From 87e674818b1035402d3b5dd3a2f55f1df25eb9e4 Mon Sep 17 00:00:00 2001 From: G Sri Harsha Date: Tue, 26 May 2020 14:44:22 +0530 Subject: Rename script2.py to video2_TaylorExpansionGeneralForm.py.py --- .../Taylor Series/script2.py | 195 --------------------- .../video2_TaylorExpansionGeneralForm.py.py | 195 +++++++++++++++++++++ 2 files changed, 195 insertions(+), 195 deletions(-) delete mode 100644 FSF-2020/calculus/series-and-transformations/Taylor Series/script2.py create mode 100644 FSF-2020/calculus/series-and-transformations/Taylor Series/video2_TaylorExpansionGeneralForm.py.py (limited to 'FSF-2020') diff --git a/FSF-2020/calculus/series-and-transformations/Taylor Series/script2.py b/FSF-2020/calculus/series-and-transformations/Taylor Series/script2.py deleted file mode 100644 index b5d0a53..0000000 --- a/FSF-2020/calculus/series-and-transformations/Taylor Series/script2.py +++ /dev/null @@ -1,195 +0,0 @@ -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 }$":RED}) - - self.play(FadeOut(self.axes),FadeOut(textFinal),FadeOut(thirdGraph),FadeOut(trTextGrup),FadeOut(mainfunction),FadeOut(fx),FadeOut(coeff[0]),FadeOut(coeff[1]),FadeOut(coeff[2])) - self.play(Write(finalFormula)) - self.wait(2) \ No newline at end of file diff --git a/FSF-2020/calculus/series-and-transformations/Taylor Series/video2_TaylorExpansionGeneralForm.py.py b/FSF-2020/calculus/series-and-transformations/Taylor Series/video2_TaylorExpansionGeneralForm.py.py new file mode 100644 index 0000000..f84cfe9 --- /dev/null +++ b/FSF-2020/calculus/series-and-transformations/Taylor Series/video2_TaylorExpansionGeneralForm.py.py @@ -0,0 +1,195 @@ +from manimlib.imports import* +import math + + +class intro(Scene): + def construct(self): + equation=TextMobject("$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) -- cgit