_tex_to_color_map({"slopes":BLUE,"even match":YELLOW})
+        bottomText5.set_color_by_tex_to_color_map({"$a_{ 1 }=0$":BLUE})
+        bottomText6.set_color_by_tex_to_color_map({"could vary":YELLOW})
+        bottomText7.set_color_by_tex_to_color_map({"rate of change of these slopes":BLUE,"same!":YELLOW})
+        bottomText8.set_color_by_tex_to_color_map({"$a_{ 2 }=-1$":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[0]))
+        self.wait(0.5)
+        self.play(ReplacementTransform(secondApprox[0],secondApprox[2]))
+        self.wait(1)
+        self.play(ReplacementTransform(bottomText3,bottomText4),FadeOut(secondApprox[2]))
+        self.wait(1)
+        self.play(Write(firstGraph),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(firstGraph,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[2],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],secondGraph))
+        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","$T_{ n }(x)$","=","$f(0)+f'(0)x+\\frac { f''(0) }{ 2! }x^2+..+\\frac { { f }^{ n }(0) }{ n! } { x }^{ n }$")
+        finalFormula.scale(0.8)
+        finalFormula.set_color_by_tex_to_color_map({"$T_{ n }(x)$":GREEN,"$f(0)+f'(0)x+\\frac { f''(0) }{ 2! }x^2+..+\\frac { { f }^{ n }(0) }{ n! } { x }^{ n }$":RED})        
+
+        self.play(FadeOut(self.axes),FadeOut(textFinal),FadeOut(secondGraph),FadeOut(trTextGrup),FadeOut(mainfunction),FadeOut(fx),FadeOut(coeff[0]),FadeOut(coeff[1]),FadeOut(coeff[2]))
+        self.play(Write(finalFormula))
+        self.wait(2)
+        # self.play(ReplacementTransform(secondApprox[2],secondApprox[3]))
+        # self.wait(0.5)
+        # self.play(ReplacementTransform(secondApprox[3],secondApprox[4]))
+        # self.wait(0.5)
+        # self.play(ReplacementTransform(secondApprox[4],secondApprox[5]))
+        # self.wait(0.5)
+        # self.play(ReplacementTransform(secondApprox[0],secondApprox[0]))
+        # self.wait(0.5)
+        # self.play(ReplacementTransform(secondApprox[0],secondApprox[0]))
+        # self.wait(0.5)
+
+
+
+