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Antiderivative of eoc9
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Grant Sanderson
parent
fbb1156108
commit
cc99bcda37
1 changed files with 295 additions and 15 deletions
310
eoc/chapter9.py
310
eoc/chapter9.py
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@ -1003,10 +1003,11 @@ class Antiderivative(AverageOfSineStart):
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CONFIG = {
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"antideriv_color" : GREEN,
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"deriv_color" : BLUE,
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"riemann_rect_dx" : 0.01,
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"y_axis_label" : "",
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"graph_origin" : 4*LEFT + DOWN,
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}
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def construct(self):
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self.force_skipping()
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self.setup_axes()
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self.add_x_axis_labels()
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self.negate_derivative_of_cosine()
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@ -1014,7 +1015,15 @@ class Antiderivative(AverageOfSineStart):
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self.apply_ftoc()
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self.show_difference_in_antiderivative()
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self.comment_on_area()
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self.write_final_answer()
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self.divide_by_pi()
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self.highlight_antiderivative_fraction()
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self.show_slope()
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self.bring_back_derivative()
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self.show_tangent_slope()
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def add_x_axis_labels(self):
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AverageOfSineStart.add_x_axis_labels(self)
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self.remove(*self.x_axis_labels[::2])
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def negate_derivative_of_cosine(self):
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cos, neg_cos, sin, neg_sin = graphs = [
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@ -1035,7 +1044,8 @@ class Antiderivative(AverageOfSineStart):
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graph.label = self.get_graph_label(
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graph, label,
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x_val = x_val,
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direction = vect
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direction = vect,
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buff = SMALL_BUFF
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)
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derivs = []
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@ -1106,7 +1116,7 @@ class Antiderivative(AverageOfSineStart):
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return 0.25*(np.floor(4*t) + smooth((4*t) % 1))
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self.play(*[
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ApplyMethod(m.fade, 0.7)
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ApplyMethod(m.fade, 0.6)
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for m in faders
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])
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self.dither()
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@ -1129,27 +1139,297 @@ class Antiderivative(AverageOfSineStart):
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**kwargs
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)
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self.play(
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FadeOut(ss_group),
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FadeOut(v_line),
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*[m.restore for m in faders]
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*map(FadeOut, [ss_group, v_line, sin_copy])
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)
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self.remove(sin_copy)
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self.dither()
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self.ss_group = ss_group
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def apply_ftoc(self):
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deriv = self.deriv
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pass
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integral = TexMobject(
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"\\int", "^\\pi", "_0", "\\sin(x)", "\\,dx"
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)
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rhs = TexMobject(
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"=", "\\big(", "-\\cos", "(", "\\pi", ")", "\\big)",
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"-", "\\big(", "-\\cos", "(", "0", ")", "\\big)",
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)
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rhs.next_to(integral, RIGHT)
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equation = VGroup(integral, rhs)
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equation.to_corner(UP+RIGHT, buff = MED_SMALL_BUFF)
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(start_pi, end_pi), (start_zero, end_zero) = start_end_pairs = [
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[
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m.get_part_by_tex(tex)
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for m in integral, rhs
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]
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for tex in "\\pi", "0"
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]
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self.revert_to_original_skipping_status()
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for tex_mob in integral, rhs:
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tex_mob.highlight_by_tex("sin", self.deriv_color)
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tex_mob.highlight_by_tex("cos", self.antideriv_color)
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tex_mob.highlight_by_tex("0", YELLOW)
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tex_mob.highlight_by_tex("\\pi", YELLOW)
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self.play(
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Write(integral),
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self.deriv.scale, 0.5,
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self.deriv.center,
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self.deriv.to_edge, LEFT, MED_SMALL_BUFF,
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self.deriv.shift, UP,
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)
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self.dither()
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self.play(FadeIn(
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VGroup(*filter(
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lambda part : part not in [end_pi, end_zero],
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rhs
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)),
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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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for start, end in start_end_pairs:
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self.play(ReplacementTransform(
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start.copy(), end,
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path_arc = np.pi/6,
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run_time = 2
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))
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self.dither()
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self.integral = integral
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self.rhs = rhs
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def show_difference_in_antiderivative(self):
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pass
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pi_point, zero_point = points = [
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self.input_to_graph_point(x, self.neg_cos)
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for x in reversed(self.bounds)
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]
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interim_point = pi_point[0]*RIGHT + zero_point[1]*UP
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pi_dot, zero_dot = dots = [
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Dot(point, color = YELLOW)
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for point in points
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]
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v_line = DashedLine(pi_point, interim_point)
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h_line = DashedLine(interim_point, zero_point)
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v_line_brace = Brace(v_line, RIGHT)
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two_height_label = v_line_brace.get_text(
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"$2$", buff = SMALL_BUFF
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)
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two_height_label.add_background_rectangle()
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pi = self.x_axis_labels[1]
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#Horrible hack
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black_pi = pi.copy().highlight(BLACK)
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self.add(black_pi, pi)
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cos_tex = self.rhs.get_part_by_tex("cos")
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self.play(ReplacementTransform(
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cos_tex.copy(), pi_dot
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))
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self.dither()
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moving_dot = pi_dot.copy()
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self.play(
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ShowCreation(v_line),
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# Animation(pi),
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pi.shift, 0.8*pi.get_width()*(LEFT+UP),
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moving_dot.move_to, interim_point,
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)
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self.play(
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ShowCreation(h_line),
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ReplacementTransform(moving_dot, zero_dot)
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)
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self.play(GrowFromCenter(v_line_brace))
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self.dither(2)
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self.play(Write(two_height_label))
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self.dither(2)
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self.v_line = v_line
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self.h_line = h_line
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self.pi_dot = pi_dot
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self.zero_dot = zero_dot
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self.v_line_brace = v_line_brace
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self.two_height_label = two_height_label
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def comment_on_area(self):
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pass
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rects = self.get_riemann_rectangles(
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self.sin,
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dx = self.riemann_rect_dx,
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stroke_width = 0,
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fill_opacity = 0.7,
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x_min = self.bounds[0],
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x_max = self.bounds[1],
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)
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area_two = TexMobject("2").replace(
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self.two_height_label
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)
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self.play(Write(rects))
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self.dither()
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self.play(area_two.move_to, rects)
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self.dither(2)
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self.rects = rects
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self.area_two = area_two
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def divide_by_pi(self):
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integral = self.integral
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rhs = self.rhs
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equals = rhs[0]
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rhs_without_eq = VGroup(*rhs[1:])
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frac_lines = VGroup(*[
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TexMobject("\\over\\,").stretch_to_fit_width(
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mob.get_width()
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).move_to(mob)
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for mob in integral, rhs_without_eq
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])
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frac_lines.shift(
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(integral.get_height()/2 + SMALL_BUFF)*DOWN
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)
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pi_minus_zeros = VGroup(*[
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TexMobject("\\pi", "-", "0").next_to(line, DOWN)
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for line in frac_lines
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])
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for tex_mob in pi_minus_zeros:
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for tex in "pi", "0":
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tex_mob.highlight_by_tex(tex, YELLOW)
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answer = TexMobject(" = \\frac{2}{\\pi}")
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answer.next_to(
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frac_lines, RIGHT,
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align_using_submobjects = True
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)
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answer.shift_onto_screen()
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self.play(
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equals.next_to, frac_lines[0].copy(), RIGHT,
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rhs_without_eq.next_to, frac_lines[1].copy(), UP,
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*map(Write, frac_lines)
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)
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self.play(*[
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ReplacementTransform(
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integral.get_part_by_tex(tex).copy(),
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pi_minus_zeros[0].get_part_by_tex(tex)
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)
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for tex in "\\pi","0"
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] + [
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Write(pi_minus_zeros[0].get_part_by_tex("-"))
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])
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self.play(*[
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ReplacementTransform(
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rhs.get_part_by_tex(
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tex, substring = False
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).copy(),
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pi_minus_zeros[1].get_part_by_tex(tex)
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)
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for tex in "\\pi", "-", "0"
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])
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self.dither(2)
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full_equation = VGroup(
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integral, frac_lines, rhs, pi_minus_zeros
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)
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background_rect = BackgroundRectangle(full_equation, fill_opacity = 1)
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background_rect.stretch_in_place(1.2, dim = 1)
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full_equation.add_to_back(background_rect)
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self.play(
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full_equation.shift,
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(answer.get_width()+MED_LARGE_BUFF)*LEFT
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)
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self.play(Write(answer))
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self.dither()
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self.antiderivative_fraction = VGroup(
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rhs_without_eq,
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frac_lines[1],
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pi_minus_zeros[1]
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)
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self.integral_fraction = VGroup(
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integral,
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frac_lines[0],
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pi_minus_zeros[0],
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equals
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)
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def highlight_antiderivative_fraction(self):
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fraction = self.antiderivative_fraction
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big_rect = Rectangle(
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stroke_width = 0,
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fill_color = BLACK,
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fill_opacity = 0.9,
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)
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big_rect.scale_to_fit_width(2*SPACE_WIDTH)
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big_rect.scale_to_fit_height(2*SPACE_HEIGHT)
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morty = Mortimer()
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morty.to_corner(DOWN+RIGHT)
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self.play(
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FadeIn(big_rect),
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FadeIn(morty),
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Animation(fraction)
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)
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self.play(morty.change, "raise_right_hand", fraction)
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self.play(Blink(morty))
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self.dither(2)
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self.play(
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FadeOut(big_rect),
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FadeOut(morty),
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Animation(fraction)
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)
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def show_slope(self):
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line = Line(
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self.zero_dot.get_center(),
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self.pi_dot.get_center(),
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)
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line.highlight(RED)
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line.scale_in_place(1.2)
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pi = TexMobject("\\pi")
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pi.next_to(self.h_line, DOWN)
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self.play(
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FadeOut(self.rects),
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FadeOut(self.area_two)
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)
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self.play(Write(pi))
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self.play(ShowCreation(line, run_time = 2))
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self.dither()
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def bring_back_derivative(self):
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self.play(
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FadeOut(self.integral_fraction),
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self.deriv.scale, 1.7,
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self.deriv.to_corner, UP+LEFT, MED_LARGE_BUFF,
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self.deriv.shift, MED_SMALL_BUFF*DOWN,
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)
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self.dither()
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def show_tangent_slope(self):
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ss_group = self.get_secant_slope_group(
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0, self.neg_cos,
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dx = 0.001,
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secant_line_color = YELLOW,
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secant_line_length = 4,
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)
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self.play(*map(ShowCreation, ss_group), run_time = 2)
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for count in range(2):
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for x in reversed(self.bounds):
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self.animate_secant_slope_group_change(
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ss_group,
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target_x = x,
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run_time = 6,
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)
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def write_final_answer(self):
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pass
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