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Up to SecantLineToTangentLine in eoc/chapter2
This commit is contained in:
Grant Sanderson
parent
91ad99eff4
commit
978a52e60f
2 changed files with 369 additions and 5 deletions
368
eoc/chapter2.py
368
eoc/chapter2.py
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@ -398,7 +398,7 @@ class GraphCarTrajectory(GraphScene):
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}
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def construct(self):
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self.setup_axes(animate = False)
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graph = self.graph_function(lambda t : 100*smooth(t/10.))
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graph = self.graph_sigmoid_trajectory_function()
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origin = self.coords_to_point(0, 0)
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self.introduce_graph(graph, origin)
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@ -406,6 +406,13 @@ class GraphCarTrajectory(GraphScene):
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self.show_velocity_graph()
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self.ask_critically_about_velocity()
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def graph_sigmoid_trajectory_function(self, **kwargs):
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graph = self.graph_function(
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lambda t : 100*smooth(t/10.),
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**kwargs
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)
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return graph
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def introduce_graph(self, graph, origin):
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h_line, v_line = [
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Line(origin, origin, color = color, stroke_width = 2)
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@ -1223,13 +1230,370 @@ class DsOverDtGraphically(GraphCarTrajectory, ZoomedScene):
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velocity_label.restore,
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)
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#Pause and reflect
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randy = Randolph()
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randy.to_corner(DOWN+LEFT).shift(2*RIGHT)
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randy.look_at(frac_line)
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self.play(FadeIn(randy))
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self.play(randy.change_mode, "pondering")
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self.dither()
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self.play(Blink(randy))
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self.play(randy.change_mode, "thinking")
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self.dither()
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self.play(Blink(randy))
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self.dither()
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class DefineTrueDerivative(Scene):
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def construct(self):
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title = TextMobject("The true derivative")
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title.to_edge(UP)
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lhs = TexMobject("\\frac{ds}{dt}(t) = ")
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VGroup(*lhs[:2]).highlight(DISTANCE_COLOR)
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VGroup(*lhs[3:5]).highlight(TIME_COLOR)
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lhs.shift(3*LEFT+UP)
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dt_rhs = self.get_fraction("dt")
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numerical_rhs_list = [
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self.get_fraction("0.%s1"%("0"*x))
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for x in range(7)
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]
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for rhs in [dt_rhs] + numerical_rhs_list:
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rhs.next_to(lhs, RIGHT)
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brace, dt_to_zero = self.get_brace_and_text(dt_rhs)
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self.add(lhs, dt_rhs)
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self.play(Write(title))
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self.dither()
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dt_rhs.save_state()
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for num_rhs in numerical_rhs_list:
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self.play(Transform(dt_rhs, num_rhs))
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self.dither()
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self.play(dt_rhs.restore)
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self.play(
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GrowFromCenter(brace),
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Write(dt_to_zero)
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)
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self.dither()
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def get_fraction(self, dt_string):
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tex_mob = TexMobject(
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"\\frac{s(t + %s) - s(t)}{%s}"%(dt_string, dt_string)
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)
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part_lengths = [
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0,
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len("s(t+"),
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1,#1 and -1 below are purely for transformation quirks
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len(dt_string)-1,
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len(")-s(t)_"),#Underscore represents frac_line
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1,
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len(dt_string)-1,
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]
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pl_cumsum = np.cumsum(part_lengths)
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result = VGroup(*[
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VGroup(*tex_mob[i1:i2])
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for i1, i2 in zip(pl_cumsum, pl_cumsum[1:])
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])
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VGroup(*result[1:3]+result[4:6]).highlight(TIME_COLOR)
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return result
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def get_brace_and_text(self, deriv_frac):
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brace = Brace(VGroup(deriv_frac), DOWN)
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dt_to_zero = brace.get_text("$dt \\to 0$")
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VGroup(*dt_to_zero[:2]).highlight(TIME_COLOR)
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return brace, dt_to_zero
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class SecantLineToTangentLine(GraphCarTrajectory, DefineTrueDerivative):
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CONFIG = {
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"start_time" : 6,
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"end_time" : 2,
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"alt_end_time" : 10,
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"start_dt" : 2,
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"end_dt" : 0.01,
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"secant_line_length" : 10,
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}
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def construct(self):
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self.setup_axes(animate = False)
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self.remove(self.y_axis_label_mob, self.x_axis_label_mob)
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self.add_derivative_definition(self.y_axis_label_mob)
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self.add_graph()
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self.show_tangent_line()
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self.best_constant_approximation_around_a_point()
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def get_ds_dt_group(self, dt, animate = False):
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points = [
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self.input_to_graph_point(time)
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for time in self.curr_time, self.curr_time+dt
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]
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dots = map(Dot, points)
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for dot in dots:
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dot.scale_in_place(0.5)
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secant_line = Line(*points)
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secant_line.highlight(VELOCITY_COLOR)
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secant_line.scale_in_place(
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self.secant_line_length/secant_line.get_length()
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)
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interim_point = points[1][0]*RIGHT + points[0][1]*UP
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dt_line = Line(points[0], interim_point, color = TIME_COLOR)
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ds_line = Line(interim_point, points[1], color = DISTANCE_COLOR)
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dt = TexMobject("dt")
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dt.highlight(TIME_COLOR)
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if dt.get_width() > dt_line.get_width():
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dt.scale(
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dt_line.get_width()/dt.get_width(),
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about_point = dt.get_top()
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)
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dt.next_to(dt_line, DOWN, buff = SMALL_BUFF)
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ds = TexMobject("ds")
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ds.highlight(DISTANCE_COLOR)
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if ds.get_height() > ds_line.get_height():
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ds.scale(
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ds_line.get_height()/ds.get_height(),
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about_point = ds.get_left()
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)
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ds.next_to(ds_line, RIGHT, buff = SMALL_BUFF)
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group = VGroup(
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secant_line,
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ds_line, dt_line,
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ds, dt,
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*dots
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)
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if animate:
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self.play(
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ShowCreation(dt_line),
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Write(dt),
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ShowCreation(dots[0]),
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)
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self.play(
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ShowCreation(ds_line),
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Write(ds),
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ShowCreation(dots[1]),
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)
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self.play(
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ShowCreation(secant_line),
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Animation(VGroup(*dots))
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)
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return group
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def add_graph(self):
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def double_smooth_graph_function(t):
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if t < 5:
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return 50*smooth(t/5.)
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else:
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return 50*(1+smooth((t-5)/5.))
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graph = self.graph_function(
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double_smooth_graph_function,
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animate = False
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)
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self.label_graph(
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graph, "s(t)",
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proportion = 1,
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direction = DOWN+RIGHT,
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buff = SMALL_BUFF,
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animate = False
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)
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def add_derivative_definition(self, target_upper_left):
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deriv_frac = self.get_fraction("dt")
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lhs = TexMobject("\\frac{ds}{dt}(t)=")
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VGroup(*lhs[:2]).highlight(DISTANCE_COLOR)
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VGroup(*lhs[3:5]).highlight(TIME_COLOR)
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lhs.next_to(deriv_frac, LEFT)
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brace, text = self.get_brace_and_text(deriv_frac)
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deriv_def = VGroup(lhs, deriv_frac, brace, text)
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deriv_word = TextMobject("Derivative")
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deriv_word.next_to(deriv_def, UP, buff = 2*MED_BUFF)
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deriv_def.add(deriv_word)
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rect = Rectangle(color = WHITE)
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rect.replace(deriv_def, stretch = True)
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rect.scale_in_place(1.2)
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deriv_def.add(rect)
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deriv_def.scale(0.7)
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deriv_def.move_to(target_upper_left, UP+LEFT)
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self.add(deriv_def)
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return deriv_def
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def show_tangent_line(self):
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self.curr_time = self.start_time
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ds_dt_group = self.get_ds_dt_group(2, animate = True)
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self.dither()
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def update_ds_dt_group(ds_dt_group, alpha):
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new_dt = interpolate(self.start_dt, self.end_dt, alpha)
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new_group = self.get_ds_dt_group(new_dt)
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Transform(ds_dt_group, new_group).update(1)
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self.play(
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UpdateFromAlphaFunc(ds_dt_group, update_ds_dt_group),
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run_time = 8
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)
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self.dither()
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def update_as_tangent_line(ds_dt_group, alpha):
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self.curr_time = interpolate(self.start_time, self.end_time, alpha)
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new_group = self.get_ds_dt_group(self.end_dt)
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Transform(ds_dt_group, new_group).update(1)
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self.play(
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UpdateFromAlphaFunc(ds_dt_group, update_as_tangent_line),
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run_time = 8,
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rate_func = there_and_back
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)
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self.dither()
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what_dt_is_not_text = self.what_this_is_not_saying()
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self.dither()
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self.play(
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UpdateFromAlphaFunc(ds_dt_group, update_ds_dt_group),
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run_time = 8,
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rate_func = lambda t : 1-there_and_back(t)
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)
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self.dither()
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self.play(FadeOut(what_dt_is_not_text))
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v_line = self.get_vertical_line_to_graph(
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self.curr_time,
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line_class = Line,
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line_kwargs = {
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"color" : MAROON_B,
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"stroke_width" : 3
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}
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)
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def v_line_update(v_line):
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v_line.put_start_and_end_on(
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self.coords_to_point(self.curr_time, 0),
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self.input_to_graph_point(self.curr_time),
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)
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return v_line
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self.play(ShowCreation(v_line))
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self.dither()
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original_end_time = self.end_time
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for end_time in self.alt_end_time, original_end_time, self.start_time:
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self.end_time = end_time
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self.play(
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UpdateFromAlphaFunc(ds_dt_group, update_as_tangent_line),
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UpdateFromFunc(v_line, v_line_update),
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run_time = abs(self.curr_time-self.end_time),
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)
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self.start_time = end_time
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self.play(FadeOut(v_line))
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def what_this_is_not_saying(self):
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phrases = [
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TextMobject(
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"$dt$", "is", "not", s
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)
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for s in "``infinitely small''", "0"
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]
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for phrase in phrases:
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phrase[0].highlight(TIME_COLOR)
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phrase[2].highlight(RED)
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phrases[0].shift(DOWN+2*RIGHT)
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phrases[1].next_to(phrases[0], DOWN, aligned_edge = LEFT)
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for phrase in phrases:
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self.play(Write(phrase))
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return VGroup(*phrases)
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def best_constant_approximation_around_a_point(self):
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words = TextMobject("""
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Best constant
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approximation
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around a point
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""")
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words.next_to(self.x_axis, UP, aligned_edge = RIGHT)
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circle = Circle(
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radius = 0.25,
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color = WHITE
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).shift(self.input_to_graph_point(self.curr_time))
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self.play(Write(words))
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self.play(ShowCreation(circle))
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self.dither()
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class LeadIntoASpecificExample(TeacherStudentsScene, SecantLineToTangentLine):
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def setup(self):
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TeacherStudentsScene.setup(self)
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def construct(self):
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dot = Dot() #Just to coordinate derivative definition
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dot.to_corner(UP+LEFT, buff = SMALL_BUFF)
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deriv_def = self.add_derivative_definition(dot)
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self.remove(deriv_def)
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self.teacher_says("An example \\\\ should help.")
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self.dither()
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self.play(
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Write(deriv_def),
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*it.chain(*[
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[pi.change_mode, "thinking", pi.look_at, dot]
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for pi in self.get_students()
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])
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)
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self.random_blink(3)
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# self.teacher_says(
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# """
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# The idea of
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# ``approaching''
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# actually makes
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# things easier
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# """,
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# height = 3,
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# target_mode = "hooray"
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# )
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# self.dither(2)
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class TCubedExample(GraphCarTrajectory):
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CONFIG = {
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"y_min" : 0,
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"y_max" : 10,
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"y_tick_frequency" : 1,
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"y_labeled_nums" : range(10),
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"x_min" : 0,
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"x_max" : 4,
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"x_labeled_nums" : range(1, 5),
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"graph_origin" : 2.5*DOWN + 6*LEFT,
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}
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def construct(self):
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self.draw_graph()
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self.show_vertical_lines()
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def draw_graph(self):
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self.setup_axes(animate = False)
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self.x_axis_label_mob.shift(0.*DOWN)
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graph = self.graph_function(lambda t : t**3, animate = True)
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self.label_graph(
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graph,
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label = "s(t) = t^3",
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proportion = 0.525,
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direction = RIGHT,
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buff = SMALL_BUFF
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)
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self.dither()
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def show_vertical_lines(self):
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for t in 1, 2:
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v_line = self.get_vertical_line_to_graph(
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t, line_kwargs = {"color" : WHITE}
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)
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brace = Brace(v_line, RIGHT)
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text = TexMobject("%d^3 = %d"%(t, t**3))
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text.next_to(brace, RIGHT)
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text.shift(0.2*UP)
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group = VGroup(v_line, brace, text)
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if t == 1:
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self.play(ShowCreation(v_line))
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self.play(
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GrowFromCenter(brace),
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Write(text)
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)
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last_group = group
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else:
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self.play(Transform(last_group, group))
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self.dither()
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self.play(FadeOut(group))
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@ -7,7 +7,7 @@ from mobject.vectorized_mobject import VGroup, VectorizedPoint
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from animation.simple_animations import Write, ShowCreation
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from topics.number_line import NumberLine
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from topics.functions import ParametricFunction
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from topics.geometry import Rectangle, DashedLine
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from topics.geometry import Rectangle, DashedLine, Line
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class GraphScene(Scene):
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CONFIG = {
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@ -145,7 +145,7 @@ class GraphScene(Scene):
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def label_graph(self, graph, label = "f(x)",
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proportion = 0.7,
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direction = LEFT,
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buff = 2*MED_BUFF,
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buff = MED_BUFF,
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animate = True
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):
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label = TexMobject(label)
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@ -187,7 +187,7 @@ class GraphScene(Scene):
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def get_vertical_line_to_graph(self,
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x,
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graph = None,
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line_class = DashedLine,
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line_class = Line,
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line_kwargs = None,
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):
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if graph is None:
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