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synced 2025-07-30 21:44:19 +08:00
Pattern for power rule in eoc3
This commit is contained in:
Grant Sanderson
@ -124,6 +124,7 @@ class ApplyMethod(Transform):
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target = copy.deepcopy(method)(*args)
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Transform.__init__(self, method.im_self, target, **kwargs)
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class FadeOut(Transform):
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CONFIG = {
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"remover" : True,
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234
eoc/chapter3.py
234
eoc/chapter3.py
@ -1008,10 +1008,236 @@ class ShowCubeDVIn3D(Scene):
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def construct(self):
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raise Exception("This scene is only here for the stage_scenes script.")
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class GraphOfXCubed(GraphScene):
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CONFIG = {
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"x_min" : -6,
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"x_max" : 6,
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"x_axis_width" : 2*SPACE_WIDTH,
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"x_labeled_nums" : range(-6, 7),
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"y_min" : -35,
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"y_max" : 35,
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"y_axis_height" : 2*SPACE_HEIGHT,
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"y_tick_frequency" : 5,
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"y_labeled_nums" : range(-30, 40, 10),
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"graph_origin" : ORIGIN,
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"dx" : 0.2,
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"deriv_x_min" : -3,
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"deriv_x_max" : 3,
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}
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def construct(self):
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self.setup_axes(animate = False)
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graph = self.get_graph(lambda x : x**3)
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label = self.get_graph_label(
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graph, "f(x) = x^3",
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direction = LEFT,
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)
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deriv_graph, full_deriv_graph = [
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self.get_derivative_graph(
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graph,
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color = DERIVATIVE_COLOR,
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x_min = low_x,
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x_max = high_x,
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)
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for low_x, high_x in [
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(self.deriv_x_min, self.deriv_x_max),
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(self.x_min, self.x_max),
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]
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]
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deriv_label = self.get_graph_label(
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deriv_graph,
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"\\frac{df}{dx}(x) = 3x^2",
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x_val = -3,
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direction = LEFT
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)
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deriv_label.shift(0.5*DOWN)
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ss_group = self.get_secant_slope_group(
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self.deriv_x_min, graph,
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dx = self.dx,
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dx_line_color = WHITE,
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df_line_color = WHITE,
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secant_line_color = YELLOW,
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)
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self.play(ShowCreation(graph))
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self.play(Write(label, run_time = 1))
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self.dither()
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self.play(Write(deriv_label, run_time = 1))
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self.play(ShowCreation(ss_group, submobject_mode = "all_at_once"))
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self.animate_secant_slope_group_change(
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ss_group,
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target_x = self.deriv_x_max,
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run_time = 10,
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added_anims = [
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ShowCreation(deriv_graph, run_time = 10)
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]
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)
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self.play(FadeIn(full_deriv_graph))
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self.dither()
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for x_val in -2, -self.dx/2, 2:
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self.animate_secant_slope_group_change(
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ss_group,
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target_x = x_val,
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run_time = 2
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)
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if x_val != -self.dx/2:
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v_line = self.get_vertical_line_to_graph(
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x_val, deriv_graph,
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line_class = DashedLine
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)
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self.play(ShowCreation(v_line))
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self.play(FadeOut(v_line))
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class PatternForPowerRule(PiCreatureScene):
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CONFIG = {
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"num_exponents" : 5,
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}
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def construct(self):
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self.introduce_pattern()
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self.generalize_pattern()
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self.show_hopping()
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def introduce_pattern(self):
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exp_range = range(1, 1+self.num_exponents)
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colors = color_gradient([BLUE_D, YELLOW], self.num_exponents)
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derivatives = VGroup()
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for exponent, color in zip(exp_range, colors):
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derivative = TexMobject(
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"\\frac{d(x^%d)}{dx} = "%exponent,
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"%d x^{%d}"%(exponent, exponent-1)
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)
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VGroup(*derivative[0][2:4]).highlight(color)
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derivatives.add(derivative)
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derivatives.arrange_submobjects(
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DOWN, aligned_edge = LEFT,
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buff = MED_LARGE_BUFF
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)
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derivatives.scale_to_fit_height(2*SPACE_HEIGHT-1)
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derivatives.to_edge(LEFT)
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self.play(FadeIn(derivatives[0]))
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for d1, d2 in zip(derivatives, derivatives[1:]):
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self.play(Transform(
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d1.copy(), d2,
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replace_mobject_with_target_in_scene = True
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))
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self.change_mode("thinking")
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self.dither()
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for derivative in derivatives[-2:]:
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derivative.save_state()
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self.play(
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derivative.scale, 2,
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derivative.next_to, derivative,
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RIGHT, SMALL_BUFF, DOWN,
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)
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self.dither(2)
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self.play(derivative.restore)
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self.remove(derivative)
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derivative.restore()
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self.add(derivative)
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self.derivatives = derivatives
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self.colors = colors
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def generalize_pattern(self):
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derivatives = self.derivatives
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power_rule = TexMobject(
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"\\frac{d (x^n)}{dx} = ",
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"nx^{n-1}"
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)
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title = TextMobject("``Power rule''")
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title.next_to(power_rule, UP, MED_LARGE_BUFF)
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lines = VGroup(*[
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Line(
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deriv.get_right(), power_rule.get_left(),
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buff = MED_SMALL_BUFF,
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color = deriv[0][2].get_color()
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)
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for deriv in derivatives
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])
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self.play(
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Transform(
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VGroup(*[d[0].copy() for d in derivatives]),
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VGroup(power_rule[0]),
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replace_mobject_with_target_in_scene = True
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),
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ShowCreation(lines),
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submobject_mode = "lagged_start",
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run_time = 2,
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)
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self.dither()
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self.play(Write(power_rule[1]))
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self.dither()
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self.play(
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Write(title),
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self.pi_creature.change_mode, "speaking"
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)
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self.dither()
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def show_hopping(self):
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exp_range = range(2, 2+self.num_exponents-1)
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self.change_mode("tired")
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for exp, color in zip(exp_range, self.colors[1:]):
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form = TexMobject(
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"x^",
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str(exp),
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"\\rightarrow",
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str(exp),
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"x^",
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str(exp-1)
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)
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form.highlight(color)
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form.to_corner(UP+RIGHT, buff = LARGE_BUFF)
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lhs = VGroup(*form[:2])
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lhs_copy = lhs.copy()
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rhs = VGroup(*form[-2:])
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arrow = form[2]
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self.play(Write(lhs))
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self.play(
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lhs_copy.move_to, rhs, DOWN+LEFT,
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Write(arrow)
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)
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self.dither()
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self.play(Transform(
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lhs_copy[1], form[3],
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path_arc = np.pi,
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rate_func = running_start,
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replace_mobject_with_target_in_scene = True
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))
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self.play(FadeIn(form[5]))
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self.dither()
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self.play(
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self.pi_creature.change_mode, "hesitant",
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self.pi_creature.look_at, lhs_copy
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)
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self.play(*map(FadeOut, [form, lhs_copy]))
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# class PowerRuleAlgebra(Scene):
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# CONFIG = {
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# "dx_color" : YELLOW,
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# }
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# def construct(self):
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# value = TextMobject("Value:")
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# x_to_n = TexMobject("x^n")
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# nudged = TextMobject("Nudged Value:")
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# x_dx_to_n = TexMobject("(x+dx)^n")
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# equals = TexMobject("equals")
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# full_product = TexMobject("(x+dx)(x+dx)(x+dx)\\cdots(x+dx)")
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# value.to_corner(UP+LEFT)
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# x_to_n.next_to(value, RIGHT)
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# nudged.next_to(value, DOWN, aligned_edge = LEFT)
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# x_dx_group = VGroup(x_dx_to_n, equals, full_product)
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# x_dx_group.arrange_submobjects(RIGHT)
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# self.add(value, nudged_value)
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@ -28,7 +28,7 @@ class GraphScene(Scene):
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"y_axis_label" : "$y$",
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"axes_color" : GREY,
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"graph_origin" : 2.5*DOWN + 4*LEFT,
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"y_axis_numbers_nudge" : 0.4*UP+0.5*LEFT,
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"y_axis_numbers_nudge" : 0.4*UP,
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"num_graph_anchor_points" : 25,
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"default_graph_colors" : [BLUE, GREEN, YELLOW],
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"default_derivative_color" : GREEN,
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@ -48,9 +48,16 @@ class GraphScene(Scene):
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)
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x_axis.shift(self.graph_origin - x_axis.number_to_point(0))
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if self.x_labeled_nums:
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x_axis.add_numbers(*self.x_labeled_nums)
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x_axis.add_numbers(*filter(
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lambda x : x != 0,
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self.x_labeled_nums
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))
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x_label = TextMobject(self.x_axis_label)
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x_label.next_to(x_axis, RIGHT+UP, buff = SMALL_BUFF)
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x_label.next_to(
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x_axis.get_tick_marks(), UP,
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aligned_edge = RIGHT,
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buff = SMALL_BUFF
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)
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x_label.shift_onto_screen()
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x_axis.add(x_label)
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self.x_axis_label_mob = x_label
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@ -69,10 +76,19 @@ class GraphScene(Scene):
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y_axis.shift(self.graph_origin-y_axis.number_to_point(0))
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y_axis.rotate(np.pi/2, about_point = y_axis.number_to_point(0))
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if self.y_labeled_nums:
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y_axis.add_numbers(*self.y_labeled_nums)
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y_axis.numbers.shift(self.y_axis_numbers_nudge)
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y_axis.add_numbers(*filter(
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lambda y : y != 0,
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self.y_labeled_nums
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))
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for mob in y_axis.numbers:
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mob.next_to(mob.get_center(), LEFT, MED_SMALL_BUFF)
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mob.shift(self.y_axis_numbers_nudge)
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y_label = TextMobject(self.y_axis_label)
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y_label.next_to(y_axis.get_top(), RIGHT, buff = MED_LARGE_BUFF)
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y_label.next_to(
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y_axis.get_tick_marks(), RIGHT,
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aligned_edge = UP,
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buff = SMALL_BUFF
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)
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y_label.shift_onto_screen()
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y_axis.add(y_label)
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self.y_axis_label_mob = y_label
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@ -90,12 +106,21 @@ class GraphScene(Scene):
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result += self.y_axis.number_to_point(y)[1]*UP
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return result
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def get_graph(self, func, color = None):
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def get_graph(
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self, func,
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color = None,
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x_min = None,
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x_max = None,
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):
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if color is None:
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color = self.default_graph_colors.next()
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if x_min is None:
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x_min = self.x_min
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if x_max is None:
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x_max = self.x_max
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def parameterized_function(alpha):
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x = interpolate(self.x_min, self.x_max, alpha)
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x = interpolate(x_min, x_max, alpha)
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return self.coords_to_point(x, func(x))
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graph = ParametricFunction(
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@ -116,13 +141,12 @@ class GraphScene(Scene):
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def slope_of_tangent(self, *args, **kwargs):
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return np.tan(self.angle_of_tangent(*args, **kwargs))
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def get_derivative_graph(self, graph, dx = 0.01, color = None):
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if color is None:
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color = self.default_derivative_color
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return self.get_graph(
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lambda x : self.slope_of_tangent(x, graph, dx) / self.space_unit_to_y,
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color = color,
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)
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def get_derivative_graph(self, graph, dx = 0.01, **kwargs):
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if "color" not in kwargs:
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kwargs["color"] = self.default_derivative_color
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def deriv(x):
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return self.slope_of_tangent(x, graph, dx) / self.space_unit_to_y
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return self.get_graph(deriv, **kwargs)
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def get_graph_label(
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self,
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@ -268,7 +268,8 @@ class Mobject(object):
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dim = np.argmax(np.abs(vect))
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buff = kwargs.get("buff", DEFAULT_MOBJECT_TO_EDGE_BUFFER)
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max_val = space_lengths[dim] - buff
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if abs(self.get_edge_center(vect)[dim]) > max_val:
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edge_center = self.get_edge_center(vect)
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if np.dot(edge_center, vect) > max_val:
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self.to_edge(vect, **kwargs)
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return self
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@ -135,7 +135,7 @@ class Line(VMobject):
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class DashedLine(Line):
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CONFIG = {
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"dashed_segment_length" : 0.15
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"dashed_segment_length" : 0.05
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}
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def __init__(self, *args, **kwargs):
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self.init_kwargs = kwargs
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