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eoc7 just finished LHopitalExample, but might redo a lot
This commit is contained in:
Grant Sanderson
@ -75,7 +75,7 @@ class Write(ShowCreation):
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if "run_time" not in kwargs:
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self.establish_run_time(mobject)
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if "lag_factor" not in kwargs:
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min_lag_factor = min(len(mobject)/2.0, 2)
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min_lag_factor = min(len(mobject.family_members_with_points())/2.0, 2)
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self.lag_factor = max(self.run_time - 1, min_lag_factor)
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ShowCreation.__init__(self, mobject, **kwargs)
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@ -185,6 +185,9 @@ class ApplyWave(Homotopy):
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"direction" : DOWN,
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"amplitude" : 0.2,
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"run_time" : 1,
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"apply_function_kwargs" : {
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"maintain_smoothness" : False,
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},
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}
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def __init__(self, mobject, **kwargs):
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digest_config(self, kwargs, locals())
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402
eoc/chapter7.py
402
eoc/chapter7.py
@ -763,7 +763,7 @@ class OtherViewsOfDx(TeacherStudentsScene):
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])
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)
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self.dither(3)
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class GraphLimitExpression(GraphScene):
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CONFIG = {
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"start_x" : 2,
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@ -1799,7 +1799,7 @@ class TheoryHeavy(TeacherStudentsScene):
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self.play(self.get_teacher().change_mode, "happy")
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self.dither(2)
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class LHopitalExample(LimitCounterExample):
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class LHopitalExample(LimitCounterExample, PiCreatureScene):
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CONFIG = {
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"graph_origin" : ORIGIN,
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"x_axis_width" : 2*SPACE_WIDTH,
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@ -1808,26 +1808,27 @@ class LHopitalExample(LimitCounterExample):
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"x_labeled_nums" : range(-6, 8, 2),
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"x_axis_label" : "$x$",
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"y_axis_height" : 2*SPACE_HEIGHT,
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"y_min" : -3,
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"y_max" : 3,
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"y_min" : -3.1,
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"y_max" : 3.1,
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"y_bottom_tick" : -4,
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"y_labeled_nums" : range(-2, 4, 2),
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"y_axis_label" : "",
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"x_color" : RED,
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"hole_radius" : 0.07,
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"big_delta" : 0.5,
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"small_delta" : 0.01,
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"dx" : 0.2,
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"dx_color" : YELLOW,
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"tex_scale_value" : 0.9,
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}
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def construct(self):
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self.force_skipping()
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self.remove(*self.get_pi_creatures())
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self.setup_axes()
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self.introduce_function()
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self.show_non_definedness_at_one()
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self.plug_in_value_close_to_one()
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self.ask_about_systematic_process()
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self.mention_derivatives_as_helpful()
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self.nudge_by_dx()
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self.show_altered_numerator_and_denominator()
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def setup_axes(self):
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GraphScene.setup_axes(self)
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@ -1865,26 +1866,32 @@ class LHopitalExample(LimitCounterExample):
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self.func_label = func_label
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def show_non_definedness_at_one(self):
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morty = Mortimer().flip().to_corner(DOWN+LEFT)
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morty = self.get_primary_pi_creature()
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words = TexMobject("\\text{Try }", "x", "=1")
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words.highlight_by_tex("x", self.x_color, substring = False)
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v_line = self.get_vertical_line_to_graph(
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1, self.graph,
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line_class = DashedLine,
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color = self.x_color
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v_line, alt_v_line = [
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self.get_vertical_line_to_graph(
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x, self.graph,
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line_class = DashedLine,
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color = self.x_color
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)
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for x in 1, -1
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]
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hole, alt_hole = [
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self.get_hole(x, self.func(x))
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for x in 1, -1
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]
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ed_group = self.get_epsilon_delta_group(
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self.big_delta, limit_x = 1,
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)
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func_1 = self.get_func_label("1")
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func_1.next_to(self.func_label, DOWN, buff = MED_LARGE_BUFF)
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rhs = TexMobject("= \\frac{0}{0}")
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rhs = TexMobject("\\Rightarrow \\frac{0}{0}")
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rhs.next_to(func_1, RIGHT)
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func_1_group = VGroup(func_1, rhs)
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hole = self.get_hole(1, self.func(1))
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ed_group = self.get_epsilon_delta_group(
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self.big_delta, limit_x = 1,
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)
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func_1_group.add_to_back(BackgroundRectangle(func_1_group))
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lim = TexMobject("\\lim", "_{x", "\\to 1}")
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lim.highlight_by_tex("x", self.x_color)
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@ -1894,6 +1901,7 @@ class LHopitalExample(LimitCounterExample):
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equals_q = TexMobject("= ???")
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equals_q.next_to(self.func_label.target, RIGHT)
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self.play(FadeIn(morty))
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self.play(PiCreatureSays(morty, words))
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self.play(
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Blink(morty),
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@ -1914,10 +1922,15 @@ class LHopitalExample(LimitCounterExample):
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self.dither()
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self.play(Write(rhs[-1]))
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self.dither()
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self.play(
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GrowFromCenter(hole),
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morty.look_at, hole
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)
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self.play(GrowFromCenter(hole))
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self.dither()
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self.play(ShowCreation(alt_v_line))
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self.play(GrowFromCenter(alt_hole))
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self.dither()
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alt_group = VGroup(alt_v_line, alt_hole)
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self.play(alt_group.set_stroke, GREY, 2)
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self.play(FocusOn(hole))
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self.dither()
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self.play(GrowFromCenter(ed_group.input_range))
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@ -1934,7 +1947,7 @@ class LHopitalExample(LimitCounterExample):
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ed_group, target_delta = self.small_delta,
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run_time = 4
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)
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self.play(Blink(morty))
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self.dither()
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self.play(
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Write(lim),
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MoveToTarget(self.func_label),
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@ -1944,29 +1957,295 @@ class LHopitalExample(LimitCounterExample):
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)
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self.dither(2)
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self.play(
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func_1_group.to_corner, DOWN+RIGHT,
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func_1_group.to_corner, UP+LEFT,
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*map(FadeOut, [morty, ed_group])
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)
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self.dither()
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self.lim_group = VGroup(lim, self.func_label, equals_q)
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for part in self.lim_group:
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part.add_to_back(BackgroundRectangle(part))
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self.func_1_group = func_1_group
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self.v_line = v_line
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def plug_in_value_close_to_one(self):
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self.revert_to_original_skipping_status()
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pass
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num = 1.00001
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result = self.func(num)
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label = self.get_func_label(num)
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rhs = TexMobject("= %.4f\\dots"%result)
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rhs.next_to(label, RIGHT)
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approx_group = VGroup(label, rhs)
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approx_group.scale_to_fit_width(SPACE_WIDTH-2*MED_LARGE_BUFF)
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approx_group.next_to(ORIGIN, UP, buff = MED_LARGE_BUFF)
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approx_group.to_edge(RIGHT)
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self.play(ReplacementTransform(
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self.func_label.copy(),
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label
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))
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self.dither()
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self.play(Write(rhs))
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self.dither()
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self.approx_group = approx_group
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def ask_about_systematic_process(self):
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pass
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morty = self.pi_creature
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morty.change_mode("plain")
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def mention_derivatives_as_helpful(self):
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pass
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self.func_1_group.save_state()
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self.play(
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FadeIn(morty),
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FadeOut(VGroup(
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*self.x_axis.numbers[:len(self.x_axis.numbers)/2]
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))
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)
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self.pi_creature_says(
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morty, "Is there a \\\\ better way?",
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bubble_kwargs = {
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"height" : 3,
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"width" : 4,
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},
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)
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self.dither(2)
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self.play(
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RemovePiCreatureBubble(
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morty, target_mode = "raise_left_hand",
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look_at_arg = self.func_1_group
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),
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self.func_1_group.scale, self.tex_scale_value,
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self.func_1_group.move_to,
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morty.get_corner(UP+LEFT), DOWN+LEFT,
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self.func_1_group.shift, MED_LARGE_BUFF*UP,
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)
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self.dither(2)
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self.play(
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morty.change_mode, "raise_right_hand",
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morty.look, UP+RIGHT,
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FadeOut(self.approx_group),
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self.func_1_group.restore,
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self.lim_group.next_to,
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morty.get_corner(UP+RIGHT), RIGHT,
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)
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self.dither(2)
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self.play(
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self.lim_group.scale, self.tex_scale_value,
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self.lim_group.next_to, ORIGIN, UP,
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self.lim_group.to_edge, LEFT,
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morty.change_mode, "plain"
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)
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self.dither()
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self.play(FadeOut(morty))
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def nudge_by_dx(self):
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pass
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nudged_v_line = self.get_vertical_line_to_graph(
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x = 1 + self.dx,
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graph = self.graph,
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line_class = DashedLine,
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color = RED
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)
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brace = Brace(
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VGroup(self.v_line, nudged_v_line),
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direction = UP,
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min_num_quads = 1,
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)
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dx_label = brace.get_text("$dx$")
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dx_label.highlight(self.dx_color)
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def show_altered_numerator_and_denominator(self):
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pass
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rect, func_1, to_zero_over_zero = self.func_1_group
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to, zero1, over, zero2 = to_zero_over_zero
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self.remove(rect)
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func_1_plus_dx = self.get_func_label(
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"(1 + dx)"
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)
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lhs_ones = []
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for part in func_1_plus_dx.get_parts_by_tex("(1 + dx)"):
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VGroup(part[0], part[2], part[5]).highlight(WHITE)
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VGroup(*part[3:5]).highlight(self.dx_color)
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lhs_ones.append(part[1])
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func_1_plus_dx.move_to(func_1, LEFT)
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nudged_rhs = TexMobject(
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"{0", "+ d\\big(\\sin(\\pi ", "x", ")\\big)",
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"\\over \\,",
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"0", "+ d\\big(", "x", "^2 - 1\\big)}"
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)
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d_sine = VGroup(*nudged_rhs[1:4])
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d_denom = VGroup(*nudged_rhs[-3:])
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nudged_rhs.highlight_by_tex("x", self.x_color)
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nudged_rhs.scale(self.tex_scale_value)
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nudged_rhs.next_to(
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func_1_plus_dx, RIGHT,
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buff = to.get_width()+MED_LARGE_BUFF
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)
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nudged_rhs_rect = BackgroundRectangle(
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nudged_rhs, fill_opacity = 1
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)
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approx = TexMobject("\\approx")
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approx.next_to(nudged_rhs, RIGHT)
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d_over_d = TexMobject(
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"{\\cos(\\pi", "x", ")", "\\pi", "\\, dx",
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"\\over\\,",
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"2\\,", "x", "\\, dx}"
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)
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d_over_d.scale(self.tex_scale_value)
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cos_dx = VGroup(*d_over_d[:5])
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two_x_dx = VGroup(*d_over_d[-3:])
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d_over_d.highlight_by_tex("x", self.x_color)
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d_over_d.highlight_by_tex("dx", self.dx_color)
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d_over_d.next_to(approx, RIGHT)
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rhs_ones = []
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rhs_xs = d_over_d.get_parts_by_tex("x", substring = False)
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for x in rhs_xs:
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one = TexMobject("\\cdot \\! 1")
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one.highlight(self.x_color)
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one.scale(self.tex_scale_value)
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one.move_to(x, DOWN+LEFT)
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rhs_ones.append(one)
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final_rhs = TexMobject("=", "\\frac{-\\pi}{2}")
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final_rhs.scale(self.tex_scale_value)
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final_rhs.next_to(d_over_d, RIGHT)
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def replace_x_by_one(index):
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self.play(FocusOn(lhs_ones[index]))
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self.play(
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Transform(
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lhs_ones[index].copy(),
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rhs_ones[index],
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path_arc = np.pi/2,
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run_time = 2,
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remover = True
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),
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Transform(
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rhs_xs[index],
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rhs_ones[index],
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run_time = 2,
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rate_func = squish_rate_func(smooth, 0.5, 1)
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)
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)
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def get_brace_shrink_anims():
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return [
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Transform(nudged_v_line, self.v_line),
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brace.stretch_to_fit_width, 0.01,
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brace.next_to, self.v_line, UP, SMALL_BUFF,
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MaintainPositionRelativeTo(dx_label, brace),
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]
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self.play(Indicate(self.v_line))
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self.play(ReplacementTransform(
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self.v_line.copy(),
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nudged_v_line
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))
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self.play(GrowFromCenter(brace))
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self.play(Write(dx_label))
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self.dither()
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self.play(FocusOn(func_1))
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self.play(Indicate(func_1))
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self.play(
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ReplacementTransform(func_1, func_1_plus_dx),
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to_zero_over_zero.next_to, func_1_plus_dx, RIGHT
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)
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self.dither()
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self.play(
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FadeIn(nudged_rhs_rect),
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ReplacementTransform(
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over, nudged_rhs.get_part_by_tex("over")
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||||
),
|
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Write(d_sine),
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*[
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ReplacementTransform(VGroup(zero), part)
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for zero, part in zip(
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[zero1, zero2],
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||||
nudged_rhs.get_parts_by_tex("0")
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||||
)
|
||||
]
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||||
)
|
||||
self.dither()
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self.play(Write(d_denom))
|
||||
self.dither()
|
||||
|
||||
#Bring in d_over_d
|
||||
self.play(
|
||||
Write(approx),
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||||
ReplacementTransform(
|
||||
nudged_rhs.get_part_by_tex("over").copy(),
|
||||
d_over_d.get_part_by_tex("over")
|
||||
),
|
||||
ReplacementTransform(d_sine.copy(), cos_dx)
|
||||
)
|
||||
replace_x_by_one(0)
|
||||
self.dither(2)
|
||||
|
||||
##Emphasize meaning of approximation
|
||||
cosine = VGroup(*cos_dx[:-1])
|
||||
d_sine_ghost, cosine_ghost = [
|
||||
mob.copy().set_fill(opacity = 0.2)
|
||||
for mob in d_sine, cosine
|
||||
]
|
||||
self.play(Indicate(d_sine))
|
||||
self.play(Transform(
|
||||
d_sine_ghost, cosine_ghost,
|
||||
run_time = 2,
|
||||
remover = True,
|
||||
))
|
||||
self.play(Indicate(cosine))
|
||||
self.dither()
|
||||
self.play(*[
|
||||
ApplyWave(mob, direction = UP, amplitude = 0.1)
|
||||
for mob in [
|
||||
d_over_d.get_part_by_tex("dx"),
|
||||
dx_label
|
||||
]
|
||||
])
|
||||
self.play(
|
||||
*get_brace_shrink_anims(),
|
||||
rate_func = there_and_back,
|
||||
run_time = 5
|
||||
)
|
||||
self.dither()
|
||||
|
||||
self.play(ReplacementTransform(d_denom.copy(), two_x_dx))
|
||||
self.dither()
|
||||
replace_x_by_one(1)
|
||||
self.dither()
|
||||
|
||||
#Cancel out dx's
|
||||
dxs = VGroup(*d_over_d.get_parts_by_tex("dx"))
|
||||
circles = VGroup(*[
|
||||
Circle(color = GREEN).replace(dx).scale_in_place(1.3)
|
||||
for dx in dxs
|
||||
])
|
||||
|
||||
self.play(ShowCreation(circles, run_time = 2))
|
||||
self.dither(2)
|
||||
self.play(*map(FadeOut, [circles, dxs]))
|
||||
self.dither(2)
|
||||
|
||||
#Write final answer
|
||||
final_ratio = final_rhs.get_part_by_tex("pi").copy()
|
||||
final_ratio.add_to_back(BackgroundRectangle(final_ratio))
|
||||
left_brace = Brace(self.v_line, LEFT)
|
||||
self.play(Write(final_rhs))
|
||||
self.play(
|
||||
*get_brace_shrink_anims(),
|
||||
run_time = 3
|
||||
)
|
||||
self.play(GrowFromCenter(left_brace))
|
||||
self.play(
|
||||
final_ratio.next_to, left_brace, LEFT, SMALL_BUFF
|
||||
)
|
||||
self.dither(2)
|
||||
|
||||
##
|
||||
|
||||
def create_pi_creature(self):
|
||||
self.pi_creature = Mortimer().flip().to_corner(DOWN+LEFT)
|
||||
return self.pi_creature
|
||||
|
||||
def func(self, x):
|
||||
if abs(x) != 1:
|
||||
return np.sin(x*np.pi) / (x**2 - 1)
|
||||
@ -2000,9 +2279,58 @@ class LHopitalExample(LimitCounterExample):
|
||||
))
|
||||
return ed_group
|
||||
|
||||
|
||||
|
||||
|
||||
class DerivativeLimitReciprocity(Scene):
|
||||
def construct(self):
|
||||
arrow = Arrow(LEFT, RIGHT, color = WHITE)
|
||||
lim = TexMobject("\\lim", "_{h", "\\to 0}")
|
||||
lim.highlight_by_tex("h", GREEN)
|
||||
lim.next_to(arrow, LEFT)
|
||||
deriv = TexMobject("{df", "\\over\\,", "dx}")
|
||||
deriv.highlight_by_tex("dx", GREEN)
|
||||
deriv.highlight_by_tex("df", YELLOW)
|
||||
deriv.next_to(arrow, RIGHT)
|
||||
|
||||
self.play(FadeIn(lim, submobject_mode = "lagged_start"))
|
||||
self.play(ShowCreation(arrow))
|
||||
self.play(FadeIn(deriv, submobject_mode = "lagged_start"))
|
||||
self.dither()
|
||||
self.play(Rotate(arrow, np.pi, run_time = 2))
|
||||
self.dither()
|
||||
|
||||
class NameLHopitalsRule(Scene):
|
||||
def construct(self):
|
||||
condition = TexMobject(
|
||||
"{f", "(", "a", ")",
|
||||
"\\over\\,",
|
||||
"g", "(", "a", ")}",
|
||||
"\\Rightarrow",
|
||||
"{\\,0\\,", "\\over\\,", "0\\,}"
|
||||
)
|
||||
condition.to_edge(UP)
|
||||
lim = TexMobject(
|
||||
"\\lim_", "{x", " \\to", "a}",
|
||||
"{f", "(", "x", ")",
|
||||
"\\over\\,",
|
||||
"g", "(", "x", ")}",
|
||||
)
|
||||
rhs = TexMobject(
|
||||
"=",
|
||||
"{ \\dfrac{df}{dx}", "(", "a", ")"
|
||||
"\\over\\,",
|
||||
"\\dfrac{dg}{dx}", "(", "a", ")}"
|
||||
)
|
||||
rhs.next_to(lim, RIGHT)
|
||||
rule = VGroup(lim, rhs)
|
||||
rule.center()
|
||||
for tex_mob in condition, lim, rhs:
|
||||
tex_mob.highlight_by_tex("f", BLUE)
|
||||
tex_mob.highlight_by_tex("g", GREEN)
|
||||
tex_mob.highlight_by_tex("a", RED, substring = False)
|
||||
tex_mob.highlight_by_tex("a}", RED)
|
||||
tex_mob.highlight_by_tex("arrow", WHITE)
|
||||
|
||||
|
||||
self.add(condition, rule)
|
||||
|
||||
|
||||
|
||||
|
||||
Reference in New Issue
Block a user