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manim/eoc/chapter10.py
Grant Sanderson c96222fec0 Up to SecondTermIntuition in eoc10
2017-04-27 12:01:33 -07:00

2935 lines
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Python
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from helpers import *
import scipy
import math
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 eoc.graph_scene import GraphScene
from eoc.chapter8 import AreaIsDerivative
from topics.common_scenes import OpeningQuote, PatreonThanks
def derivative(func, x, n = 1, dx = 0.01):
samples = [func(x + (k - n/2)*dx) for k in range(n+1)]
while len(samples) > 1:
samples = [
(s_plus_dx - s)/dx
for s, s_plus_dx in zip(samples, samples[1:])
]
return samples[0]
def taylor_approximation(func, highest_term, center_point = 0):
derivatives = [
derivative(func, center_point, n = n)
for n in range(highest_term + 1)
]
coefficients = [
d/math.factorial(n)
for n, d in enumerate(derivatives)
]
return lambda x : sum([
c*((x-center_point)**n)
for n, c in enumerate(coefficients)
])
class Chapter10OpeningQuote(OpeningQuote):
CONFIG = {
"quote" : [
"For me, mathematics is a collection of ",
"examples", "; a ",
"theorem", " is a statement about a collection of ",
"examples", " and the purpose of proving ",
"theorems", " is to classify and explain the ",
"examples", "."
],
"quote_arg_separator" : "",
"highlighted_quote_terms" : {
"examples" : BLUE,
},
"author" : "John B. Conway",
"fade_in_kwargs" : {
"run_time" : 7,
}
}
class ExampleApproximation(GraphScene):
CONFIG = {
"function" : lambda x : np.exp(-x**2),
"function_tex" : "e^{-x^2}",
"function_color" : BLUE,
"order_sequence" : [0, 2, 4],
"center_point" : 0,
"approximation_terms" : ["1 ", "-x^2", "+\\frac{1}{2}x^4"],
"approximation_color" : GREEN,
"x_min" : -3,
"x_max" : 3,
"y_min" : -1,
"y_max" : 2,
"graph_origin" : DOWN + 2*LEFT,
}
def construct(self):
self.setup_axes()
func_graph = self.get_graph(
self.function,
self.function_color,
)
approx_graphs = [
self.get_graph(
taylor_approximation(self.function, n),
self.approximation_color
)
for n in self.order_sequence
]
near_text = TextMobject(
"Near %s $= %d$"%(
self.x_axis_label, self.center_point
)
)
near_text.to_corner(UP + RIGHT)
near_text.add_background_rectangle()
equation = TexMobject(
self.function_tex,
"\\approx",
*self.approximation_terms
)
equation.next_to(near_text, DOWN, MED_LARGE_BUFF)
equation.to_edge(RIGHT)
near_text.next_to(equation, UP, MED_LARGE_BUFF)
equation.highlight_by_tex(
self.function_tex, self.function_color,
substring = False
)
approx_terms = VGroup(*[
equation.get_part_by_tex(tex, substring = False)
for tex in self.approximation_terms
])
approx_terms.set_fill(
self.approximation_color,
opacity = 0,
)
equation.add_background_rectangle()
approx_graph = VectorizedPoint(
self.input_to_graph_point(self.center_point, func_graph)
)
self.play(
ShowCreation(func_graph, run_time = 2),
Animation(equation),
Animation(near_text),
)
for graph, term in zip(approx_graphs, approx_terms):
self.play(
Transform(approx_graph, graph, run_time = 2),
Animation(equation),
Animation(near_text),
term.set_fill, None, 1,
)
self.dither()
self.dither(2)
class ExampleApproximationWithSine(ExampleApproximation):
CONFIG = {
"function" : np.sin,
"function_tex" : "\\sin(x)",
"order_sequence" : [1, 3, 5],
"center_point" : 0,
"approximation_terms" : [
"x",
"-\\frac{1}{6}x^3",
"+\\frac{1}{120}x^5",
],
"approximation_color" : GREEN,
"x_min" : -2*np.pi,
"x_max" : 2*np.pi,
"x_tick_frequency" : np.pi/2,
"y_min" : -2,
"y_max" : 2,
"graph_origin" : DOWN + 2*LEFT,
}
class ExampleApproximationWithExp(ExampleApproximation):
CONFIG = {
"function" : np.exp,
"function_tex" : "e^x",
"order_sequence" : [1, 2, 3, 4],
"center_point" : 0,
"approximation_terms" : [
"1 + x",
"+\\frac{1}{2}x^2",
"+\\frac{1}{6}x^3",
"+\\frac{1}{24}x^4",
],
"approximation_color" : GREEN,
"x_min" : -3,
"x_max" : 4,
"y_min" : -1,
"y_max" : 10,
"graph_origin" : 2*DOWN + 3*LEFT,
}
class Pendulum(ReconfigurableScene):
CONFIG = {
"anchor_point" : 3*UP + 4*LEFT,
"radius" : 4,
"weight_radius" : 0.2,
"angle" : np.pi/6,
}
def construct(self):
self.draw_pendulum()
self.show_oscillation()
self.show_height()
self.get_angry_at_cosine()
self.substitute_approximation()
self.show_confusion()
def draw_pendulum(self):
pendulum = self.get_pendulum()
ceiling = self.get_ceiling()
self.add(ceiling)
self.play(ShowCreation(pendulum.line))
self.play(DrawBorderThenFill(pendulum.weight, run_time = 1))
self.pendulum = pendulum
def show_oscillation(self):
trajectory_dots = self.get_trajectory_dots()
kwargs = self.get_swing_kwargs()
self.play(
ShowCreation(
trajectory_dots,
rate_func = None,
run_time = kwargs["run_time"]
),
Rotate(self.pendulum, -2*self.angle, **kwargs),
)
for m in 2, -2, 2:
self.play(Rotate(self.pendulum, m*self.angle, **kwargs))
self.dither()
def show_height(self):
v_line = self.get_v_line()
h_line = self.get_h_line()
radius_brace = self.get_radius_brace()
height_brace = self.get_height_brace()
height_tex = self.get_height_brace_tex(height_brace)
arc, theta = self.get_arc_and_theta()
height_tex_R = height_tex.get_part_by_tex("R")
height_tex_theta = height_tex.get_part_by_tex("\\theta")
to_write = VGroup(*[
part
for part in height_tex
if part not in [height_tex_R, height_tex_theta]
])
self.play(
ShowCreation(h_line),
GrowFromCenter(height_brace)
)
self.play(
ShowCreation(v_line),
ShowCreation(arc),
Write(theta),
)
self.play(
GrowFromCenter(radius_brace)
)
self.dither(2)
self.play(
Write(to_write),
ReplacementTransform(
radius_brace[-1].copy(),
height_tex_R
),
ReplacementTransform(
theta.copy(),
height_tex_theta
),
run_time = 2
)
self.dither(2)
self.arc = arc
self.theta = theta
self.height_tex_R = height_tex_R
self.cosine = VGroup(*[
height_tex.get_part_by_tex(tex)
for tex in "cos", "theta", ")"
])
self.one_minus = VGroup(*[
height_tex.get_part_by_tex(tex)
for tex in "\\big(1-", "\\big)"
])
def get_angry_at_cosine(self):
cosine = self.cosine
morty = Mortimer()
morty.to_corner(DOWN+RIGHT)
cosine.generate_target()
cosine.save_state()
cosine.target.next_to(morty, UP)
self.play(FadeIn(morty))
self.play(
MoveToTarget(cosine),
morty.change, "angry", cosine.target,
)
self.dither()
self.play(Blink(morty))
self.dither()
self.morty = morty
def substitute_approximation(self):
morty = self.morty
cosine = self.cosine
cosine.generate_target()
cosine_approx = self.get_cosine_approx()
cosine_approx.next_to(cosine, UP+RIGHT)
cosine_approx.to_edge(RIGHT)
cosine.target.next_to(
cosine_approx, LEFT,
align_using_submobjects = True
)
kwargs = self.get_swing_kwargs()
self.play(
FadeIn(
cosine_approx,
run_time = 2,
submobject_mode = "lagged_start"
),
MoveToTarget(cosine),
morty.change, "pondering", cosine_approx
)
self.dither()
self.play(
ApplyMethod(
cosine_approx.theta_squared_over_two.copy().next_to,
self.height_tex_R,
run_time = 2,
),
FadeOut(self.one_minus),
morty.look_at, self.height_tex_R,
)
self.play(morty.change, "thinking", self.height_tex_R)
self.transition_to_alt_config(
angle = np.pi/12,
transformation_kwargs = {"run_time" : 2},
)
def show_confusion(self):
randy = Randolph(color = BLUE_C)
randy.scale(0.8)
randy.next_to(self.cosine, DOWN+LEFT)
randy.to_edge(DOWN)
self.play(FadeIn(randy))
self.play(
randy.change, "confused", self.cosine
)
self.play(randy.look_at, self.height_tex_R)
self.dither()
self.play(randy.look_at, self.cosine)
self.play(Blink(randy))
self.dither()
#######
def get_pendulum(self):
line = Line(
self.anchor_point,
self.anchor_point + self.radius*DOWN,
color = WHITE
)
weight = Circle(
radius = self.weight_radius,
fill_color = GREY,
fill_opacity = 1,
stroke_width = 0,
)
weight.move_to(line.get_end())
result = VGroup(line, weight)
result.rotate(
self.angle,
about_point = self.anchor_point
)
result.line = line
result.weight = weight
return result
def get_ceiling(self):
line = Line(LEFT, RIGHT, color = GREY)
line.scale(SPACE_WIDTH)
line.move_to(self.anchor_point[1]*UP)
return line
def get_trajectory_dots(self, n_dots = 40, color = YELLOW):
arc_angle = np.pi/6
proportions = self.swing_rate_func(
np.linspace(0, 1, n_dots)
)
angles = -2*arc_angle*proportions
angle_to_point = lambda a : np.cos(a)*RIGHT + np.sin(a)*UP
dots = VGroup(*[
# Line(*map(angle_to_point, pair))
Dot(angle_to_point(angle), radius = 0.005)
for angle in angles
])
dots.highlight(color)
dots.scale(self.radius)
dots.rotate(-np.pi/2 + arc_angle)
dots.shift(self.anchor_point)
return dots
def get_v_line(self):
return DashedLine(
self.anchor_point,
self.anchor_point + self.radius*DOWN,
color = WHITE
)
def get_h_line(self, color = BLUE):
start = self.anchor_point + self.radius*DOWN
end = start + self.radius*np.sin(self.angle)*RIGHT
return Line(start, end, color = color)
def get_radius_brace(self):
v_line = self.get_v_line()
brace = Brace(v_line, RIGHT)
brace.rotate(self.angle, about_point = self.anchor_point)
brace.add(brace.get_text("$R$", buff = SMALL_BUFF))
return brace
def get_height_brace(self):
h_line = self.get_h_line()
height = (1 - np.cos(self.angle))*self.radius
line = Line(
h_line.get_end(),
h_line.get_end() + height*UP,
)
brace = Brace(line, RIGHT)
return brace
def get_height_brace_tex(self, brace):
tex_mob = TexMobject(
"R", "\\big(1-", "\\cos(", "\\theta", ")", "\\big)"
)
tex_mob.highlight_by_tex("theta", YELLOW)
tex_mob.next_to(brace, RIGHT)
return tex_mob
def get_arc_and_theta(self):
arc = Arc(
start_angle = -np.pi/2,
angle = self.angle,
color = YELLOW
)
theta = TexMobject("\\theta")
theta.highlight(YELLOW)
theta.next_to(
arc.point_from_proportion(0.5),
DOWN, SMALL_BUFF
)
for mob in arc, theta:
mob.shift(self.anchor_point)
return arc, theta
def get_cosine_approx(self):
approx = TexMobject(
"\\approx 1 - ", "{\\theta", "^2", "\\over", "2}"
)
approx.highlight_by_tex("theta", YELLOW)
approx.theta_squared_over_two = VGroup(*approx[-4:])
return approx
def get_swing_kwargs(self):
return {
"about_point" : self.anchor_point,
"run_time" : 1.7,
"rate_func" : self.swing_rate_func,
}
def swing_rate_func(self, t):
return (1-np.cos(np.pi*t))/2.0
class ExampleApproximationWithCos(ExampleApproximationWithSine):
CONFIG = {
"function" : np.cos,
"function_tex" : "\\cos(\\theta)",
"order_sequence" : [0, 2],
"approximation_terms" : [
"1",
"-\\frac{1}{2} \\theta ^2",
],
"x_axis_label" : "$\\theta$",
"y_axis_label" : "",
"x_axis_width" : 13,
"graph_origin" : DOWN,
}
def construct(self):
ExampleApproximationWithSine.construct(self)
randy = Randolph(color = BLUE_C)
randy.to_corner(DOWN+LEFT)
high_graph = self.get_graph(lambda x : 4)
v_lines, alt_v_lines = [
VGroup(*[
self.get_vertical_line_to_graph(
u*dx, high_graph,
line_class = DashedLine,
color = YELLOW
)
for u in -1, 1
])
for dx in 0.01, 0.7
]
self.play(*map(ShowCreation, v_lines), run_time = 2)
self.play(Transform(
v_lines, alt_v_lines,
run_time = 2,
))
self.play(FadeIn(randy))
self.play(PiCreatureBubbleIntroduction(
randy, "How...?",
bubble_class = ThoughtBubble,
look_at_arg = self.graph_origin,
target_mode = "confused"
))
self.dither(2)
self.play(Blink(randy))
self.dither()
def setup_axes(self):
GraphScene.setup_axes(self)
x_val_label_pairs = [
(-np.pi, "-\\pi"),
(np.pi, "\\pi"),
(2*np.pi, "2\\pi"),
]
self.x_axis_labels = VGroup()
for x_val, label in x_val_label_pairs:
tex = TexMobject(label)
tex.next_to(self.coords_to_point(x_val, 0), DOWN)
self.add(tex)
self.x_axis_labels.add(tex)
class ConstructQuadraticApproximation(ExampleApproximationWithCos):
CONFIG = {
"x_axis_label" : "$x$",
"colors" : [BLUE, YELLOW, GREEN],
}
def construct(self):
self.setup_axes()
self.add_cosine_graph()
self.add_quadratic_graph()
self.introduce_quadratic_constants()
self.show_value_at_zero()
self.set_c0_to_one()
self.let_c1_and_c2_vary()
self.show_tangent_slope()
self.compute_cosine_derivative()
self.compute_polynomial_derivative()
self.let_c2_vary()
self.point_out_negative_concavity()
self.compute_cosine_second_derivative()
self.show_matching_curvature()
self.show_matching_tangent_lines()
self.compute_polynomial_second_derivative()
self.box_final_answer()
def add_cosine_graph(self):
cosine_label = TexMobject("\\cos(x)")
cosine_label.to_corner(UP+LEFT)
cosine_graph = self.get_graph(np.cos)
dot = Dot(color = WHITE)
dot.move_to(cosine_label)
for mob in cosine_label, cosine_graph:
mob.highlight(self.colors[0])
def update_dot(dot):
dot.move_to(cosine_graph.points[-1])
return dot
self.play(Write(cosine_label, run_time = 1))
self.play(dot.move_to, cosine_graph.points[0])
self.play(
ShowCreation(cosine_graph),
UpdateFromFunc(dot, update_dot),
run_time = 4
)
self.play(FadeOut(dot))
self.cosine_label = cosine_label
self.cosine_graph = cosine_graph
def add_quadratic_graph(self):
quadratic_graph = self.get_quadratic_graph()
self.play(ReplacementTransform(
self.cosine_graph.copy(),
quadratic_graph,
run_time = 3
))
self.quadratic_graph = quadratic_graph
def introduce_quadratic_constants(self):
quadratic_tex = self.get_quadratic_tex("c_0", "c_1", "c_2")
const_terms = quadratic_tex.get_parts_by_tex("c")
free_to_change = TextMobject("Free to change")
free_to_change.next_to(const_terms, DOWN, LARGE_BUFF)
arrows = VGroup(*[
Arrow(
free_to_change.get_top(),
const.get_bottom(),
tip_length = 0.75*Arrow.CONFIG["tip_length"],
color = const.get_color()
)
for const in const_terms
])
alt_consts_list = [
(0, -1, -0.25),
(1, -1, -0.25),
(1, 0, -0.25),
(),
]
self.play(FadeIn(
quadratic_tex,
run_time = 3,
submobject_mode = "lagged_start"
))
self.play(
FadeIn(free_to_change),
*map(ShowCreation, arrows)
)
self.play(*[
ApplyMethod(
const.scale_in_place, 0.8,
run_time = 2,
rate_func = squish_rate_func(there_and_back, a, a + 0.75)
)
for const, a in zip(const_terms, np.linspace(0, 0.25, len(const_terms)))
])
for alt_consts in alt_consts_list:
self.change_quadratic_graph(
self.quadratic_graph, *alt_consts
)
self.dither()
self.quadratic_tex = quadratic_tex
self.free_to_change_group = VGroup(free_to_change, *arrows)
self.free_to_change_group.arrows = arrows
def show_value_at_zero(self):
arrow, x_equals_0 = ax0_group = self.get_arrow_x_equals_0_group()
ax0_group.next_to(
self.cosine_label, RIGHT,
align_using_submobjects = True
)
one = TexMobject("1")
one.next_to(arrow, RIGHT)
one.save_state()
one.move_to(self.cosine_label)
one.set_fill(opacity = 0)
v_line = self.get_vertical_line_to_graph(
0, self.cosine_graph,
line_class = DashedLine,
color = YELLOW
)
self.play(ShowCreation(v_line))
self.play(
ShowCreation(arrow),
Write(x_equals_0, run_time = 2)
)
self.play(one.restore)
self.dither()
self.v_line = v_line
self.equals_one_group = VGroup(arrow, x_equals_0, one)
def set_c0_to_one(self):
poly_at_zero = self.get_quadratic_tex(
"c_0", "c_1", "c_2", arg = "0"
)
poly_at_zero.next_to(self.quadratic_tex, DOWN)
equals_c0 = TexMobject("=", "c_0", "+0")
equals_c0.highlight_by_tex("c_0", self.colors[0])
equals_c0.next_to(
poly_at_zero.get_part_by_tex("="), DOWN,
buff = MED_LARGE_BUFF,
aligned_edge = LEFT
)
poly_group = VGroup(
equals_c0,
poly_at_zero,
self.quadratic_tex,
)
poly_group_target = VGroup(
TexMobject("=", "1", "+0").highlight_by_tex("1", self.colors[0]),
self.get_quadratic_tex("1", "c_1", "c_2", arg = "0"),
self.get_quadratic_tex("1", "c_1", "c_2"),
)
for start, target in zip(poly_group, poly_group_target):
target.move_to(start)
self.play(FadeOut(self.free_to_change_group))
self.play(ReplacementTransform(
self.quadratic_tex.copy(),
poly_at_zero
))
self.dither(2)
self.play(FadeIn(equals_c0))
self.dither(2)
self.play(Transform(
poly_group, poly_group_target,
run_time = 2,
submobject_mode = "lagged_start"
))
self.dither(2)
self.play(*map(FadeOut, [poly_at_zero, equals_c0]))
self.free_to_change_group.remove(
self.free_to_change_group.arrows[0]
)
self.play(FadeIn(self.free_to_change_group))
def let_c1_and_c2_vary(self):
alt_consts_list = [
(1, 1, -0.25),
(1, -1, -0.25),
(1, -1, 0.25),
(1, 1, -0.1),
]
for alt_consts in alt_consts_list:
self.change_quadratic_graph(
self.quadratic_graph,
*alt_consts
)
self.dither()
def show_tangent_slope(self):
graph_point_at_zero = self.input_to_graph_point(
0, self.cosine_graph
)
tangent_line = self.get_tangent_line(0, self.cosine_graph)
self.play(ShowCreation(tangent_line))
self.change_quadratic_graph(
self.quadratic_graph, 1, 0, -0.1
)
self.dither()
self.change_quadratic_graph(
self.quadratic_graph, 1, 1, -0.1
)
self.dither(2)
self.change_quadratic_graph(
self.quadratic_graph, 1, 0, -0.1
)
self.dither(2)
self.tangent_line = tangent_line
def compute_cosine_derivative(self):
derivative, rhs = self.get_cosine_derivative()
self.play(FadeIn(
VGroup(derivative, *rhs[:2]),
run_time = 2,
submobject_mode = "lagged_start"
))
self.dither(2)
self.play(Write(VGroup(*rhs[2:])), run_time = 2)
self.dither()
self.play(Rotate(
self.tangent_line, np.pi/12,
in_place = True,
run_time = 3,
rate_func = wiggle
))
self.dither()
def compute_polynomial_derivative(self):
derivative = self.get_quadratic_derivative("c_1", "c_2")
derivative_at_zero = self.get_quadratic_derivative(
"c_1", "c_2", arg = "0"
)
equals_c1 = TexMobject("=", "c_1", "+0")
equals_c1.next_to(
derivative_at_zero.get_part_by_tex("="), DOWN,
buff = MED_LARGE_BUFF,
aligned_edge = LEFT,
)
equals_c1.highlight_by_tex("c_1", self.colors[1])
poly_group = VGroup(
equals_c1,
derivative,
self.quadratic_tex
)
poly_group_target = VGroup(
TexMobject("=", "0", "+0").highlight_by_tex(
"0", self.colors[1], substring = False
),
self.get_quadratic_derivative("0", "c_2", arg = "0"),
self.get_quadratic_tex("1", "0", "c_2")
)
for start, target in zip(poly_group, poly_group_target):
target.move_to(start)
self.play(FadeOut(self.free_to_change_group))
self.play(FadeIn(
derivative,
run_time = 3,
submobject_mode = "lagged_start"
))
self.dither()
self.play(Transform(
derivative, derivative_at_zero,
run_time = 2,
submobject_mode = "lagged_start"
))
self.dither(2)
self.play(Write(equals_c1))
self.dither(2)
self.play(Transform(
poly_group, poly_group_target,
run_time = 3,
submobject_mode = "lagged_start"
))
self.dither(2)
self.play(*map(FadeOut, poly_group[:-1]))
self.free_to_change_group.remove(
self.free_to_change_group.arrows[1]
)
self.play(FadeIn(self.free_to_change_group))
def let_c2_vary(self):
alt_c2_values = [-1, -0.05, 1, -0.2]
for alt_c2 in alt_c2_values:
self.change_quadratic_graph(
self.quadratic_graph,
1, 0, alt_c2
)
self.dither()
def point_out_negative_concavity(self):
partial_cosine_graph = self.get_graph(
np.cos,
x_min = -1,
x_max = 1,
color = PINK
)
self.play(ShowCreation(partial_cosine_graph, run_time = 2))
self.dither()
for x, run_time in (-1, 2), (1, 4):
self.play(self.get_tangent_line_change_anim(
self.tangent_line, x, self.cosine_graph,
run_time = run_time
))
self.dither()
self.play(*map(FadeOut, [
partial_cosine_graph, self.tangent_line
]))
def compute_cosine_second_derivative(self):
second_deriv, rhs = self.get_cosine_second_derivative()
self.play(FadeIn(
VGroup(second_deriv, *rhs[1][:2]),
run_time = 2,
submobject_mode = "lagged_start"
))
self.dither(3)
self.play(Write(VGroup(*rhs[1][2:]), run_time = 2))
self.dither()
def show_matching_curvature(self):
alt_consts_list = [
(1, 1, -0.2),
(1, 0, -0.2),
(1, 0, -0.5),
]
for alt_consts in alt_consts_list:
self.change_quadratic_graph(
self.quadratic_graph,
*alt_consts
)
self.dither()
def show_matching_tangent_lines(self):
graphs = [self.quadratic_graph, self.cosine_graph]
tangent_lines = [
self.get_tangent_line(0, graph, color = color)
for graph, color in zip(graphs, [WHITE, YELLOW])
]
tangent_change_anims = [
self.get_tangent_line_change_anim(
line, np.pi/2, graph,
run_time = 6,
rate_func = there_and_back,
)
for line, graph in zip(tangent_lines, graphs)
]
self.play(*map(ShowCreation, tangent_lines))
self.play(*tangent_change_anims)
self.play(*map(FadeOut, tangent_lines))
def compute_polynomial_second_derivative(self):
c2s = ["c_2", "\\text{\\tiny $\\left(-\\frac{1}{2}\\right)$}"]
derivs = [
self.get_quadratic_derivative("0", c2)
for c2 in c2s
]
second_derivs = [
TexMobject(
"{d^2 P \\over dx^2}", "(x)", "=", "2", c2
)
for c2 in c2s
]
for deriv, second_deriv in zip(derivs, second_derivs):
second_deriv[0].scale(
0.7, about_point = second_deriv[0].get_right()
)
second_deriv[-1].highlight(self.colors[-1])
second_deriv.next_to(
deriv, DOWN,
buff = MED_LARGE_BUFF,
aligned_edge = LEFT
)
poly_group = VGroup(
second_derivs[0],
derivs[0],
self.quadratic_tex
)
poly_group_target = VGroup(
second_derivs[1],
derivs[1],
self.get_quadratic_tex("1", "0", c2s[1])
)
for tex_mob in poly_group_target:
tex_mob.get_part_by_tex(c2s[1]).shift(SMALL_BUFF*UP)
self.play(FadeOut(self.free_to_change_group))
self.play(FadeIn(derivs[0]))
self.dither(2)
self.play(Write(second_derivs[0]))
self.dither(2)
self.play(Transform(
poly_group, poly_group_target,
run_time = 3,
submobject_mode = "lagged_start"
))
self.dither(3)
def box_final_answer(self):
box = Rectangle(stroke_color = PINK)
box.stretch_to_fit_width(
self.quadratic_tex.get_width() + MED_LARGE_BUFF
)
box.stretch_to_fit_height(
self.quadratic_tex.get_height() + MED_LARGE_BUFF
)
box.move_to(self.quadratic_tex)
self.play(ShowCreation(box, run_time = 2))
self.dither(2)
######
def change_quadratic_graph(self, graph, *args, **kwargs):
transformation_kwargs = {}
transformation_kwargs["run_time"] = kwargs.pop("run_time", 2)
transformation_kwargs["rate_func"] = kwargs.pop("rate_func", smooth)
new_graph = self.get_quadratic_graph(*args, **kwargs)
self.play(Transform(graph, new_graph, **transformation_kwargs))
graph.underlying_function = new_graph.underlying_function
def get_quadratic_graph(self, c0 = 1, c1 = 0, c2 = -0.5):
return self.get_graph(
lambda x : c0 + c1*x + c2*x**2,
color = self.colors[2]
)
def get_quadratic_tex(self, c0, c1, c2, arg = "x"):
tex_mob = TexMobject(
"P(", arg, ")", "=",
c0, "+", c1, arg, "+", c2, arg, "^2"
)
for tex, color in zip([c0, c1, c2], self.colors):
tex_mob.highlight_by_tex(tex, color)
tex_mob.to_corner(UP+RIGHT)
return tex_mob
def get_quadratic_derivative(self, c1, c2, arg = "x"):
result = TexMobject(
"{dP \\over dx}", "(", arg, ")", "=",
c1, "+", "2", c2, arg
)
result[0].scale(0.7, about_point = result[0].get_right())
for index, color in zip([5, 8], self.colors[1:]):
result[index].highlight(color)
if hasattr(self, "quadratic_tex"):
result.next_to(
self.quadratic_tex, DOWN,
buff = MED_LARGE_BUFF,
aligned_edge = LEFT
)
return result
def get_arrow_x_equals_0_group(self):
arrow = Arrow(LEFT, RIGHT)
x_equals_0 = TexMobject("x = 0")
x_equals_0.scale(0.75)
x_equals_0.next_to(arrow.get_center(), UP, 2*SMALL_BUFF)
x_equals_0.shift(SMALL_BUFF*LEFT)
return VGroup(arrow, x_equals_0)
def get_tangent_line(self, x, graph, color = YELLOW):
tangent_line = Line(LEFT, RIGHT, color = color)
tangent_line.rotate(self.angle_of_tangent(x, graph))
tangent_line.scale(2)
tangent_line.move_to(self.input_to_graph_point(x, graph))
return tangent_line
def get_tangent_line_change_anim(self, tangent_line, new_x, graph, **kwargs):
start_x = self.x_axis.point_to_number(
tangent_line.get_center()
)
def update(tangent_line, alpha):
x = interpolate(start_x, new_x, alpha)
new_line = self.get_tangent_line(
x, graph, color = tangent_line.get_color()
)
Transform(tangent_line, new_line).update(1)
return tangent_line
return UpdateFromAlphaFunc(tangent_line, update, **kwargs)
def get_cosine_derivative(self):
if not hasattr(self, "cosine_label"):
self.cosine_label = TexMobject("\\cos(x)")
self.cosine_label.to_corner(UP+LEFT)
derivative = TexMobject(
"{d(", "\\cos", ")", "\\over", "dx}", "(0)",
)
derivative.highlight_by_tex("\\cos", self.colors[0])
derivative.scale(0.7)
derivative.next_to(
self.cosine_label, DOWN,
buff = MED_LARGE_BUFF,
aligned_edge = LEFT
)
rhs = TexMobject("=", "-\\sin(0)", "=", "0")
rhs.highlight_by_tex("\\sin", self.colors[1])
rhs.scale(0.75)
rhs.next_to(
derivative, RIGHT,
align_using_submobjects = True
)
self.cosine_derivative = VGroup(derivative, rhs)
return self.cosine_derivative
def get_cosine_second_derivative(self):
if not hasattr(self, "cosine_derivative"):
self.get_cosine_derivative()
second_deriv = TexMobject(
"{d^2(", "\\cos", ")", "\\over", "dx^2}",
"(", "0", ")",
)
second_deriv.highlight_by_tex("cos", self.colors[0])
second_deriv.highlight_by_tex("-\\cos", self.colors[2])
second_deriv.scale(0.75)
second_deriv.add_background_rectangle()
second_deriv.next_to(
self.cosine_derivative, DOWN,
buff = MED_LARGE_BUFF,
aligned_edge = LEFT
)
rhs = TexMobject("=", "-\\cos(0)", "=", "-1")
rhs.highlight_by_tex("cos", self.colors[2])
rhs.scale(0.8)
rhs.next_to(
second_deriv, RIGHT,
align_using_submobjects = True
)
rhs.add_background_rectangle()
self.cosine_second_derivative = VGroup(second_deriv, rhs)
return self.cosine_second_derivative
class ReflectOnQuadraticApproximation(TeacherStudentsScene):
def construct(self):
self.show_example_approximation()
self.add_polynomial()
self.show_c0()
self.show_c1()
self.show_c2()
def show_example_approximation(self):
approx_at_x, approx_at_point = [
TexMobject(
"\\cos(", s, ")", "\\approx",
"1 - \\frac{1}{2}", "(", s, ")", "^2"
).next_to(self.get_students(), UP, 2)
for s in "x", "0.1",
]
approx_rhs = TexMobject("=", "0.995")
approx_rhs.next_to(approx_at_point, RIGHT)
real_result = TexMobject(
"\\cos(", "0.1", ")", "=",
"%.7f"%np.cos(0.1)
)
real_result.shift(
approx_rhs.get_part_by_tex("=").get_center() -\
real_result.get_part_by_tex("=").get_center()
)
for mob in approx_at_point, real_result:
mob.highlight_by_tex("0.1", YELLOW)
real_result.set_fill(opacity = 0)
self.play(
Write(approx_at_x, run_time = 2),
self.teacher.change_mode, "raise_right_hand"
)
self.dither(2)
self.play(ReplacementTransform(
approx_at_x, approx_at_point,
))
self.dither()
self.play(Write(approx_rhs))
self.dither(2)
self.play(
real_result.shift, 1.5*DOWN,
real_result.set_fill, None, 1,
)
self.change_student_modes(*["hooray"]*3)
self.dither(2)
self.change_student_modes(
*["plain"]*3,
added_anims = map(FadeOut, [
approx_at_point, approx_rhs, real_result
]),
look_at_arg = approx_at_x
)
def add_polynomial(self):
polynomial = self.get_polynomial()
const_terms = polynomial.get_parts_by_tex("c")
self.play(
Write(polynomial),
self.teacher.change, "pondering"
)
self.dither(2)
self.play(*[
ApplyMethod(
const.shift, MED_LARGE_BUFF*UP,
run_time = 2,
rate_func = squish_rate_func(there_and_back, a, a+0.7)
)
for const, a in zip(const_terms, np.linspace(0, 0.3, len(const_terms)))
])
self.dither()
self.const_terms = const_terms
self.polynomial = polynomial
def show_c0(self):
c0 = self.polynomial.get_part_by_tex("c_0")
c0.save_state()
equation = TexMobject("P(0) = \\cos(0)")
equation.to_corner(UP+RIGHT)
new_polynomial = self.get_polynomial(c0 = "1")
self.play(c0.shift, UP)
self.play(Write(equation))
self.dither()
self.play(Transform(self.polynomial, new_polynomial))
self.play(FadeOut(equation))
def show_c1(self):
c1 = self.polynomial.get_part_by_tex("c_1")
c1.save_state()
equation = TexMobject(
"\\frac{dP}{dx}(0) = \\frac{d(\\cos)}{dx}(0)"
)
equation.to_corner(UP+RIGHT)
new_polynomial = self.get_polynomial(c0 = "1", c1 = "0")
self.play(c1.shift, UP)
self.play(Write(equation))
self.dither()
self.play(Transform(self.polynomial, new_polynomial))
self.dither()
self.play(FadeOut(equation))
def show_c2(self):
c2 = self.polynomial.get_part_by_tex("c_2")
c2.save_state()
equation = TexMobject(
"\\frac{d^2 P}{dx^2}(0) = \\frac{d^2(\\cos)}{dx^2}(0)"
)
equation.to_corner(UP+RIGHT)
alt_c2_tex = "\\text{\\tiny $\\left(-\\frac{1}{2}\\right)$}"
new_polynomial = self.get_polynomial(
c0 = "1", c1 = "0", c2 = alt_c2_tex
)
new_polynomial.get_part_by_tex(alt_c2_tex).shift(SMALL_BUFF*UP)
self.play(c2.shift, UP)
self.play(FadeIn(equation))
self.dither(2)
self.play(Transform(self.polynomial, new_polynomial))
self.dither(2)
self.play(FadeOut(equation))
#####
def get_polynomial(self, c0 = "c_0", c1 = "c_1", c2 = "c_2"):
polynomial = TexMobject(
"P(x) = ", c0, "+", c1, "x", "+", c2, "x^2"
)
colors = ConstructQuadraticApproximation.CONFIG["colors"]
for tex, color in zip([c0, c1, c2], colors):
polynomial.highlight_by_tex(tex, color, substring = False)
polynomial.next_to(self.teacher, UP, LARGE_BUFF)
polynomial.to_edge(RIGHT)
return polynomial
class ReflectionOnQuadraticSupplement(ConstructQuadraticApproximation):
def construct(self):
self.setup_axes()
self.add(self.get_graph(np.cos, color = self.colors[0]))
quadratic_graph = self.get_quadratic_graph()
self.add(quadratic_graph)
self.dither()
for c0 in 0, 2, 1:
self.change_quadratic_graph(
quadratic_graph,
c0 = c0
)
self.dither(2)
for c1 in 1, -1, 0:
self.change_quadratic_graph(
quadratic_graph,
c1 = c1
)
self.dither(2)
for c2 in -0.1, -1, -0.5:
self.change_quadratic_graph(
quadratic_graph,
c2 = c2
)
self.dither(2)
class SimilarityOfChangeBehavior(ConstructQuadraticApproximation):
def construct(self):
colors = [YELLOW, WHITE]
max_x = np.pi/2
self.setup_axes()
cosine_graph = self.get_graph(np.cos, color = self.colors[0])
quadratic_graph = self.get_quadratic_graph()
graphs = VGroup(cosine_graph, quadratic_graph)
dots = VGroup()
for graph, color in zip(graphs, colors):
dot = Dot(color = color)
dot.move_to(self.input_to_graph_point(0, graph))
dot.graph = graph
dots.add(dot)
def update_dot(dot, alpha):
x = interpolate(0, max_x, alpha)
dot.move_to(self.input_to_graph_point(x, dot.graph))
dot_anims = [
UpdateFromAlphaFunc(dot, update_dot, run_time = 3)
for dot in dots
]
tangent_lines = VGroup(*[
self.get_tangent_line(0, graph, color)
for graph, color in zip(graphs, colors)
])
tangent_line_movements = [
self.get_tangent_line_change_anim(
line, max_x, graph,
run_time = 5,
)
for line, graph in zip(tangent_lines, graphs)
]
self.add(cosine_graph, quadratic_graph)
self.play(FadeIn(dots))
self.play(*dot_anims)
self.play(
FadeIn(tangent_lines),
FadeOut(dots)
)
self.play(*tangent_line_movements + dot_anims, run_time = 6)
self.play(*map(FadeOut, [tangent_lines, dots]))
self.dither()
class MoreTerms(TeacherStudentsScene):
def construct(self):
self.teacher_says(
"More terms!",
target_mode = "surprised",
)
self.change_student_modes(*["hooray"]*3)
self.dither(3)
class CubicAndQuarticApproximations(ConstructQuadraticApproximation):
CONFIG = {
"colors": [BLUE, YELLOW, GREEN, RED, MAROON_B],
}
def construct(self):
self.add_background()
self.take_third_derivative_of_cubic()
self.show_third_derivative_of_cosine()
self.set_c3_to_zero()
self.show_cubic_curves()
self.add_quartic_term()
self.show_fourth_derivative_of_cosine()
self.take_fourth_derivative_of_quartic()
self.solve_for_c4()
self.show_quartic_approximation()
def add_background(self):
self.setup_axes()
self.cosine_graph = self.get_graph(
np.cos, color = self.colors[0]
)
self.quadratic_graph = self.get_quadratic_graph()
self.big_rect = Rectangle(
height = 2*SPACE_HEIGHT,
width = 2*SPACE_WIDTH,
stroke_width = 0,
fill_color = BLACK,
fill_opacity = 0.5,
)
self.add(
self.cosine_graph, self.quadratic_graph,
self.big_rect
)
self.cosine_label = TexMobject("\\cos", "(0)", "=1")
self.cosine_label.highlight_by_tex("cos", self.colors[0])
self.cosine_label.scale(0.75)
self.cosine_label.to_corner(UP+LEFT)
self.add(self.cosine_label)
self.add(self.get_cosine_derivative())
self.add(self.get_cosine_second_derivative())
self.polynomial = TexMobject(
"P(x)=", "1", "-\\frac{1}{2}", "x^2"
)
self.polynomial.highlight_by_tex("1", self.colors[0])
self.polynomial.highlight_by_tex("-\\frac{1}{2}", self.colors[2])
self.polynomial.to_corner(UP+RIGHT)
self.polynomial.quadratic_part = VGroup(
*self.polynomial[1:]
)
self.add(self.polynomial)
def take_third_derivative_of_cubic(self):
polynomial = self.polynomial
plus_cubic_term = TexMobject("+\\,", "c_3", "x^3")
plus_cubic_term.next_to(polynomial, RIGHT)
plus_cubic_term.to_edge(RIGHT, buff = LARGE_BUFF)
plus_cubic_term.highlight_by_tex("c_3", self.colors[3])
plus_cubic_copy = plus_cubic_term.copy()
polynomial.generate_target()
polynomial.target.next_to(plus_cubic_term, LEFT)
self.play(FocusOn(polynomial))
self.play(
MoveToTarget(polynomial),
GrowFromCenter(plus_cubic_term)
)
self.dither()
brace = Brace(polynomial.quadratic_part, DOWN)
third_derivative = TexMobject(
"\\frac{d^3 P}{dx^3}(x) = ", "0"
)
third_derivative.shift(
brace.get_bottom() + MED_SMALL_BUFF*DOWN -\
third_derivative.get_part_by_tex("0").get_top()
)
self.play(Write(third_derivative[0]))
self.play(GrowFromCenter(brace))
self.play(ReplacementTransform(
polynomial.quadratic_part.copy(),
VGroup(third_derivative[1])
))
self.dither(2)
self.play(plus_cubic_copy.next_to, third_derivative, RIGHT)
derivative_term = self.take_derivatives_of_monomial(
VGroup(*plus_cubic_copy[1:])
)
third_derivative.add(plus_cubic_copy[0], derivative_term)
self.plus_cubic_term = plus_cubic_term
self.polynomial_third_derivative = third_derivative
self.polynomial_third_derivative_brace = brace
def show_third_derivative_of_cosine(self):
cosine_third_derivative = self.get_cosine_third_derivative()
dot = Dot(fill_opacity = 0.5)
dot.move_to(self.polynomial_third_derivative)
self.play(
dot.move_to, cosine_third_derivative,
dot.set_fill, None, 0
)
self.play(ReplacementTransform(
self.cosine_second_derivative.copy(),
cosine_third_derivative
))
self.dither(2)
dot.set_fill(opacity = 0.5)
self.play(
dot.move_to, self.polynomial_third_derivative.get_right(),
dot.set_fill, None, 0,
)
self.dither()
def set_c3_to_zero(self):
c3s = VGroup(
self.polynomial_third_derivative[-1][-1],
self.plus_cubic_term.get_part_by_tex("c_3")
)
zeros = VGroup(*[
TexMobject("0").move_to(c3)
for c3 in c3s
])
zeros.highlight(self.colors[3])
zeros.shift(SMALL_BUFF*UP)
zeros[0].shift(0.25*SMALL_BUFF*(UP+LEFT))
self.play(Transform(
c3s, zeros,
run_time = 2,
submobject_mode = "lagged_start"
))
self.dither(2)
def show_cubic_curves(self):
real_graph = self.quadratic_graph
real_graph.save_state()
graph = real_graph.copy()
graph.save_state()
alt_graphs = [
self.get_graph(func, color = real_graph.get_color())
for func in [
lambda x : x*(x-1)*(x+1),
lambda x : 1 - 0.5*(x**2) + 0.2*(x**3)
]
]
self.play(FadeIn(graph))
real_graph.set_stroke(width = 0)
for alt_graph in alt_graphs:
self.play(Transform(graph, alt_graph, run_time = 2))
self.dither()
self.play(graph.restore, run_time = 2)
real_graph.restore()
self.play(FadeOut(graph))
def add_quartic_term(self):
polynomial = self.polynomial
plus_quartic_term = TexMobject("+\\,", "c_4", "x^4")
plus_quartic_term.next_to(polynomial, RIGHT)
plus_quartic_term.highlight_by_tex("c_4", self.colors[4])
self.play(*map(FadeOut, [
self.plus_cubic_term,
self.polynomial_third_derivative,
self.polynomial_third_derivative_brace,
]))
self.play(Write(plus_quartic_term))
self.dither()
self.plus_quartic_term = plus_quartic_term
def show_fourth_derivative_of_cosine(self):
cosine_fourth_derivative = self.get_cosine_fourth_derivative()
self.play(FocusOn(self.cosine_third_derivative))
self.play(ReplacementTransform(
self.cosine_third_derivative.copy(),
cosine_fourth_derivative
))
self.dither(3)
def take_fourth_derivative_of_quartic(self):
quartic_term = VGroup(*self.plus_quartic_term.copy()[1:])
fourth_deriv_lhs = TexMobject("{d^4 P \\over dx^4}(x)", "=")
fourth_deriv_lhs.next_to(
self.polynomial, DOWN,
buff = MED_LARGE_BUFF,
aligned_edge = LEFT
)
alt_rhs = TexMobject("=", "24 \\cdot", "c_4")
alt_rhs.next_to(
fourth_deriv_lhs.get_part_by_tex("="), DOWN,
buff = LARGE_BUFF,
aligned_edge = LEFT
)
alt_rhs.highlight_by_tex("c_4", self.colors[4])
self.play(Write(fourth_deriv_lhs))
self.play(
quartic_term.next_to, fourth_deriv_lhs, RIGHT
)
self.dither()
fourth_deriv_rhs = self.take_derivatives_of_monomial(quartic_term)
self.dither()
self.play(Write(alt_rhs))
self.dither()
self.fourth_deriv_lhs = fourth_deriv_lhs
self.fourth_deriv_rhs = fourth_deriv_rhs
self.fourth_deriv_alt_rhs = alt_rhs
def solve_for_c4(self):
c4s = VGroup(
self.fourth_deriv_alt_rhs.get_part_by_tex("c_4"),
self.fourth_deriv_rhs[-1],
self.plus_quartic_term.get_part_by_tex("c_4")
)
fraction = TexMobject("\\text{\\small $\\frac{1}{24}$}")
fraction.highlight(self.colors[4])
fractions = VGroup(*[
fraction.copy().move_to(c4, LEFT)
for c4 in c4s
])
fractions.shift(SMALL_BUFF*UP)
x_to_4 = self.plus_quartic_term.get_part_by_tex("x^4")
x_to_4.generate_target()
x_to_4.target.shift(MED_SMALL_BUFF*RIGHT)
self.play(
Transform(
c4s, fractions,
run_time = 3,
submobject_mode = "lagged_start",
),
MoveToTarget(x_to_4, run_time = 2)
)
self.dither(3)
def show_quartic_approximation(self):
real_graph = self.quadratic_graph
graph = real_graph.copy()
quartic_graph = self.get_graph(
lambda x : 1 - (x**2)/2.0 + (x**4)/24.0,
color = graph.get_color(),
)
tex_mobs = VGroup(*[
self.polynomial,
self.fourth_deriv_rhs,
self.fourth_deriv_alt_rhs,
self.cosine_label,
self.cosine_derivative,
self.cosine_second_derivative,
self.cosine_third_derivative[1],
])
for tex_mob in tex_mobs:
tex_mob.add_to_back(BackgroundRectangle(tex_mob))
self.play(FadeIn(graph))
real_graph.set_stroke(width = 0)
self.play(
Transform(
graph, quartic_graph,
run_time = 3,
),
Animation(tex_mobs)
)
self.dither(3)
####
def take_derivatives_of_monomial(self, term, *added_anims):
"""
Must be a group of pure TexMobjects,
last part must be of the form x^n
"""
n = int(term[-1].get_tex_string()[-1])
curr_term = term
added_anims_iter = iter(added_anims)
for k in range(n, 0, -1):
exponent = curr_term[-1][-1]
exponent_copy = exponent.copy()
front_num = TexMobject("%d \\cdot"%k)
front_num.move_to(curr_term[0][0], LEFT)
new_monomial = TexMobject("x^%d"%(k-1))
new_monomial.replace(curr_term[-1])
Transform(curr_term[-1], new_monomial).update(1)
curr_term.generate_target()
curr_term.target.shift(
(front_num.get_width()+SMALL_BUFF)*RIGHT
)
curr_term[-1][-1].set_fill(opacity = 0)
possibly_added_anims = []
try:
possibly_added_anims.append(added_anims_iter.next())
except:
pass
self.play(
ApplyMethod(
exponent_copy.replace, front_num[0],
path_arc = np.pi,
),
Write(
front_num[1],
rate_func = squish_rate_func(smooth, 0.5, 1)
),
MoveToTarget(curr_term),
*possibly_added_anims,
run_time = 2
)
self.remove(exponent_copy)
self.add(front_num)
curr_term = VGroup(front_num, *curr_term)
self.dither()
self.play(FadeOut(curr_term[-1]))
return VGroup(*curr_term[:-1])
def get_cosine_third_derivative(self):
if not hasattr(self, "cosine_second_derivative"):
self.get_cosine_second_derivative()
third_deriv = TexMobject(
"{d^3(", "\\cos", ")", "\\over", "dx^3}",
"(", "0", ")",
)
third_deriv.highlight_by_tex("cos", self.colors[0])
third_deriv.highlight_by_tex("-\\cos", self.colors[3])
third_deriv.scale(0.75)
third_deriv.add_background_rectangle()
third_deriv.next_to(
self.cosine_second_derivative, DOWN,
buff = MED_LARGE_BUFF,
aligned_edge = LEFT
)
rhs = TexMobject("=", "\\sin(0)", "=", "0")
rhs.highlight_by_tex("sin", self.colors[3])
rhs.scale(0.8)
rhs.next_to(
third_deriv, RIGHT,
align_using_submobjects = True
)
rhs.add_background_rectangle()
rhs.background_rectangle.scale_in_place(1.2)
self.cosine_third_derivative = VGroup(third_deriv, rhs)
return self.cosine_third_derivative
def get_cosine_fourth_derivative(self):
if not hasattr(self, "cosine_third_derivative"):
self.get_cosine_third_derivative()
fourth_deriv = TexMobject(
"{d^4(", "\\cos", ")", "\\over", "dx^4}",
"(", "0", ")",
)
fourth_deriv.highlight_by_tex("cos", self.colors[0])
fourth_deriv.scale(0.75)
fourth_deriv.add_background_rectangle()
fourth_deriv.next_to(
self.cosine_third_derivative, DOWN,
buff = MED_LARGE_BUFF,
aligned_edge = LEFT
)
rhs = TexMobject("=", "\\cos(0)", "=", "1")
rhs.highlight_by_tex("cos", self.colors[4])
rhs.scale(0.8)
rhs.next_to(
fourth_deriv, RIGHT,
align_using_submobjects = True
)
rhs.add_background_rectangle()
rhs.background_rectangle.scale_in_place(1.2)
self.cosine_fourth_derivative = VGroup(fourth_deriv, rhs)
return self.cosine_fourth_derivative
class NoticeAFewThings(TeacherStudentsScene):
def construct(self):
self.teacher_says(
"Notice a few things",
target_mode = "hesitant"
)
self.dither(3)
class FactorialTerms(CubicAndQuarticApproximations):
def construct(self):
lhs_list = [
TexMobject(
"{d%s"%s, "\\over", "dx%s}"%s, "(", "c_8", "x^8", ")="
)
for i in range(9)
for s in ["^%d"%i if i > 1 else ""]
]
for lhs in lhs_list:
lhs.highlight_by_tex("c_8", YELLOW)
lhs.next_to(ORIGIN, LEFT)
lhs_list[0].set_fill(opacity = 0)
added_anims = [
ReplacementTransform(
start_lhs, target_lhs,
rate_func = squish_rate_func(smooth, 0, 0.5)
)
for start_lhs, target_lhs in zip(lhs_list, lhs_list[1:])
]
term = TexMobject("c_8", "x^8")
term.next_to(lhs[-1], RIGHT)
term.highlight_by_tex("c_8", YELLOW)
self.add(term)
self.dither()
result = self.take_derivatives_of_monomial(term, *added_anims)
factorial_term = VGroup(*result[:-1])
brace = Brace(factorial_term)
eight_factorial = brace.get_text("$8!$")
coefficient = result[-1]
words = TextMobject(
"Set", "$c_8$",
"$ = \\frac{\\text{Desired derivative value}}{8!}"
)
words.highlight_by_tex("c_8", YELLOW)
words.shift(2*UP)
self.play(
GrowFromCenter(brace),
Write(eight_factorial)
)
self.play(
ReplacementTransform(
coefficient.copy(),
words.get_part_by_tex("c_8")
),
Write(words),
)
self.dither(2)
class HigherTermsDontMessUpLowerTerms(Scene):
CONFIG = {
"colors" : CubicAndQuarticApproximations.CONFIG["colors"][::2],
}
def construct(self):
self.add_polynomial()
self.show_second_derivative()
def add_polynomial(self):
c0_tex = "1"
c2_tex = "\\text{\\small $\\left(-\\frac{1}{2}\\right)$}"
c4_tex = "c_4"
polynomial = TexMobject(
"P(x) = ",
c0_tex, "+",
c2_tex, "x^2", "+",
c4_tex, "x^4",
)
polynomial.shift(2*LEFT + UP)
c0, c2, c4 = [
polynomial.get_part_by_tex(tex)
for tex in c0_tex, c2_tex, c4_tex
]
for term, color in zip([c0, c2, c4], self.colors):
term.highlight(color)
arrows = VGroup(*[
Arrow(
c4.get_top(), c.get_top(),
path_arc = arc,
color = c.get_color()
)
for c, arc in (c2, 0.9*np.pi), (c0, np.pi)
])
no_affect_words = TextMobject(
"Doesn't affect \\\\ previous terms"
)
no_affect_words.next_to(arrows, RIGHT)
no_affect_words.shift(MED_SMALL_BUFF*(UP+LEFT))
self.add(*polynomial[:-2])
self.dither()
self.play(Write(VGroup(*polynomial[-2:])))
self.play(
Write(no_affect_words),
ShowCreation(arrows),
run_time = 3
)
self.dither(2)
self.polynomial = polynomial
self.c0_tex = c0_tex
self.c2_tex = c2_tex
self.c4_tex = c4_tex
def show_second_derivative(self):
second_deriv = TexMobject(
"{d^2 P \\over dx^2}(", "0", ")", "=",
"2", self.c2_tex, "+",
"3 \\cdot 4", self.c4_tex, "(", "0", ")", "^2"
)
second_deriv.highlight_by_tex(self.c2_tex, self.colors[1])
second_deriv.highlight_by_tex(self.c4_tex, self.colors[2])
second_deriv.highlight_by_tex("0", YELLOW)
second_deriv.next_to(
self.polynomial, DOWN,
buff = MED_LARGE_BUFF,
aligned_edge = LEFT
)
higher_terms = VGroup(*second_deriv[-6:])
brace = Brace(higher_terms, DOWN)
equals_zero = brace.get_text("=0")
second_deriv.save_state()
second_deriv.move_to(self.polynomial, LEFT)
second_deriv.set_fill(opacity = 0)
self.play(second_deriv.restore)
self.dither()
self.play(GrowFromCenter(brace))
self.dither()
self.play(Write(equals_zero))
self.dither(3)
class ApproximateNearNewPoint(CubicAndQuarticApproximations):
CONFIG = {
"target_approx_centers" : [-np.pi/2, np.pi/2, np.pi],
}
def construct(self):
self.setup_axes()
self.add_cosine_graph()
self.shift_approximation_center()
self.show_polynomials()
def add_cosine_graph(self):
self.cosine_graph = self.get_graph(
np.cos, self.colors[0]
)
self.add(self.cosine_graph)
def shift_approximation_center(self):
quartic_graph = self.get_quartic_approximation(0)
dot = Dot(color = YELLOW)
dot.move_to(self.coords_to_point(0, 1))
v_line = self.get_vertical_line_to_graph(
self.target_approx_centers[-1], self.cosine_graph,
line_class = DashedLine,
color = YELLOW
)
pi = self.x_axis_labels[1]
pi.add_background_rectangle()
self.play(
ReplacementTransform(
self.cosine_graph.copy(),
quartic_graph,
),
DrawBorderThenFill(dot, run_time = 1)
)
for target, rt in zip(self.target_approx_centers, [3, 4, 4]):
self.change_approximation_center(
quartic_graph, dot, target, run_time = rt
)
self.play(
ShowCreation(v_line),
Animation(pi)
)
self.dither()
def change_approximation_center(self, graph, dot, target, **kwargs):
start = self.x_axis.point_to_number(dot.get_center())
def update_quartic(graph, alpha):
new_a = interpolate(start, target, alpha)
new_graph = self.get_quartic_approximation(new_a)
Transform(graph, new_graph).update(1)
return graph
def update_dot(dot, alpha):
new_x = interpolate(start, target, alpha)
dot.move_to(self.input_to_graph_point(new_x, self.cosine_graph))
self.play(
UpdateFromAlphaFunc(graph, update_quartic),
UpdateFromAlphaFunc(dot, update_dot),
**kwargs
)
def show_polynomials(self):
poly_around_pi = self.get_polynomial("(x-\\pi)", "\\pi")
poly_around_pi.to_corner(UP+LEFT)
randy = Randolph()
randy.to_corner(DOWN+LEFT)
self.play(FadeIn(
poly_around_pi,
run_time = 4,
submobject_mode = "lagged_start"
))
self.dither(2)
self.play(FadeIn(randy))
self.play(randy.change, "confused", poly_around_pi)
self.play(Blink(randy))
self.dither(2)
self.play(randy.change_mode, "happy")
self.dither(2)
###
def get_polynomial(self, arg, center_tex):
result = TexMobject(
"P_{%s}(x)"%center_tex, "=", "c_0", *it.chain(*[
["+", "c_%d"%d, "%s^%d"%(arg, d)]
for d in range(1, 5)
])
)
for d, color in enumerate(self.colors):
result.highlight_by_tex("c_%d"%d, color)
result.scale(0.85)
result.add_background_rectangle()
return result
def get_quartic_approximation(self, a):
coefficients = [
derivative(np.cos, a, n)
for n in range(5)
]
func = lambda x : sum([
(c/math.factorial(n))*(x - a)**n
for n, c in enumerate(coefficients)
])
return self.get_graph(func, color = GREEN)
class TranslationOfInformation(CubicAndQuarticApproximations):
def construct(self):
self.add_background()
self.mention_information_exchange()
self.show_derivative_pattern()
self.show_polynomial()
self.name_taylor_polynomial()
self.draw_new_function_graph()
self.write_general_function_derivative()
self.replace_coefficients_in_generality()
self.walk_through_terms()
self.show_polynomial_around_a()
def add_background(self):
self.setup_axes()
self.cosine_graph = self.get_graph(
np.cos, color = self.colors[0]
)
self.add(self.cosine_graph)
def mention_information_exchange(self):
deriv_info = TextMobject(
"Derivative \\\\ information \\\\ at a point"
)
deriv_info.next_to(ORIGIN, LEFT, LARGE_BUFF)
deriv_info.to_edge(UP)
output_info = TextMobject(
"Output \\\\ information \\\\ near that piont"
)
output_info.next_to(ORIGIN, RIGHT, LARGE_BUFF)
output_info.to_edge(UP)
arrow = Arrow(deriv_info, output_info)
center_v_line = self.get_vertical_line_to_graph(
0, self.cosine_graph,
line_class = DashedLine,
color = YELLOW
)
outer_v_lines = VGroup(*[
center_v_line.copy().shift(vect)
for vect in LEFT, RIGHT
])
outer_v_lines.highlight(GREEN)
dot = Dot(color = YELLOW)
dot.move_to(center_v_line.get_top())
dot.save_state()
dot.move_to(deriv_info)
dot.set_fill(opacity = 0)
quadratic_graph = self.get_quadratic_graph()
self.play(Write(deriv_info, run_time = 2))
self.play(dot.restore)
self.play(ShowCreation(center_v_line))
self.dither()
self.play(ShowCreation(arrow))
self.play(Write(output_info, run_time = 2))
self.play(ReplacementTransform(
VGroup(center_v_line).copy(),
outer_v_lines
))
self.play(ReplacementTransform(
self.cosine_graph.copy(),
quadratic_graph
), Animation(dot))
for x in -1, 1, 0:
start_x = self.x_axis.point_to_number(dot.get_center())
self.play(UpdateFromAlphaFunc(
dot,
lambda d, a : d.move_to(self.input_to_graph_point(
interpolate(start_x, x, a),
self.cosine_graph
)),
run_time = 2
))
self.dither()
self.play(*map(FadeOut, [
deriv_info, arrow, output_info, outer_v_lines
]))
self.quadratic_graph = quadratic_graph
self.v_line = center_v_line
self.dot = dot
def show_derivative_pattern(self):
derivs_at_x, derivs_at_zero = [
VGroup(*[
TexMobject(tex, "(", arg, ")")
for tex in [
"\\cos", "-\\sin",
"-\\cos", "\\sin", "\\cos"
]
])
for arg in "x", "0"
]
arrows = VGroup(*[
Arrow(
UP, ORIGIN,
color = WHITE,
buff = 0,
tip_length = MED_SMALL_BUFF
)
for d in derivs_at_x
])
group = VGroup(*it.chain(*zip(
derivs_at_x,
arrows
)))
group.add(TexMobject("\\vdots"))
group.arrange_submobjects(DOWN, buff = SMALL_BUFF)
group.scale_to_fit_height(2*SPACE_HEIGHT - MED_LARGE_BUFF)
group.to_edge(LEFT)
for dx, d0, color in zip(derivs_at_x, derivs_at_zero, self.colors):
for d in dx, d0:
d.highlight(color)
d0.replace(dx)
rhs_group = VGroup(*[
TexMobject("=", "%d"%d).scale(0.7).next_to(deriv, RIGHT)
for deriv, d in zip(derivs_at_zero, [1, 0, -1, 0, 1])
])
derivative_values = VGroup(*[
rhs[1] for rhs in rhs_group
])
for value, color in zip(derivative_values, self.colors):
value.highlight(color)
zeros = VGroup(*[
deriv.get_part_by_tex("0")
for deriv in derivs_at_zero
])
self.play(FadeIn(derivs_at_x[0]))
self.dither()
for start_d, arrow, target_d in zip(group[::2], group[1::2], group[2::2]):
self.play(
ReplacementTransform(
start_d.copy(), target_d
),
ShowCreation(arrow)
)
self.dither()
self.dither()
self.play(ReplacementTransform(
derivs_at_x, derivs_at_zero
))
self.dither()
self.play(*map(Write, rhs_group))
self.dither()
for rhs in rhs_group:
self.play(Indicate(rhs[1]), color = WHITE)
self.dither()
self.play(*[
ReplacementTransform(
zero.copy(), self.dot,
run_time = 3,
rate_func = squish_rate_func(smooth, a, a+0.4)
)
for zero, a in zip(zeros, np.linspace(0, 0.6, len(zeros)))
])
self.dither()
self.cosine_derivative_group = VGroup(
derivs_at_zero, arrows, group[-1], rhs_group
)
self.derivative_values = derivative_values
def show_polynomial(self):
derivative_values = self.derivative_values.copy()
polynomial = self.get_polynomial("x", 1, 0, -1, 0, 1)
polynomial.to_corner(UP+RIGHT)
monomial = TexMobject("\\frac{1}{4!}", "x^4")
monomial = VGroup(VGroup(monomial[0]), monomial[1])
monomial.next_to(polynomial, DOWN, LARGE_BUFF)
self.play(*[
Transform(
dv, pc,
run_time = 2,
path_arc = np.pi/2
)
for dv, pc, a in zip(
derivative_values,
polynomial.coefficients,
np.linspace(0, 0.6, len(derivative_values))
)
])
self.play(
Write(polynomial, run_time = 5),
Animation(derivative_values)
)
self.remove(derivative_values)
self.dither(2)
to_fade = self.take_derivatives_of_monomial(monomial)
self.play(FadeOut(to_fade))
self.dither()
self.polynomial = polynomial
def name_taylor_polynomial(self):
brace = Brace(
VGroup(
self.polynomial.coefficients,
self.polynomial.factorials
),
DOWN
)
name = brace.get_text("``Taylor polynomial''")
name.shift(MED_SMALL_BUFF*RIGHT)
quartic_graph = self.get_graph(
lambda x : 1 - (x**2)/2.0 + (x**4)/24.0,
color = GREEN,
x_min = -3.2,
x_max = 3.2,
)
quartic_graph.highlight(self.colors[4])
self.play(GrowFromCenter(brace))
self.play(Write(name))
self.dither()
self.play(
Transform(
self.quadratic_graph, quartic_graph,
run_time = 2
),
Animation(self.dot)
)
self.dither(2)
self.taylor_name_group = VGroup(brace, name)
def draw_new_function_graph(self):
def func(x):
return (np.sin(x**2 + x)+0.5)*np.exp(-x**2)
graph = self.get_graph(
func, color = self.colors[0]
)
self.play(*map(FadeOut, [
self.cosine_derivative_group,
self.cosine_graph,
self.quadratic_graph,
self.v_line,
self.dot
]))
self.play(ShowCreation(graph))
self.graph = graph
def write_general_function_derivative(self):
derivs_at_x, derivs_at_zero, derivs_at_a = deriv_lists = [
VGroup(*[
TexMobject("\\text{$%s$}"%args[0], *args[1:])
for args in [
("f", "(", arg, ")"),
("\\frac{df}{dx}", "(", arg, ")"),
("\\frac{d^2 f}{dx^2}", "(", arg, ")"),
("\\frac{d^3 f}{dx^3}", "(", arg, ")"),
("\\frac{d^4 f}{dx^4}", "(", arg, ")"),
]
])
for arg in "x", "0", "a"
]
derivs_at_x.arrange_submobjects(DOWN, buff = MED_LARGE_BUFF)
derivs_at_x.scale_to_fit_height(2*SPACE_HEIGHT - MED_LARGE_BUFF)
derivs_at_x.to_edge(LEFT)
zeros = VGroup(*[
deriv.get_part_by_tex("0")
for deriv in derivs_at_zero
])
self.dot.move_to(self.input_to_graph_point(0, self.graph))
self.v_line.put_start_and_end_on(
self.graph_origin, self.dot.get_center()
)
for color, dx, d0, da in zip(self.colors, *deriv_lists):
for d in dx, d0, da:
d.highlight(color)
d.add_background_rectangle()
d0.replace(dx)
da.replace(dx)
self.play(FadeIn(derivs_at_x[0]))
self.dither()
for start, target in zip(derivs_at_x, derivs_at_x[1:]):
self.play(ReplacementTransform(
start.copy(), target
))
self.dither()
self.dither()
self.play(ReplacementTransform(
derivs_at_x, derivs_at_zero,
))
self.play(ReplacementTransform(
zeros.copy(), self.dot,
run_time = 2,
submobject_mode = "lagged_start"
))
self.play(ShowCreation(self.v_line))
self.dither()
self.derivs_at_zero = derivs_at_zero
self.derivs_at_a = derivs_at_a
def replace_coefficients_in_generality(self):
new_polynomial = self.get_polynomial("x", *[
tex_mob.get_tex_string()
for tex_mob in self.derivs_at_zero[:-1]
])
new_polynomial.to_corner(UP+RIGHT)
polynomial_fourth_term = VGroup(
*self.polynomial[-7:-1]
)
self.polynomial.remove(*polynomial_fourth_term)
self.play(
ReplacementTransform(
self.polynomial, new_polynomial,
run_time = 2,
submobject_mode = "lagged_start"
),
FadeOut(polynomial_fourth_term),
FadeOut(self.taylor_name_group),
)
self.polynomial = new_polynomial
self.dither(3)
def walk_through_terms(self):
func = self.graph.underlying_function
approx_graphs = [
self.get_graph(
taylor_approximation(func, n),
color = WHITE
)
for n in range(7)
]
for graph, color in zip(approx_graphs, self.colors):
graph.highlight(color)
left_mob = self.polynomial.coefficients[0]
right_mobs = list(self.polynomial.factorials)
right_mobs.append(self.polynomial[-1])
braces = [
Brace(
VGroup(left_mob, *right_mobs[:n]),
DOWN
)
for n in range(len(approx_graphs))
]
brace = braces[0]
brace.stretch_to_fit_width(MED_LARGE_BUFF)
approx_graph = approx_graphs[0]
self.polynomial.add_background_rectangle()
self.play(GrowFromCenter(brace))
self.play(ShowCreation(approx_graph))
self.dither()
for new_brace, new_graph in zip(braces[1:], approx_graphs[1:]):
self.play(Transform(brace, new_brace))
self.play(
Transform(approx_graph, new_graph, run_time = 2),
Animation(self.polynomial),
Animation(self.dot),
)
self.dither()
self.play(FadeOut(brace))
self.approx_graph = approx_graph
self.approx_order = len(approx_graphs) - 1
def show_polynomial_around_a(self):
new_polynomial = self.get_polynomial("(x-a)", *[
tex_mob.get_tex_string()
for tex_mob in self.derivs_at_a[:-2]
])
new_polynomial.to_corner(UP+RIGHT)
new_polynomial.add_background_rectangle()
polynomial_third_term = VGroup(
*self.polynomial[1][-7:-1]
)
self.polynomial[1].remove(*polynomial_third_term)
group = VGroup(self.approx_graph, self.dot, self.v_line)
def get_update_function(target_x):
def update(group, alpha):
graph, dot, line = group
start_x = self.x_axis.point_to_number(dot.get_center())
x = interpolate(start_x, target_x, alpha)
graph_point = self.input_to_graph_point(x, self.graph)
dot.move_to(graph_point)
line.put_start_and_end_on(
self.coords_to_point(x, 0),
graph_point,
)
new_approx_graph = self.get_graph(
taylor_approximation(
self.graph.underlying_function,
self.approx_order,
center_point = x
),
color = graph.get_color()
)
Transform(graph, new_approx_graph).update(1)
return VGroup(graph, dot, line)
return update
self.play(
UpdateFromAlphaFunc(
group, get_update_function(1), run_time = 2
),
Animation(self.polynomial),
Animation(polynomial_third_term)
)
self.dither()
self.play(Transform(
self.derivs_at_zero,
self.derivs_at_a
))
self.play(
Transform(self.polynomial, new_polynomial),
FadeOut(polynomial_third_term)
)
self.dither()
for x in -1, np.pi/6:
self.play(
UpdateFromAlphaFunc(
group, get_update_function(x),
),
Animation(self.polynomial),
run_time = 4,
)
self.dither()
#####
def get_polynomial(self, arg, *coefficients):
result = TexMobject(
"P(x) = ", str(coefficients[0]), *list(it.chain(*[
["+", str(c), "{%s"%arg, "^%d"%n, "\\over", "%d!}"%n]
for n, c in zip(it.count(1), coefficients[1:])
])) + [
"+ \\cdots"
]
)
result.scale(0.8)
coefs = VGroup(result[1], *result[3:-1:6])
for coef, color in zip(coefs, self.colors):
coef.highlight(color)
result.coefficients = coefs
result.factorials = VGroup(*result[7::6])
return result
class ExpPolynomial(TranslationOfInformation, ExampleApproximationWithExp):
CONFIG = {
"x_tick_frequency" : 1,
"x_leftmost_tick" : -3,
"graph_origin" : 2*(DOWN+LEFT),
"y_axis_label" : "",
}
def construct(self):
self.setup_axes()
self.add_graph()
self.show_derivatives()
self.show_polynomial()
self.walk_through_terms()
def add_graph(self):
graph = self.get_graph(np.exp)
e_to_x = self.get_graph_label(graph, "e^x")
self.play(
ShowCreation(graph),
Write(
e_to_x,
rate_func = squish_rate_func(smooth, 0.5, 1)
),
run_time = 2
)
self.dither()
self.graph = graph
self.e_to_x = e_to_x
def show_derivatives(self):
self.e_to_x.generate_target()
derivs_at_x, derivs_at_zero = [
VGroup(*[
TexMobject("e^%s"%s).highlight(c)
for c in self.colors
])
for s in "x", "0"
]
derivs_at_x.submobjects[0] = self.e_to_x.target
arrows = VGroup(*[
Arrow(
UP, ORIGIN,
color = WHITE,
buff = SMALL_BUFF,
tip_length = 0.2,
)
for d in derivs_at_x
])
group = VGroup(*it.chain(*zip(
derivs_at_x,
arrows
)))
group.add(TexMobject("\\vdots"))
group.arrange_submobjects(DOWN, buff = 2*SMALL_BUFF)
group.scale_to_fit_height(2*SPACE_HEIGHT - MED_LARGE_BUFF)
group.to_edge(LEFT)
for dx, d0 in zip(derivs_at_x, derivs_at_zero):
for d in dx, d0:
d.add_background_rectangle()
d0.replace(dx)
rhs_group = VGroup(*[
TexMobject("=", "1").scale(0.7).next_to(deriv, RIGHT)
for deriv in derivs_at_zero
])
derivative_values = VGroup(*[
rhs[1] for rhs in rhs_group
])
for value, color in zip(derivative_values, self.colors):
value.highlight(color)
for arrow in arrows:
d_dx = TexMobject("d/dx")
d_dx.scale(0.5)
d_dx.next_to(arrow, RIGHT, SMALL_BUFF)
d_dx.shift(SMALL_BUFF*UP)
arrow.add(d_dx)
self.play(MoveToTarget(self.e_to_x))
derivs_at_x.submobjects[0] = self.e_to_x
for start_d, arrow, target_d in zip(group[::2], group[1::2], group[2::2]):
self.play(
ReplacementTransform(
start_d.copy(), target_d
),
Write(arrow, run_time = 1)
)
self.dither()
self.dither()
self.play(ReplacementTransform(
derivs_at_x, derivs_at_zero
))
self.dither()
self.play(*map(Write, rhs_group))
self.derivative_values = derivative_values
def show_polynomial(self):
derivative_values = self.derivative_values.copy()
polynomial = self.get_polynomial("x", 1, 1, 1, 1, 1)
polynomial.to_corner(UP+RIGHT)
##These are to make the walk_through_terms method work
self.polynomial = polynomial.copy()
self.dot = Dot(fill_opacity = 0)
###
polynomial.add_background_rectangle()
self.play(*[
Transform(
dv, pc,
run_time = 2,
path_arc = np.pi/2
)
for dv, pc in zip(
derivative_values,
polynomial.coefficients,
)
])
self.play(
Write(polynomial, run_time = 4),
Animation(derivative_values)
)
####
def setup_axes(self):
GraphScene.setup_axes(self)
class ShowSecondTerm(TeacherStudentsScene):
def construct(self):
colors = CubicAndQuarticApproximations.CONFIG["colors"]
polynomial = TexMobject(
"f(a)", "+",
"\\frac{df}{dx}(a)", "(x - a)", "+",
"\\frac{d^2 f}{dx^2}(a)", "(x - a)^2"
)
for tex, color in zip(["f(a)", "df", "d^2 f"], colors):
polynomial.highlight_by_tex(tex, color)
polynomial.next_to(self.teacher, UP+LEFT)
polynomial.shift(MED_LARGE_BUFF*UP)
second_term = VGroup(*polynomial[-2:])
box = Rectangle(color = colors[2])
box.replace(second_term, stretch = True)
box.stretch_in_place(1.1, 0)
box.stretch_in_place(1.2, 1)
words = TextMobject("Geometric view")
words.next_to(box, UP)
self.play(Write(polynomial))
self.play(
ShowCreation(box),
FadeIn(words),
self.teacher.change_mode, "raise_right_hand"
)
self.dither(3)
class SecondTermIntuition(AreaIsDerivative):
CONFIG = {
"func" : lambda x : x*(x-1)*(x-2) + 2,
"num_rects" : 300,
"t_max" : 2.3,
"x_max" : 4,
"x_labeled_nums" : None,
"x_axis_label" : "",
"y_labeled_nums" : None,
"y_axis_label" : "",
"y_min" : -1,
"y_max" : 5,
"y_tick_frequency" : 1,
"variable_point_label" : "x",
"area_opacity" : 1,
"default_riemann_start_color" : BLUE_E,
"dx" : 0.15,
"skip_reconfiguration" : False,
}
def setup(self):
GraphScene.setup(self)
ReconfigurableScene.setup(self)
self.foreground_mobjects = []
def construct(self):
self.setup_axes()
self.introduce_area()
self.write_derivative()
self.nudge_end_point()
self.point_out_triangle()
self.relabel_start_and_end()
self.compute_triangle_area()
self.walk_through_taylor_terms()
def introduce_area(self):
graph = self.v_graph = self.get_graph(
self.func, color = WHITE,
)
self.foreground_mobjects.append(graph)
area = self.area = self.get_area(0, self.t_max)
func_name = TexMobject("f_{\\text{area}}(x)")
func_name.move_to(self.coords_to_point(0.6, 1))
self.foreground_mobjects.append(func_name)
self.add(graph, area, func_name)
self.add_T_label(self.t_max)
if not self.skip_animations:
for target in 1.6, 2.7, self.t_max:
self.change_area_bounds(
new_t_max = target,
run_time = 3,
)
self.func_name = func_name
def write_derivative(self):
deriv = TexMobject("\\frac{df_{\\text{area}}}{dx}(x)")
deriv.next_to(
self.input_to_graph_point(2.7, self.v_graph),
RIGHT
)
deriv.shift_onto_screen()
self.play(ApplyWave(self.v_graph, direction = UP))
self.play(Write(deriv, run_time = 2))
self.dither()
self.deriv_label = deriv
def nudge_end_point(self):
dark_area = self.area.copy()
dark_area.set_fill(BLACK, opacity = 0.5)
curr_x = self.x_axis.point_to_number(self.area.get_right())
new_x = curr_x + self.dx
rect = Rectangle(
stroke_width = 0,
fill_color = YELLOW,
fill_opacity = 0.75
)
rect.replace(
VGroup(
VectorizedPoint(self.coords_to_point(new_x, 0)),
self.right_v_line,
),
stretch = True
)
dx_brace = Brace(rect, DOWN, buff = 0)
dx_label = dx_brace.get_text("$dx$", buff = SMALL_BUFF)
dx_label_group = VGroup(dx_label, dx_brace)
height_brace = Brace(rect, LEFT, buff = SMALL_BUFF)
self.change_area_bounds(new_t_max = new_x)
self.play(
FadeIn(dark_area),
*map(Animation, self.foreground_mobjects)
)
self.play(
FadeOut(self.T_label_group),
FadeIn(dx_label_group),
FadeIn(rect),
FadeIn(height_brace)
)
self.dither()
if not self.skip_reconfiguration:
self.transition_to_alt_config(
dx = self.dx/10.0,
run_time = 3,
)
self.play(FadeOut(height_brace))
self.dx_label_group = dx_label_group
self.rect = rect
self.dark_area = dark_area
def point_out_triangle(self):
triangle = Polygon(LEFT, ORIGIN, UP)
triangle.set_stroke(width = 0)
triangle.set_fill(MAROON_B, opacity = 1)
triangle.replace(
Line(
self.rect.get_corner(UP+LEFT),
self.right_v_line.get_top()
),
stretch = True
)
circle = Circle(color = RED)
circle.scale_to_fit_height(triangle.get_height())
circle.replace(triangle, dim_to_match = 1)
circle.scale_in_place(1.3)
self.play(DrawBorderThenFill(triangle))
self.play(ShowCreation(circle))
self.play(FadeOut(circle))
self.triangle = triangle
def relabel_start_and_end(self):
dx_label, dx_brace = self.dx_label_group
x_minus_a = TexMobject("(x-a)")
x_minus_a.scale(0.7)
x_minus_a.move_to(dx_label)
labels = VGroup()
arrows = VGroup()
for vect, tex in (LEFT, "a"), (RIGHT, "x"):
point = self.rect.get_corner(DOWN+vect)
label = TexMobject(tex)
label.next_to(point, DOWN+vect)
label.shift(LARGE_BUFF*vect)
arrow = Arrow(
label.get_corner(UP-vect),
point,
buff = SMALL_BUFF,
tip_length = 0.2,
color = WHITE
)
labels.add(label)
arrows.add(arrow)
for label, arrow in zip(labels, arrows):
self.play(
Write(label),
ShowCreation(arrow)
)
self.dither()
self.dither()
self.play(ReplacementTransform(
dx_label, x_minus_a
))
self.dither()
self.x_minus_a = x_minus_a
def compute_triangle_area(self):
triangle = self.triangle.copy()
tex_scale_factor = 0.7
base_line = Line(*[
triangle.get_corner(DOWN+vect)
for vect in LEFT, RIGHT
])
base_line.highlight(RED)
base_label = TextMobject("Base = ", "$(x-a)$")
base_label.scale(tex_scale_factor)
base_label.next_to(base_line, DOWN+RIGHT, MED_LARGE_BUFF)
base_label.shift(SMALL_BUFF*UP)
base_term = base_label[1].copy()
base_arrow = Arrow(
base_label.get_left(),
base_line.get_center(),
buff = SMALL_BUFF,
color = base_line.get_color(),
tip_length = 0.2
)
height_brace = Brace(triangle, RIGHT, buff = SMALL_BUFF)
height_labels = [
TexMobject("\\text{Height} = ", s, "(x-a)")
for s in [
"(\\text{Slope})",
"\\frac{d^2 f_{\\text{area}}}{dx^2}(a)"
]
]
for label in height_labels:
label.scale(tex_scale_factor)
label.next_to(height_brace, RIGHT)
height_term = VGroup(*height_labels[1][1:]).copy()
self.play(
FadeIn(base_label),
ShowCreation(base_arrow),
ShowCreation(base_line)
)
self.dither(2)
self.play(
GrowFromCenter(height_brace),
Write(height_labels[0])
)
self.dither(2)
self.play(ReplacementTransform(*height_labels))
self.dither(2)
#Show area formula
equals_half = TexMobject("=\\frac{1}{2}")
equals_half.scale(tex_scale_factor)
group = VGroup(triangle, equals_half, height_term, base_term)
group.generate_target()
group.target.arrange_submobjects(RIGHT, buff = SMALL_BUFF)
group.target[-1].next_to(
group.target[-2], RIGHT,
buff = SMALL_BUFF,
align_using_submobjects = True
)
group.target[1].shift(0.02*DOWN)
group.target.to_corner(UP+RIGHT)
exp_2 = TexMobject("2").scale(0.8*tex_scale_factor)
exp_2.next_to(
group.target[-2], UP+RIGHT,
buff = 0,
align_using_submobjects = True
)
equals_half.scale(0.1)
equals_half.move_to(triangle)
equals_half.set_fill(opacity = 0)
self.play(
FadeOut(self.deriv_label),
MoveToTarget(group, run_time = 2)
)
self.dither(2)
self.play(Transform(
group[-1], exp_2
))
self.dither(2)
def walk_through_taylor_terms(self):
mini_area, mini_rect, mini_triangle = [
mob.copy()
for mob in self.dark_area, self.rect, self.triangle
]
mini_area.set_fill(BLUE_E, opacity = 1)
mini_area.scale_to_fit_height(1)
mini_rect.scale_to_fit_height(1)
mini_triangle.scale_to_fit_height(0.5)
geometric_taylor = VGroup(
TexMobject("f(x) \\approx "), mini_area,
TexMobject("+"), mini_rect,
TexMobject("+"), mini_triangle,
)
geometric_taylor.arrange_submobjects(
RIGHT, buff = MED_SMALL_BUFF
)
geometric_taylor.to_corner(UP+LEFT)
analytic_taylor = TexMobject(
"f(x) \\approx", "f(a)", "+",
"\\frac{df}{dx}(a)(x-a)", "+",
"\\frac{1}{2}\\frac{d^2 f}{dx^2}(a)(x-a)^2"
)
analytic_taylor.highlight_by_tex("f(a)", BLUE)
analytic_taylor.highlight_by_tex("df", YELLOW)
analytic_taylor.highlight_by_tex("d^2 f", MAROON_B)
analytic_taylor.scale(0.7)
analytic_taylor.next_to(
geometric_taylor, DOWN,
aligned_edge = LEFT
)
for part in analytic_taylor:
part.add_to_back(BackgroundRectangle(part))
new_func_name = TexMobject("f_{\\text{area}}(a)")
new_func_name.replace(self.func_name)
self.play(FadeIn(
geometric_taylor,
run_time = 2,
submobject_mode = "lagged_start"
))
self.dither()
self.play(
FadeIn(VGroup(*analytic_taylor[:3])),
self.dark_area.set_fill, BLUE_E, 1,
Transform(self.func_name, new_func_name)
)
self.dither()
self.play(
self.rect.scale_in_place, 0.5,
rate_func = there_and_back
)
self.play(FadeIn(VGroup(*analytic_taylor[3:5])))
self.dither(2)
self.play(
self.triangle.scale_in_place, 0.5,
rate_func = there_and_back
)
self.play(FadeIn(VGroup(*analytic_taylor[5:])))
self.dither(3)
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