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Beginning L'Hopital's example of eoc7
This commit is contained in:
Grant Sanderson
@ -75,7 +75,8 @@ 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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self.lag_factor = max(self.run_time - 1, 2)
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min_lag_factor = min(len(mobject)/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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def establish_run_time(self, mobject):
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330
eoc/chapter7.py
330
eoc/chapter7.py
@ -1118,6 +1118,7 @@ class GraphLimitExpression(GraphScene):
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def get_epsilon_delta_group(
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self,
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delta,
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limit_x = 0,
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dashed_line_stroke_width = 3,
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dashed_line_length = 2*SPACE_HEIGHT,
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input_range_color = YELLOW,
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@ -1127,12 +1128,13 @@ class GraphLimitExpression(GraphScene):
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result = VGroup()
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kwargs.pop("self")
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result.delta = kwargs.pop("delta")
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result.limit_x = kwargs.pop("limit_x")
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result.kwargs = kwargs
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dashed_line = DashedLine(
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ORIGIN, dashed_line_length*RIGHT,
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stroke_width = dashed_line_stroke_width
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)
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x_values = [-delta, delta]
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x_values = [limit_x-delta, limit_x+delta]
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x_axis_points = [self.coords_to_point(x, 0) for x in x_values]
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result.delta_lines = VGroup(*[
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dashed_line.copy().rotate(np.pi/2).move_to(
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@ -1140,12 +1142,11 @@ class GraphLimitExpression(GraphScene):
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)
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for point in x_axis_points
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])
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result.epsilon_lines = VGroup(*[
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dashed_line.copy().move_to(
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self.coords_to_point(0, self.func(x))
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if self.func(limit_x) < 0:
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result.delta_lines.rotate(
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np.pi, RIGHT,
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about_point = result.delta_lines.get_bottom()
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)
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for x in x_values
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])
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basically_zero = 0.00001
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result.input_range, result.output_range = [
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VGroup(*[
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@ -1156,12 +1157,19 @@ class GraphLimitExpression(GraphScene):
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x_max = x_max,
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)
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for x_min, x_max in [
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(-delta, -basically_zero),
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(basically_zero, delta),
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(limit_x-delta, limit_x-basically_zero),
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(limit_x+basically_zero, limit_x+delta),
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]
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]).set_stroke(width = input_range_stroke_width)
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for func in lambda h : 0, self.func
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for func in (lambda h : 0), self.func
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]
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result.epsilon_lines = VGroup(*[
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dashed_line.copy().move_to(
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self.coords_to_point(limit_x, 0)[0]*RIGHT+\
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result.output_range.get_edge_center(vect)[1]*UP
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)
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for vect in DOWN, UP
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])
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result.digest_mobject_attrs()
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return result
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@ -1171,12 +1179,14 @@ class GraphLimitExpression(GraphScene):
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**kwargs
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):
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added_anims = kwargs.get("added_anims", [])
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limit_x = epsilon_delta_group.limit_x
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start_delta = epsilon_delta_group.delta
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ed_group_kwargs = epsilon_delta_group.kwargs
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def update_ed_group(ed_group, alpha):
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delta = interpolate(start_delta, target_delta, alpha)
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new_group = self.get_epsilon_delta_group(
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delta, **ed_group_kwargs
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delta, limit_x = limit_x,
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**ed_group_kwargs
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)
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Transform(ed_group, new_group).update(1)
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return ed_group
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@ -1202,6 +1212,7 @@ class LimitCounterExample(GraphLimitExpression):
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"graph_origin" : DOWN,
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"graph_color" : BLUE,
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"hole_radius" : 0.075,
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"smaller_hole_radius" : 0.04,
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"big_delta" : 1.5,
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"small_delta" : 0.05,
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}
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@ -1323,6 +1334,7 @@ class LimitCounterExample(GraphLimitExpression):
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ed_group, target_delta = self.small_delta,
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run_time = 6
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)
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self.hole_radius = self.smaller_hole_radius
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brace = Brace(self.ed_group.epsilon_lines, RIGHT, buff = SMALL_BUFF)
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brace_text = brace.get_text("Can't get \\\\ smaller", buff = SMALL_BUFF)
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@ -1582,7 +1594,7 @@ class EpsilonDeltaExample(GraphLimitExpression, ZoomedScene):
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result.braces = VGroup(*[
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brace.copy().scale_to_fit_height(
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group.get_height()
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).next_to(group, RIGHT)
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).next_to(group, RIGHT, SMALL_BUFF)
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for i in 1, 2
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for group in [VGroup(lines[0], lines[i])]
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])
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@ -1627,6 +1639,7 @@ class EpsilonDeltaExample(GraphLimitExpression, ZoomedScene):
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class EpsilonDeltaCounterExample(LimitCounterExample, EpsilonDeltaExample):
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def construct(self):
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self.hole_radius = 0.04
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self.add_func()
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self.setup_axes()
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self.draw_graph()
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@ -1668,6 +1681,7 @@ class EpsilonDeltaCounterExample(LimitCounterExample, EpsilonDeltaExample):
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ed_group.output_range,
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run_time = 2
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))
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self.remove(self.graph_holes)
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self.play(*map(GrowFromCenter, ed_group.epsilon_lines))
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self.dither(2)
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self.animate_epsilon_delta_group_change(
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@ -1683,6 +1697,7 @@ class EpsilonDeltaCounterExample(LimitCounterExample, EpsilonDeltaExample):
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vects = [
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self.coords_to_point(0, 2) - self.coords_to_point(0, 1.5),
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self.coords_to_point(0, 1) - self.coords_to_point(0, 2),
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self.coords_to_point(0, 1.5) - self.coords_to_point(0, 1),
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]
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for vect in vects:
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self.play(self.epsilon_group.shift, vect)
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@ -1698,13 +1713,292 @@ class EpsilonDeltaCounterExample(LimitCounterExample, EpsilonDeltaExample):
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rate_func = lambda t : 0.2*there_and_back(2*t%1)
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)
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class TheoryHeavy(TeacherStudentsScene):
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def construct(self):
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lhs = TexMobject(
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"{df", "\\over \\,", "dx}", "(", "x", ")"
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)
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equals = TexMobject("=")
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rhs = TexMobject(
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"\\lim", "_{h", "\\to 0}",
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"{f", "(", "x", "+", "h", ")", "-", "f", "(", "x", ")",
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"\\over \\,", "h}"
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)
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derivative = VGroup(lhs, equals, rhs)
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derivative.arrange_submobjects(RIGHT)
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for tex_mob in derivative:
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tex_mob.highlight_by_tex("x", RED)
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tex_mob.highlight_by_tex("h", GREEN)
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tex_mob.highlight_by_tex("dx", GREEN)
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tex_mob.highlight_by_tex("f", YELLOW)
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derivative.next_to(self.get_pi_creatures(), UP, buff = MED_LARGE_BUFF)
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lim = rhs.get_part_by_tex("lim")
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epsilon_delta = TexMobject("(\\epsilon, \\delta)")
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epsilon_delta.next_to(lim, UP, buff = 1.5*LARGE_BUFF)
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arrow = Arrow(epsilon_delta, lim, color = WHITE)
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self.student_says(
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"Too much theory!",
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target_mode = "angry",
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write_kwargs = {"run_time" : 2},
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)
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self.dither()
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student = self.get_students()[1]
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Scene.play(self,
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Write(lhs),
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FadeOut(student.bubble),
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FadeOut(student.bubble.content),
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*[
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ApplyFunction(
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lambda pi : pi.change_mode("pondering").look_at(epsilon_delta),
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pi
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)
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for pi in self.get_pi_creatures()
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]
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)
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part_tex_pairs = [
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("df", "f"),
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("over", "+"),
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("over", "-"),
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("over", "to"),
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("over", "over"),
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("dx", "h"),
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("(", "("),
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("x", "x"),
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(")", ")"),
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]
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self.play(Write(equals), Write(lim), *[
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ReplacementTransform(
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VGroup(*lhs.get_parts_by_tex(t1)).copy(),
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VGroup(*rhs.get_parts_by_tex(t2)),
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run_time = 2,
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rate_func = squish_rate_func(smooth, alpha, alpha+0.5)
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)
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for (t1, t2), alpha in zip(
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part_tex_pairs,
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np.linspace(0, 0.5, len(part_tex_pairs))
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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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Write(epsilon_delta),
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ShowCreation(arrow)
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)
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self.dither(3)
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derivative.add(epsilon_delta, arrow)
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self.student_says(
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"How do you \\\\ compute limits?",
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student_index = 2,
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added_anims = [
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derivative.scale, 0.8,
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derivative.to_corner, UP+LEFT
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]
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)
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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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CONFIG = {
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"graph_origin" : ORIGIN,
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"x_axis_width" : 2*SPACE_WIDTH,
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"x_min" : -5,
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"x_max" : 5,
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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_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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}
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def construct(self):
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self.force_skipping()
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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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self.x_axis_label_mob.highlight(self.x_color)
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def introduce_function(self):
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graph = self.get_graph(self.func)
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colored_graph = graph.copy().highlight(YELLOW)
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func_label = self.get_func_label()
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func_label.next_to(ORIGIN, RIGHT, buff = LARGE_BUFF)
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func_label.to_edge(UP)
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x_copy = self.x_axis_label_mob.copy()
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self.play(
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Write(func_label),
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Transform(
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x_copy, VGroup(*func_label.get_parts_by_tex("x")),
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remover = True
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)
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)
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self.play(ShowCreation(
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graph,
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run_time = 3,
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rate_func = None
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))
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self.dither(4) ## Overly oscillation
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self.play(ShowCreation(colored_graph, run_time = 2))
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self.dither()
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self.play(ShowCreation(graph, run_time = 2))
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self.remove(colored_graph)
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self.dither()
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self.graph = graph
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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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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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)
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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.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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lim = TexMobject("\\lim", "_{x", "\\to 1}")
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lim.highlight_by_tex("x", self.x_color)
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lim.move_to(self.func_label, LEFT)
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self.func_label.generate_target()
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self.func_label.target.next_to(lim, RIGHT)
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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(PiCreatureSays(morty, words))
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self.play(
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Blink(morty),
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ShowCreation(v_line)
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)
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self.play(
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RemovePiCreatureBubble(
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morty, target_mode = "pondering",
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look_at_arg = func_1
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),
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ReplacementTransform(
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self.func_label.copy(),
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func_1
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)
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)
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self.dither(2)
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self.play(Write(VGroup(*rhs[:-1])))
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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.dither()
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self.play(GrowFromCenter(ed_group.input_range))
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self.play(
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ReplacementTransform(
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ed_group.input_range.copy(),
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ed_group.output_range
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),
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*map(ShowCreation, ed_group.delta_lines)
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)
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self.play(*map(GrowFromCenter, ed_group.epsilon_lines))
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self.play(morty.change_mode, "thinking")
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self.animate_epsilon_delta_group_change(
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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.play(
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Write(lim),
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MoveToTarget(self.func_label),
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Write(equals_q),
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morty.change_mode, "confused",
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morty.look_at, lim
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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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*map(FadeOut, [morty, ed_group])
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)
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self.dither()
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|
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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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def ask_about_systematic_process(self):
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pass
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def mention_derivatives_as_helpful(self):
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pass
|
||||
|
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def nudge_by_dx(self):
|
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pass
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def show_altered_numerator_and_denominator(self):
|
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pass
|
||||
|
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##
|
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|
||||
def func(self, x):
|
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if abs(x) != 1:
|
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return np.sin(x*np.pi) / (x**2 - 1)
|
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else:
|
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return np.pi*np.cos(x*np.pi) / (2*x)
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||||
|
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def get_func_label(self, argument = "x"):
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in_tex = "{%s}"%str(argument)
|
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result = TexMobject(
|
||||
"{\\sin(\\pi ", in_tex, ") \\over ",
|
||||
in_tex, "^2 - 1}"
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||||
)
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||||
result.highlight_by_tex(in_tex, self.x_color)
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return result
|
||||
|
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def get_epsilon_delta_group(self, delta, **kwargs):
|
||||
ed_group = GraphLimitExpression.get_epsilon_delta_group(self, delta, **kwargs)
|
||||
color = ed_group.kwargs["input_range_color"]
|
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radius = min(delta/2, self.hole_radius)
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||||
pairs = [
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# (ed_group.input_range[0], (1, 0)),
|
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(ed_group.input_range[1], (1, 0)),
|
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# (ed_group.output_range[0], (1, self.func(1))),
|
||||
(ed_group.output_range[1], (1, self.func(1))),
|
||||
]
|
||||
for mob, coords in pairs:
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mob.add(self.get_hole(
|
||||
*coords,
|
||||
color = color,
|
||||
radius = radius
|
||||
))
|
||||
return ed_group
|
||||
|
||||
|
||||
|
||||
|
||||
@ -200,7 +200,7 @@ class VMobject(Mobject):
|
||||
but will be tracked in a separate special list for when
|
||||
it comes time to display.
|
||||
"""
|
||||
subpath_mobject = self.copy()##Really helps to be of the same class
|
||||
subpath_mobject = self.copy() ##Really helps to be of the same class
|
||||
subpath_mobject.is_subpath = True
|
||||
subpath_mobject.set_points(points)
|
||||
self.add(subpath_mobject)
|
||||
|
||||
@ -89,9 +89,9 @@ class PiCreature(SVGMobject):
|
||||
color = self.color
|
||||
)
|
||||
new_self.scale_to_fit_height(self.get_height())
|
||||
new_self.shift(self.eyes.get_center() - new_self.eyes.get_center())
|
||||
if self.is_flipped() ^ new_self.is_flipped():
|
||||
new_self.flip()
|
||||
new_self.shift(self.eyes.get_center() - new_self.eyes.get_center())
|
||||
if hasattr(self, "purposeful_looking_direction"):
|
||||
new_self.look(self.purposeful_looking_direction)
|
||||
Transform(self, new_self).update(1)
|
||||
|
||||
Reference in New Issue
Block a user