manim/eoc/exp_footnote.py

from helpers import *

from mobject.tex_mobject import TexMobject
from mobject import Mobject
from mobject.image_mobject import ImageMobject
from mobject.vectorized_mobject import *

from animation.animation import Animation
from animation.transform import *
from animation.simple_animations import *
from animation.playground import *
from topics.geometry import *
from topics.characters import *
from topics.functions import *
from topics.fractals import *
from topics.number_line import *
from topics.combinatorics import *
from topics.numerals import *
from topics.three_dimensions import *
from topics.objects import *
from scene import Scene
from scene.zoomed_scene import ZoomedScene
from scene.reconfigurable_scene import ReconfigurableScene
from camera import Camera
from mobject.svg_mobject import *
from mobject.tex_mobject import *

from topics.common_scenes import OpeningQuote

from eoc.graph_scene import *

class ExpFootnoteOpeningQuote(OpeningQuote):
    CONFIG = {
        "quote" : [
        "Who has not be amazed to learn that the function",
        "$y = e^x$,", "like a phoenix rising again from its own",
        "ashes, is its own derivative?",
        ],
        "highlighted_quote_terms" : {
            "$y = e^x$" : MAROON_B
        },
        "author" : "Francois le Lionnais"
    }

class LastVideo(TeacherStudentsScene):
    def construct(self):
        series = VideoSeries()
        series.to_edge(UP)
        last_video = series[2]
        last_video.save_state()
        this_video = series[3]

        known_formulas = VGroup(*map(TexMobject, [
            "\\frac{d(x^n)}{dx} = nx^{n-1}",
            "\\frac{d(\\sin(x))}{dx} = \\cos(x)",
        ]))
        known_formulas.arrange_submobjects(
            DOWN, buff = MED_LARGE_BUFF,
        )
        known_formulas.scale_to_fit_height(2.5)
        exp_question = TexMobject("2^x", ", 7^x", ", e^x", " ???")

        last_video_brace = Brace(last_video)
        known_formulas.next_to(last_video_brace, DOWN)
        last_video_brace.save_state()
        last_video_brace.shift(3*LEFT)
        last_video_brace.set_fill(opacity = 0)

        self.add(series)
        self.play(
            last_video_brace.restore,
            last_video.highlight, YELLOW,
            self.get_teacher().change_mode, "raise_right_hand",
        )
        self.play(Write(known_formulas))
        self.dither()
        self.student_says(
            exp_question, student_index = -1,
            added_anims = [self.get_teacher().change_mode, "pondering"]
        )
        self.dither(2)
        self.play(known_formulas.replace, last_video)
        self.play(last_video_brace.next_to, this_video, DOWN)
        self.play(
            last_video.restore,
            this_video.highlight, YELLOW
        )
        self.play(       
            exp_question.next_to, last_video_brace, DOWN,
            FadeOut(self.get_students()[-1].bubble),
        )
        self.change_student_modes(
            *["pondering"]*3,
            look_at_arg = exp_question
        )
        self.dither()

class PopulationSizeGraphVsPopulationMassGraph(Scene):
    def construct(self):
        pass

class DoublingPopulation(PiCreatureScene):
    CONFIG = {
        "time_color" : YELLOW,
        "pi_creature_grid_dimensions" : (8, 8),
        "pi_creature_grid_height" : 6,
    }
    
    def construct(self):
        self.remove(self.get_pi_creatures())
        self.introduce_expression()
        self.introduce_pi_creatures()
        self.count_through_days()
        self.ask_about_dM_dt()
        self.growth_per_day()
        self.relate_growth_rate_to_pop_size()

    def introduce_expression(self):
        f_x = TexMobject("f(x)", "=", "2^x")
        f_t = TexMobject("f(t)", "=", "2^t")
        P_t = TexMobject("P(t)", "=", "2^t")
        M_t = TexMobject("M(t)", "=", "2^t")
        functions = VGroup(f_x, f_t, P_t, M_t)
        for function in functions:
            function.scale(1.2)
            function.to_corner(UP+LEFT)
        for function in functions[1:]:
            for i, j in (0, 2), (2, 1):
                function[i][j].highlight(self.time_color)

        t_expression = TexMobject("t", "=", "\\text{Time (in days)}")
        t_expression.to_corner(UP+RIGHT)
        t_expression[0].highlight(self.time_color)

        pop_brace, mass_brace = [
            Brace(function[0], DOWN)
            for function in P_t, M_t
        ]
        for brace, word in (pop_brace, "size"), (mass_brace, "mass"):
            text = brace.get_text("Population %s"%word)
            text.to_edge(LEFT)
            brace.text = text

        self.play(Write(f_x))
        self.dither()
        self.play(
            Transform(f_x, f_t),
            FadeIn(
                t_expression,
                run_time = 2,
                submobject_mode = "lagged_start"
            )
        )
        self.play(Transform(f_x, P_t))
        self.play(
            GrowFromCenter(pop_brace),
            Write(pop_brace.text, run_time = 2)
        )
        self.dither(2)

        self.function = f_x
        self.pop_brace = pop_brace
        self.t_expression = t_expression
        self.mass_function = M_t
        self.mass_brace = mass_brace

    def introduce_pi_creatures(self):
        creatures = self.get_pi_creatures()
        total_num_days = self.get_num_days()
        num_start_days = 4

        self.reset()
        for x in range(num_start_days):
            self.let_one_day_pass()
        self.dither()
        self.play(
            Transform(self.function, self.mass_function),
            Transform(self.pop_brace, self.mass_brace),
            Transform(self.pop_brace.text, self.mass_brace.text),
        )
        self.dither()
        for x in range(total_num_days-num_start_days):
            self.let_one_day_pass()
            self.dither()
        self.joint_blink(shuffle = False)
        self.dither()

    def count_through_days(self):
        self.reset()
        brace = self.get_population_size_descriptor()
        days_to_let_pass = 3

        self.play(GrowFromCenter(brace))
        self.dither()
        for x in range(days_to_let_pass):
            self.let_one_day_pass()
            new_brace = self.get_population_size_descriptor()
            self.play(Transform(brace, new_brace))
            self.dither()

        self.population_size_descriptor = brace

    def ask_about_dM_dt(self):
        dM_dt_question = TexMobject("{dM", "\\over dt}", "=", "???")
        dM, dt, equals, q_marks = dM_dt_question
        dM_dt_question.next_to(self.function, DOWN, buff = LARGE_BUFF)
        dM_dt_question.to_edge(LEFT)

        self.play(
            FadeOut(self.pop_brace),
            FadeOut(self.pop_brace.text),
            Write(dM_dt_question)
        )
        self.dither(3)
        for mob in dM, dt:
            self.play(Indicate(mob))
            self.dither()

        self.dM_dt_question = dM_dt_question

    def growth_per_day(self):
        day_to_day, frac = self.get_from_day_to_day_label()

        self.play(
            FadeOut(self.dM_dt_question),
            FadeOut(self.population_size_descriptor),
            FadeIn(day_to_day)
        )
        rect = self.let_day_pass_and_highlight_new_creatures(frac)

        for x in range(2):
            new_day_to_day, new_frac = self.get_from_day_to_day_label()
            self.play(*map(FadeOut, [rect, frac]))
            frac = new_frac
            self.play(Transform(day_to_day, new_day_to_day))
            rect = self.let_day_pass_and_highlight_new_creatures(frac)
        self.play(*map(FadeOut, [rect, frac, day_to_day]))

    def let_day_pass_and_highlight_new_creatures(self, frac):
        num_new_creatures = 2**self.get_curr_day()

        self.let_one_day_pass()
        new_creatures = VGroup(
            *self.get_on_screen_pi_creatures()[-num_new_creatures:]
        )
        rect = Rectangle(
            color = GREEN,
            fill_color = BLUE,
            fill_opacity = 0.3,
        )
        rect.replace(new_creatures, stretch = True)
        rect.stretch_to_fit_height(rect.get_height()+MED_SMALL_BUFF)
        rect.stretch_to_fit_width(rect.get_width()+MED_SMALL_BUFF)
        self.play(DrawBorderThenFill(rect))
        self.play(Write(frac))
        self.dither()
        return rect

    def relate_growth_rate_to_pop_size(self):
        false_deriv = TexMobject(
            "{d(2^t) ", "\\over dt}", "= 2^t"
        )
        difference_eq = TexMobject(
            "{ {2^{t+1} - 2^t} \\over", "1}", "= 2^t"
        )
        real_deriv = TexMobject(
            "{ {2^{t+dt} - 2^t} \\over", "dt}", "= \\, ???"
        )
        VGroup(
            false_deriv[0][3], 
            false_deriv[2][-1],
            difference_eq[0][1],
            difference_eq[0][-2],
            difference_eq[2][-1],
            difference_eq[2][-1],
            real_deriv[0][1],
            real_deriv[0][-2],
        ).highlight(YELLOW)
        VGroup(
            difference_eq[0][3],
            difference_eq[1][-1],
            real_deriv[0][3],
            real_deriv[0][4],
            real_deriv[1][-2],
            real_deriv[1][-1],
        ).highlight(GREEN)

        expressions = [false_deriv, difference_eq, real_deriv]
        text_arg_list = [
            ("Tempting", "...",),
            ("Rate of change", "\\\\ over one full day"),
            ("Rate of change", "\\\\ in a small time"),
        ]
        for expression, text_args in zip(expressions, text_arg_list):
            expression.next_to(
                self.function, DOWN, 
                buff = LARGE_BUFF,
                aligned_edge = LEFT,
            )
            expression.brace = Brace(expression, DOWN)
            expression.brace_text = expression.brace.get_text(*text_args)

        time = self.t_expression[-1]
        new_time = TexMobject("3")
        new_time.move_to(time, LEFT)

        fading_creatures = VGroup(*self.get_on_screen_pi_creatures()[8:])


        self.play(*map(FadeIn, [
            false_deriv, false_deriv.brace, false_deriv.brace_text
        ]))
        self.dither()
        self.play(
            Transform(time, new_time),
            FadeOut(fading_creatures)
        )
        self.dither()
        for x in range(3):
            self.let_one_day_pass(run_time = 2)
            self.dither(2)

        for expression in difference_eq, real_deriv:
            expression.brace_text[1].highlight(GREEN)
            self.play(
                Transform(false_deriv, expression),
                Transform(false_deriv.brace, expression.brace),
                Transform(false_deriv.brace_text, expression.brace_text),
            )
            self.dither(3)
        self.reset()
        for x in range(self.get_num_days()):
            self.let_one_day_pass()
        self.dither()

        rect = Rectangle(color = YELLOW)
        rect.replace(real_deriv)
        rect.stretch_to_fit_width(rect.get_width()+MED_SMALL_BUFF)
        rect.stretch_to_fit_height(rect.get_height()+MED_SMALL_BUFF)
        self.play(*map(FadeOut, [
            false_deriv.brace, false_deriv.brace_text
        ]))
        self.play(ShowCreation(rect))
        self.play(*[
            ApplyFunction(
                lambda pi : pi.change_mode("pondering").look_at(real_deriv),
                pi,
                run_time = 2,
                rate_func = squish_rate_func(smooth, a, a+0.5)
            )
            for pi in self.get_pi_creatures()
            for a in [0.5*random.random()]
        ])
        self.dither(3)

    ###########

    def create_pi_creatures(self):
        width, height = self.pi_creature_grid_dimensions
        creature_array = VGroup(*[
            VGroup(*[
                PiCreature(mode = "plain")
                for y in range(height)
            ]).arrange_submobjects(UP, buff = MED_LARGE_BUFF)
            for x in range(width)
        ]).arrange_submobjects(RIGHT, buff = MED_LARGE_BUFF)
        creatures = VGroup(*it.chain(*creature_array))
        creatures.scale_to_fit_height(self.pi_creature_grid_height)
        creatures.to_corner(DOWN+RIGHT)

        colors = color_gradient([BLUE, GREEN, GREY_BROWN], len(creatures))
        random.shuffle(colors)
        for creature, color in zip(creatures, colors):
            creature.set_color(color)

        return creatures

    def reset(self):
        time = self.t_expression[-1]
        faders = [time] + list(self.get_on_screen_pi_creatures())
        new_time = TexMobject("0")
        new_time.next_to(self.t_expression[-2], RIGHT)
        first_creature = self.get_pi_creatures()[0]

        self.play(*map(FadeOut, faders))
        self.play(*map(FadeIn, [first_creature, new_time]))
        self.t_expression.submobjects[-1] = new_time

    def let_one_day_pass(self, run_time = 2):
        all_creatures = self.get_pi_creatures()
        on_screen_creatures = self.get_on_screen_pi_creatures()
        low_i = len(on_screen_creatures)
        high_i = min(2*low_i, len(all_creatures))
        new_creatures = VGroup(*all_creatures[low_i:high_i])

        to_children_anims = []
        growing_anims = []
        for old_pi, pi in zip(on_screen_creatures, new_creatures):
            pi.save_state()
            child = pi.copy()
            child.scale(0.25, about_point = child.get_bottom())
            child.eyes.scale(1.5, about_point = child.eyes.get_bottom())
            pi.move_to(old_pi)
            pi.set_fill(opacity = 0)

            index = list(new_creatures).index(pi)
            prop = float(index)/len(new_creatures)
            alpha  = np.clip(len(new_creatures)/8.0, 0, 0.5)
            rate_func = squish_rate_func(
                smooth, alpha*prop, alpha*prop+(1-alpha)
            )

            to_child_anim = Transform(pi, child, rate_func = rate_func)
            to_child_anim.update(1)
            growing_anim = ApplyMethod(pi.restore, rate_func = rate_func)
            to_child_anim.update(0)

            to_children_anims.append(to_child_anim)
            growing_anims.append(growing_anim)

        time = self.t_expression[-1]
        total_new_creatures = len(on_screen_creatures) + len(new_creatures)
        new_time = TexMobject(str(int(np.log2(total_new_creatures))))
        new_time.move_to(time, LEFT)

        growing_anims.append(Transform(time, new_time))

        self.play(*to_children_anims, run_time = run_time/2.0)
        self.play(*growing_anims, run_time = run_time/2.0)
        
    def get_num_pi_creatures_on_screen(self):
        mobjects = self.get_mobjects()
        return sum([
            pi in mobjects for pi in self.get_pi_creatures()
        ])

    def get_population_size_descriptor(self):
        on_screen_creatures = self.get_on_screen_pi_creatures()
        brace = Brace(on_screen_creatures, LEFT)
        n = len(on_screen_creatures)
        label = brace.get_text(
            "$2^%d$"%int(np.log2(n)),
            "$=%d$"%n,
        )
        brace.add(label)
        return brace

    def get_num_days(self):
        x, y = self.pi_creature_grid_dimensions
        return int(np.log2(x*y))

    def get_curr_day(self):
        return int(np.log2(len(self.get_on_screen_pi_creatures())))

    def get_from_day_to_day_label(self):
        curr_day = self.get_curr_day()
        top_words = TextMobject(
            "From day", str(curr_day), 
            "to", str(curr_day+1), ":"
        )
        top_words.scale_to_fit_width(4)
        top_words.next_to(
            self.function, DOWN,
            buff = MED_LARGE_BUFF,
            aligned_edge = LEFT,
        )
        top_words[1].highlight(GREEN)

        bottom_words = TexMobject(
            str(2**curr_day),
            "\\text{ creatures}", "\\over {1 \\text{ day}}"
        )
        bottom_words[0].highlight(GREEN)
        bottom_words.next_to(top_words, DOWN, buff = MED_LARGE_BUFF)

        return top_words, bottom_words

class GraphOfTwoToT(GraphScene):
    CONFIG = {
        "x_axis_label" : "$t$",
        "y_axis_label" : "$M$",
        "x_labeled_nums" : range(1, 7),
        "y_labeled_nums" : range(8, 40, 8),
        "x_max" : 6,
        "y_min" : 0,
        "y_max" : 32,
        "y_tick_frequency" : 2,
        "graph_origin" : 2.5*DOWN + 5*LEFT,
    }
    def construct(self):
        self.setup_axes()
        example_t = 3
        graph = self.get_graph(lambda t : 2**t, color = BLUE_C)
        self.graph = graph
        graph_label = self.get_graph_label(
            graph, "M(t) = 2^t",
            direction = LEFT,
        )
        label_group = self.get_label_group(example_t)
        v_line, brace, height_label, ss_group, slope_label = label_group
        self.animate_secant_slope_group_change(
            ss_group,
            target_dx = 1,
            run_time = 0
        )
        self.remove(ss_group)

        #Draw graph and revert to tangent
        self.play(ShowCreation(graph))
        self.play(Write(graph_label))
        self.dither()
        self.play(Write(ss_group))
        self.dither()
        for target_dx in 0.01, 1, 0.01:
            self.animate_secant_slope_group_change(
                ss_group,
                target_dx = target_dx
            )
            self.dither()

        #Mark up with values

        self.play(ShowCreation(v_line))
        self.play(
            GrowFromCenter(brace),
            Write(height_label, run_time = 1)
        )
        self.dither()
        self.play(
            FadeIn(
                slope_label, 
                run_time = 4,
                submobject_mode = "lagged_start"
            ),
            ReplacementTransform(
                height_label.copy(),
                slope_label.get_part_by_tex("2^")
            )
        )
        self.dither()

        #Vary value
        threes = VGroup(height_label[1], slope_label[2][1])
        ts = VGroup(*[
            TexMobject("t").highlight(YELLOW).scale(0.75).move_to(three)
            for three in threes
        ])
        self.play(Transform(threes, ts))

        alt_example_t = example_t+1
        def update_label_group(group, alpha):
            t = interpolate(example_t, alt_example_t, alpha)
            new_group = self.get_label_group(t)
            Transform(group, new_group).update(1)
            for t, three in zip(ts, threes):
                t.move_to(three)
            Transform(threes, ts).update(1)
            return group

        self.play(UpdateFromAlphaFunc(
            label_group, update_label_group,
            run_time = 3,
        ))
        self.play(UpdateFromAlphaFunc(
            label_group, update_label_group,
            run_time = 3,
            rate_func = lambda t : 1 - 1.5*smooth(t)
        ))

    def get_label_group(self, t):
        graph = self.graph

        v_line = self.get_vertical_line_to_graph(
            t, graph,
            color = YELLOW,
        )
        brace = Brace(v_line, RIGHT)
        height_label = brace.get_text("$2^%d$"%t)

        ss_group = self.get_secant_slope_group(
            t, graph, dx = 0.01,
            df_label = "dM",
            dx_label = "dt",
            dx_line_color = GREEN,
            secant_line_color = RED,
        )
        slope_label = TexMobject(
            "\\text{Slope}", "=", 
            "2^%d"%t,
            "(%.7f\\dots)"%np.log(2)
        )
        slope_label.next_to(
            ss_group.secant_line.point_from_proportion(0.65),
            DOWN+RIGHT,
            buff = 0
        )
        slope_label.highlight_by_tex("Slope", RED)
        return VGroup(
            v_line, brace, height_label,
            ss_group, slope_label
        )

class AnalyzeExponentRatio(PiCreatureScene):
    CONFIG = {
        "base" : 2,
        "base_str" : "2",
    }
    def construct(self):
        base_str = self.base_str

        func_def = TexMobject("M(", "t", ")", "= ", "%s^"%base_str, "t")
        func_def.to_corner(UP+LEFT)
        self.add(func_def)

        ratio = TexMobject(
            "{ {%s^"%base_str, "{t", "+", "dt}", "-", 
            "%s^"%base_str, "t}",
            "\\over \\,", "dt}"
        )
        ratio.shift(UP+LEFT)

        lhs = TexMobject("{dM", "\\over \\,", "dt}", "(", "t", ")", "=")
        lhs.next_to(ratio, LEFT)


        two_to_t_plus_dt = VGroup(*ratio[:4])
        two_to_t = VGroup(*ratio[5:7])
        two_to_t_two_to_dt = TexMobject(
            "%s^"%base_str, "t", 
            "%s^"%base_str, "{dt}"
        )
        two_to_t_two_to_dt.move_to(two_to_t_plus_dt, DOWN+LEFT)
        exp_prop_brace = Brace(two_to_t_two_to_dt, UP)

        one = TexMobject("1")
        one.move_to(ratio[5], DOWN)
        lp, rp = parens = TexMobject("()")
        parens.stretch(1.3, 1)
        parens.scale_to_fit_height(ratio.get_height())
        lp.next_to(ratio, LEFT, buff = 0)
        rp.next_to(ratio, RIGHT, buff = 0)

        extracted_two_to_t = TexMobject("%s^"%base_str, "t")
        extracted_two_to_t.next_to(lp, LEFT, buff = SMALL_BUFF)

        expressions = [
            ratio, two_to_t_two_to_dt, 
            extracted_two_to_t, lhs, func_def
        ]
        for expression in expressions:
            expression.highlight_by_tex("t", YELLOW)
            expression.highlight_by_tex("dt", GREEN)

        #Apply exponential property
        self.play(
            Write(ratio), Write(lhs),
            self.pi_creature.change_mode, "raise_right_hand"
        )
        self.dither(2)
        self.play(
            two_to_t_plus_dt.next_to, exp_prop_brace, UP,
            self.pi_creature.change_mode, "pondering"
        )
        self.play(
            ReplacementTransform(
                two_to_t_plus_dt.copy(), two_to_t_two_to_dt,
                run_time = 2,
                path_arc = np.pi,
            ),
            FadeIn(exp_prop_brace)
        )
        self.dither(2)

        #Talk about exponential property
        add_exp_rect, mult_rect = rects = [
            Rectangle(
                stroke_color = BLUE,
                stroke_width = 2,
            ).replace(mob).scale_in_place(1.1)
            for mob in [
                VGroup(*two_to_t_plus_dt[1:]),
                two_to_t_two_to_dt
            ]
        ]
        words = VGroup(*[
            TextMobject(s, " ideas")
            for s in "Additive", "Multiplicative"
        ])
        words[0].move_to(words[1], LEFT)
        words.highlight(BLUE)
        words.next_to(two_to_t_plus_dt, RIGHT, buff = 1.5*LARGE_BUFF)
        arrows = VGroup(*[
            Arrow(word.get_left(), rect, color = words.get_color())
            for word, rect in zip(words, rects)
        ])

        self.play(ShowCreation(add_exp_rect))
        self.dither()
        self.play(ReplacementTransform(
            add_exp_rect.copy(), mult_rect
        ))
        self.dither()
        self.change_mode("happy")
        self.play(Write(words[0], run_time = 2))
        self.play(ShowCreation(arrows[0]))
        self.dither()
        self.play(
            Transform(*words),
            Transform(*arrows),
        )
        self.dither(2)
        self.play(*map(FadeOut, [
            words[0], arrows[0], add_exp_rect, mult_rect,
            two_to_t_plus_dt, exp_prop_brace,
        ]))

        #Factor out 2^t
        self.play(*[
            FadeIn(
                mob,
                run_time = 2,
                rate_func = squish_rate_func(smooth, 0.5, 1)
            )
            for mob in one, lp, rp
        ] + [
            ReplacementTransform(
                mob, extracted_two_to_t,
                path_arc = np.pi/2,
                run_time = 2,
            )
            for mob in two_to_t, VGroup(*two_to_t_two_to_dt[:2])
        ] + [
            lhs.next_to, extracted_two_to_t, LEFT
        ])
        self.change_mode("pondering")
        shifter = VGroup(ratio[4], one, *two_to_t_two_to_dt[2:])
        stretcher = VGroup(lp, ratio[7], rp)
        self.play(
            shifter.next_to, ratio[7], UP,
            stretcher.stretch_in_place, 0.9, 0
        )
        self.dither(2)

        #Ask about dt -> 0
        brace = Brace(VGroup(extracted_two_to_t, ratio), DOWN)
        alt_brace = Brace(parens, DOWN)
        dt_to_zero = TexMobject("dt", "\\to 0")
        dt_to_zero.highlight_by_tex("dt", GREEN)
        dt_to_zero.next_to(brace, DOWN)

        self.play(GrowFromCenter(brace))
        self.play(Write(dt_to_zero))
        self.dither(2)

        #Who cares
        randy = Randolph()
        randy.scale(0.7)
        randy.to_edge(DOWN)

        self.play(
            FadeIn(randy),
            self.pi_creature.change_mode, "plain",
        )
        self.play(PiCreatureSays(
            randy, "Who cares?", 
            bubble_kwargs = {"direction" : LEFT},
            target_mode = "angry",
        ))
        self.dither(2)
        self.play(
            RemovePiCreatureBubble(randy),
            FadeOut(randy),
            self.pi_creature.change_mode, "hooray",
            self.pi_creature.look_at, parens
        )
        self.play(
            Transform(brace, alt_brace),
            dt_to_zero.next_to, alt_brace, DOWN
        )
        self.dither()

        #Highlight separation
        rects = [
            Rectangle(
                stroke_color = color,
                stroke_width = 2,
            ).replace(mob, stretch = True).scale_in_place(1.1)
            for mob, color in [
                (VGroup(parens, dt_to_zero), GREEN), 
                (extracted_two_to_t, YELLOW),
            ]
        ]
        self.play(ShowCreation(rects[0]))
        self.dither(2)
        self.play(ReplacementTransform(rects[0].copy(), rects[1]))
        self.change_mode("happy")
        self.dither()
        self.play(*map(FadeOut, rects))

        #Plug in specific values
        static_constant = self.try_specific_dt_values()
        constant = static_constant.copy()

        #Replace with actual constant
        limit_term = VGroup(
            brace, dt_to_zero, ratio[4], one, rects[0],
            *ratio[7:]+two_to_t_two_to_dt[2:]
        )
        self.play(FadeIn(rects[0]))
        self.play(limit_term.to_corner, DOWN+LEFT)
        self.play(
            lp.stretch, 0.5, 1,
            lp.stretch, 0.8, 0,
            lp.next_to, extracted_two_to_t[0], RIGHT,
            rp.stretch, 0.5, 1,
            rp.stretch, 0.8, 0,
            rp.next_to, lp, RIGHT, SMALL_BUFF,
            rp.shift, constant.get_width()*RIGHT,
            constant.next_to, extracted_two_to_t[0], RIGHT, MED_LARGE_BUFF
        )
        self.dither()
        self.change_mode("confused")
        self.dither()

        #Indicate distinction between dt group and t group again
        for mob in limit_term, extracted_two_to_t:
            self.play(FocusOn(mob))
            self.play(Indicate(mob))
        self.dither()

        #hold_final_value
        derivative = VGroup(
            lhs, extracted_two_to_t, parens, constant
        )
        func_def_rhs = VGroup(*func_def[-2:]).copy()
        func_lp, func_rp = func_parens = TexMobject("()")
        func_parens.set_fill(opacity = 0)
        func_lp.next_to(func_def_rhs[0], LEFT, buff = 0)
        func_rp.next_to(func_lp, RIGHT, buff = func_def_rhs.get_width())
        func_def_rhs.add(func_parens)
        M = lhs[0][1]

        self.play(
            FadeOut(M),
            func_def_rhs.move_to, M, LEFT,
            func_def_rhs.set_fill, None, 1,
        )
        lhs[0].submobjects[1] = func_def_rhs
        self.dither()
        self.play(
            derivative.next_to, self.pi_creature, UP,
            derivative.to_edge, RIGHT,
            self.pi_creature.change_mode, "raise_right_hand"
        )
        self.dither(2)
        for mob in extracted_two_to_t, constant:
            self.play(Indicate(mob))
            self.dither()
        self.dither(2)

    def try_specific_dt_values(self):
        expressions = []
        for num_zeros in [1, 2, 4, 7]:
            dt_str = "0." + num_zeros*"0" + "1"
            dt_num = float(dt_str)
            output_num = (self.base**dt_num - 1) / dt_num
            output_str = "%.7f\\dots"%output_num

            expression = TexMobject(
                "{%s^"%self.base_str, "{%s}"%dt_str, "-1", 
                "\\over \\,", "%s}"%dt_str, 
                "=", output_str
            )
            expression.highlight_by_tex(dt_str, GREEN)
            expression.highlight_by_tex(output_str, BLUE)
            expression.to_corner(UP+RIGHT)
            expressions.append(expression)

        curr_expression = expressions[0]
        self.play(
            Write(curr_expression),
            self.pi_creature.change_mode, "pondering"
        )
        self.dither(2)
        for expression in expressions[1:]:
            self.play(Transform(curr_expression, expression))
            self.dither(2)
        return curr_expression[-1]

class ExponentRatioWithThree(AnalyzeExponentRatio):
    CONFIG = {
        "base" : 3,
        "base_str" : "3",
    }

class ExponentRatioWithSeven(AnalyzeExponentRatio):
    CONFIG = {
        "base" : 7,
        "base_str" : "7",
    }

class ExponentRatioWithE(AnalyzeExponentRatio):
    CONFIG = {
        "base" : np.exp(1),
        "base_str" : "e",
    }

class AskAboutConstantOne(TeacherStudentsScene):
    def construct(self):
        note = TexMobject(
            "{ d(a^", "t", ")", "\\over \\,", "dt}", 
            "=", "a^", "t", "(\\text{Some constant})"
        )
        note.highlight_by_tex("t", YELLOW)
        note.highlight_by_tex("dt", GREEN)
        note.highlight_by_tex("constant", BLUE)
        note.to_corner(UP+LEFT)
        self.add(note)

        self.student_says(
            "Is there a base where\\\\",
            "that constant is 1?"
        )
        self.change_student_modes(
            "pondering", "raise_right_hand", "thinking",
            # look_at_arg = self.get_students()[1].bubble
        )
        self.dither(2)
        self.play(FadeOut(note[-1], run_time = 3))
        self.dither()

        self.teacher_says(
            "There is!\\\\",
            "$e = 2.71828\\dots$",
            target_mode = "hooray"
        )
        self.change_student_modes(*["confused"]*3)
        self.dither(3)

class NaturalLog(Scene):
    def construct(self):
        expressions = VGroup(*map(self.get_expression, [2, 3, 7]))
        expressions.arrange_submobjects(DOWN, buff = MED_SMALL_BUFF)
        expressions.to_edge(LEFT)

        self.play(FadeIn(
            expressions, 
            run_time = 3, 
            submobject_mode = "lagged_start"
        ))
        self.dither()
        self.play(
            expressions.set_fill, None, 1,
            run_time = 2,
            submobject_mode = "lagged_start"
        )
        self.dither()
        for i in 0, 2, 1:
            self.show_natural_loggedness(expressions[i])

    def show_natural_loggedness(self, expression):
        base, constant = expression[1], expression[-3]

        log_constant, exp_constant = constant.copy(), constant.copy()
        log_base, exp_base = base.copy(), base.copy()
        log_equals, exp_equals = map(TexMobject, "==")

        ln = TexMobject("\\ln(2)")
        log_base.move_to(ln[-2])        
        ln.remove(ln[-2])
        log_equals.next_to(ln, LEFT)
        log_constant.next_to(log_equals, LEFT)
        log_expression = VGroup(
            ln, log_constant, log_equals, log_base
        )

        e = TexMobject("e")
        exp_constant.scale(0.7)
        exp_constant.next_to(e, UP+RIGHT, buff = 0)
        exp_base.next_to(exp_equals, RIGHT)
        VGroup(exp_base, exp_equals).next_to(
            VGroup(e, exp_constant), RIGHT, 
            aligned_edge = DOWN
        )
        exp_expression = VGroup(
            e, exp_constant, exp_equals, exp_base
        )

        for group, vect in (log_expression, UP), (exp_expression, DOWN):
            group.to_edge(RIGHT)
            group.shift(vect)

        self.play(
            ReplacementTransform(base.copy(), log_base),
            ReplacementTransform(constant.copy(), log_constant),
            run_time = 2
        )
        self.play(Write(ln), Write(log_equals))
        self.dither()
        self.play(
            ReplacementTransform(
                log_expression.copy(),
                exp_expression,
                run_time = 2,
            )
        )
        self.dither(2)

        ln_a = expression[-1]
        self.play(
            FadeOut(expression[-2]),
            FadeOut(constant),
            ln_a.move_to, constant, LEFT,
            ln_a.highlight, BLUE
        )
        self.dither()
        self.play(*map(FadeOut, [log_expression, exp_expression]))
        self.dither()

    def get_expression(self, base):
        expression = TexMobject(
            "{d(", "%d^"%base, "t", ")", "\\over \\,", "dt}",
            "=", "%d^"%base, "t", "(%.4f\\dots)"%np.log(base),
        )
        expression.highlight_by_tex("t", YELLOW)
        expression.highlight_by_tex("dt", GREEN)
        expression.highlight_by_tex("\\dots", BLUE)

        brace = Brace(expression.get_part_by_tex("\\dots"), UP)
        brace_text = brace.get_text("$\\ln(%d)$"%base)
        for mob in brace, brace_text:
            mob.set_fill(opacity = 0)

        expression.add(brace, brace_text)
        return expression

class NextVideo(TeacherStudentsScene):
    def construct(self):
        series = VideoSeries()
        series.to_edge(UP)
        this_video = series[3]
        next_video = series[4]
        brace = Brace(this_video, DOWN)
        this_video.save_state()
        this_video.highlight(YELLOW)

        this_tex = TexMobject(
            "{d(", "a^t", ") \\over dt} = ", "a^t", "\\ln(a)"
        )
        this_tex[1][1].highlight(YELLOW)
        this_tex[3][1].highlight(YELLOW)
        this_tex.next_to(brace, DOWN)

        next_tex = VGroup(*map(TextMobject, [
            "Chain rule", "Product rule", "$\\vdots$"
        ]))
        next_tex.arrange_submobjects(DOWN)
        next_tex.next_to(brace, DOWN)
        next_tex.shift(
            next_video.get_center()[0]*RIGHT\
            -next_tex.get_center()[0]*RIGHT
        )

        self.add(series, brace, *this_tex[:3])
        self.change_student_modes(
            "confused", "pondering", "erm",
            look_at_arg = this_tex
        )
        self.play(ReplacementTransform(
            this_tex[1].copy(), this_tex[3]
        ))
        self.dither()
        self.play(
            Write(this_tex[4]),
            ReplacementTransform(
                this_tex[3][0].copy(),
                this_tex[4][3],
                path_arc = np.pi,
                remover = True
            )
        )
        self.dither(2)
        self.play(this_tex.replace, this_video)
        self.play(
            brace.next_to, next_video, DOWN,
            this_video.restore,
            Animation(this_tex),
            next_video.highlight, YELLOW,
            Write(next_tex),
            self.get_teacher().change_mode, "raise_right_hand"
        )
        self.change_student_modes(
            *["pondering"]*3,
            look_at_arg = next_tex
        )
        self.dither(3)