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AnalyzeExponentRatio of exp_footnote
This commit is contained in:
Grant Sanderson
parent
01630a4958
commit
b22b25c5a8
1 changed files with 300 additions and 12 deletions
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@ -224,6 +224,7 @@ class DoublingPopulation(PiCreatureScene):
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self.play(
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FadeOut(self.dM_dt_question),
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FadeOut(self.population_size_descriptor),
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FadeIn(day_to_day)
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)
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rect = self.let_day_pass_and_highlight_new_creatures(frac)
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@ -238,11 +239,8 @@ class DoublingPopulation(PiCreatureScene):
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def let_day_pass_and_highlight_new_creatures(self, frac):
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num_new_creatures = 2**self.get_curr_day()
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brace = self.population_size_descriptor
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self.let_one_day_pass()
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new_brace = self.get_population_size_descriptor()
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self.play(Transform(brace, new_brace))
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new_creatures = VGroup(
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*self.get_on_screen_pi_creatures()[-num_new_creatures:]
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)
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@ -291,7 +289,7 @@ class DoublingPopulation(PiCreatureScene):
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expressions = [false_deriv, difference_eq, real_deriv]
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text_arg_list = [
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("Tempting", "...",),
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("Rate of change", "\\\\ per day"),
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("Rate of change", "\\\\ over one full day"),
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("Rate of change", "\\\\ in a small time"),
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]
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for expression, text_args in zip(expressions, text_arg_list):
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@ -309,7 +307,6 @@ class DoublingPopulation(PiCreatureScene):
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fading_creatures = VGroup(*self.get_on_screen_pi_creatures()[8:])
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brace = self.population_size_descriptor
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self.play(*map(FadeIn, [
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false_deriv, false_deriv.brace, false_deriv.brace_text
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@ -319,13 +316,9 @@ class DoublingPopulation(PiCreatureScene):
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Transform(time, new_time),
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FadeOut(fading_creatures)
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)
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new_brace = self.get_population_size_descriptor()
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self.play(Transform(brace, new_brace))
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self.dither()
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for x in range(3):
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self.let_one_day_pass(run_time = 2)
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new_brace = self.get_population_size_descriptor()
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self.play(Transform(brace, new_brace))
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self.dither(2)
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for expression in difference_eq, real_deriv:
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@ -336,7 +329,6 @@ class DoublingPopulation(PiCreatureScene):
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Transform(false_deriv.brace_text, expression.brace_text),
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)
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self.dither(3)
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self.play(FadeOut(brace))
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self.reset()
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for x in range(self.get_num_days()):
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self.let_one_day_pass()
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@ -484,8 +476,304 @@ class DoublingPopulation(PiCreatureScene):
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return top_words, bottom_words
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class AnalyzeExponentRatio(PiCreatureScene):
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CONFIG = {
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"base" : 2,
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"base_str" : "2",
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}
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def construct(self):
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base_str = self.base_str
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func_def = TexMobject("M(", "t", ")", "= ", "%s^"%base_str, "t")
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func_def.to_corner(UP+LEFT)
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self.add(func_def)
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ratio = TexMobject(
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"{ {%s^"%base_str, "{t", "+", "dt}", "-",
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"%s^"%base_str, "t}",
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"\\over \\,", "dt}"
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)
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ratio.shift(UP+LEFT)
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lhs = TexMobject("{dM", "\\over \\,", "dt}", "(", "t", ")", "=")
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lhs.next_to(ratio, LEFT)
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two_to_t_plus_dt = VGroup(*ratio[:4])
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two_to_t = VGroup(*ratio[5:7])
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two_to_t_two_to_dt = TexMobject(
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"%s^"%base_str, "t",
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"%s^"%base_str, "{dt}"
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)
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two_to_t_two_to_dt.move_to(two_to_t_plus_dt, DOWN+LEFT)
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exp_prop_brace = Brace(two_to_t_two_to_dt, UP)
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one = TexMobject("1")
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one.move_to(ratio[5], DOWN)
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lp, rp = parens = TexMobject("()")
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parens.stretch(1.3, 1)
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parens.scale_to_fit_height(ratio.get_height())
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lp.next_to(ratio, LEFT, buff = 0)
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rp.next_to(ratio, RIGHT, buff = 0)
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extracted_two_to_t = TexMobject("%s^"%base_str, "t")
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extracted_two_to_t.next_to(lp, LEFT, buff = SMALL_BUFF)
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expressions = [
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ratio, two_to_t_two_to_dt,
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extracted_two_to_t, lhs, func_def
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]
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for expression in expressions:
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expression.highlight_by_tex("t", YELLOW)
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expression.highlight_by_tex("dt", GREEN)
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#Apply exponential property
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self.play(
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Write(ratio), Write(lhs),
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self.pi_creature.change_mode, "raise_right_hand"
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)
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self.dither(2)
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self.play(
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two_to_t_plus_dt.next_to, exp_prop_brace, UP,
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self.pi_creature.change_mode, "pondering"
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)
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self.play(
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ReplacementTransform(
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two_to_t_plus_dt.copy(), two_to_t_two_to_dt,
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run_time = 2,
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path_arc = np.pi,
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),
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FadeIn(exp_prop_brace)
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)
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self.dither(2)
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#Talk about exponential property
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add_exp_rect, mult_rect = rects = [
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Rectangle(
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stroke_color = BLUE,
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stroke_width = 2,
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).replace(mob).scale_in_place(1.1)
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for mob in [
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VGroup(*two_to_t_plus_dt[1:]),
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two_to_t_two_to_dt
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]
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]
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words = VGroup(*[
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TextMobject(s, " ideas")
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for s in "Additive", "Multiplicative"
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])
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words[0].move_to(words[1], LEFT)
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words.highlight(BLUE)
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words.next_to(two_to_t_plus_dt, RIGHT, buff = 1.5*LARGE_BUFF)
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arrows = VGroup(*[
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Arrow(word.get_left(), rect, color = words.get_color())
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for word, rect in zip(words, rects)
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])
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self.play(ShowCreation(add_exp_rect))
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self.dither()
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self.play(ReplacementTransform(
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add_exp_rect.copy(), mult_rect
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))
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self.dither()
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self.change_mode("happy")
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self.play(Write(words[0], run_time = 2))
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self.play(ShowCreation(arrows[0]))
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self.dither()
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self.play(
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Transform(*words),
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Transform(*arrows),
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)
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self.dither(2)
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self.play(*map(FadeOut, [
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words[0], arrows[0], add_exp_rect, mult_rect,
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two_to_t_plus_dt, exp_prop_brace,
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]))
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#Factor out 2^t
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self.play(*[
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FadeIn(
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mob,
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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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for mob in one, lp, rp
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] + [
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ReplacementTransform(
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mob, extracted_two_to_t,
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path_arc = np.pi/2,
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run_time = 2,
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)
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for mob in two_to_t, VGroup(*two_to_t_two_to_dt[:2])
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] + [
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lhs.next_to, extracted_two_to_t, LEFT
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])
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self.change_mode("pondering")
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shifter = VGroup(ratio[4], one, *two_to_t_two_to_dt[2:])
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stretcher = VGroup(lp, ratio[7], rp)
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self.play(
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shifter.next_to, ratio[7], UP,
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stretcher.stretch_in_place, 0.9, 0
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)
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self.dither(2)
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#Ask about dt -> 0
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brace = Brace(VGroup(extracted_two_to_t, ratio), DOWN)
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alt_brace = Brace(parens, DOWN)
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dt_to_zero = TexMobject("dt", "\\to 0")
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dt_to_zero.highlight_by_tex("dt", GREEN)
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dt_to_zero.next_to(brace, DOWN)
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self.play(GrowFromCenter(brace))
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self.play(Write(dt_to_zero))
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self.dither(2)
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#Who cares
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randy = Randolph()
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randy.scale(0.7)
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randy.to_edge(DOWN)
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self.play(
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FadeIn(randy),
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self.pi_creature.change_mode, "plain",
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)
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self.play(PiCreatureSays(
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randy, "Who cares?",
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bubble_kwargs = {"direction" : LEFT},
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target_mode = "angry",
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))
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self.dither(2)
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self.play(
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RemovePiCreatureBubble(randy),
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FadeOut(randy),
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self.pi_creature.change_mode, "hooray",
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self.pi_creature.look_at, parens
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)
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self.play(
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Transform(brace, alt_brace),
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dt_to_zero.next_to, alt_brace, DOWN
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)
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self.dither()
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#Highlight separation
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rects = [
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Rectangle(
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stroke_color = color,
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stroke_width = 2,
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).replace(mob, stretch = True).scale_in_place(1.1)
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for mob, color in [
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(VGroup(parens, dt_to_zero), GREEN),
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(extracted_two_to_t, YELLOW),
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]
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]
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self.play(ShowCreation(rects[0]))
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self.dither(2)
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self.play(ReplacementTransform(rects[0].copy(), rects[1]))
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self.change_mode("happy")
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self.dither()
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self.play(*map(FadeOut, rects))
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#Plug in specific values
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static_constant = self.try_specific_dt_values()
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constant = static_constant.copy()
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#Replace with actual constant
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limit_term = VGroup(
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brace, dt_to_zero, ratio[4], one, rects[0],
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*ratio[7:]+two_to_t_two_to_dt[2:]
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)
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self.play(FadeIn(rects[0]))
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self.play(limit_term.to_corner, DOWN+LEFT)
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self.play(
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lp.stretch, 0.5, 1,
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lp.stretch, 0.8, 0,
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lp.next_to, extracted_two_to_t[0], RIGHT,
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rp.stretch, 0.5, 1,
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rp.stretch, 0.8, 0,
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rp.next_to, lp, RIGHT, SMALL_BUFF,
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rp.shift, constant.get_width()*RIGHT,
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constant.next_to, extracted_two_to_t[0], RIGHT, MED_LARGE_BUFF
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)
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self.dither()
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self.change_mode("confused")
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self.dither()
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#hold_final_value
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derivative = VGroup(
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lhs, extracted_two_to_t, parens, constant
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)
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func_def_rhs = VGroup(*func_def[-2:]).copy()
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func_lp, func_rp = func_parens = TexMobject("()")
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func_parens.set_fill(opacity = 0)
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func_lp.next_to(func_def_rhs[0], LEFT, buff = 0)
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func_rp.next_to(func_lp, RIGHT, buff = func_def_rhs.get_width())
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func_def_rhs.add(func_parens)
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M = lhs[0][1]
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self.play(
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FadeOut(M),
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func_def_rhs.move_to, M, LEFT,
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func_def_rhs.set_fill, None, 1,
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)
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lhs[0].submobjects[1] = func_def_rhs
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self.dither()
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self.play(
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derivative.next_to, self.pi_creature, UP,
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derivative.to_edge, RIGHT,
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self.pi_creature.change_mode, "raise_right_hand"
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)
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self.dither(2)
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for mob in extracted_two_to_t, constant:
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self.play(Indicate(mob))
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self.dither()
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self.dither(2)
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def try_specific_dt_values(self):
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expressions = []
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for num_zeros in [1, 2, 4, 7]:
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dt_str = "0." + num_zeros*"0" + "1"
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dt_num = float(dt_str)
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output_num = (self.base**dt_num - 1) / dt_num
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output_str = "%.7f\\dots"%output_num
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expression = TexMobject(
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"{%s^"%self.base_str, "{%s}"%dt_str, "-1",
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"\\over \\,", "%s}"%dt_str,
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"=", output_str
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)
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expression.highlight_by_tex(dt_str, GREEN)
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expression.highlight_by_tex(output_str, BLUE)
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expression.to_corner(UP+RIGHT)
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expressions.append(expression)
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curr_expression = expressions[0]
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self.play(
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Write(curr_expression),
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self.pi_creature.change_mode, "pondering"
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)
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self.dither(2)
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for expression in expressions[1:]:
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self.play(Transform(curr_expression, expression))
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self.dither(2)
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return curr_expression[-1]
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class ExponentRatioWithThree(AnalyzeExponentRatio):
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CONFIG = {
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"base" : 3,
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"base_str" : "3",
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}
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class ExponentRatioWithSeven(AnalyzeExponentRatio):
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CONFIG = {
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"base" : 3,
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"base_str" : "3",
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}
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class ExponentRatioWithE(AnalyzeExponentRatio):
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CONFIG = {
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"base" : np.exp(1),
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"base_str" : "e",
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}
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