from manim import * class EpsilonDeltaLimit(Scene): def construct(self): # Title and formal definition title = Tex(r"Epsilon-Delta Definition of Limit") title.to_edge(UP) self.play(Write(title)) self.wait(0.5) formal = MathTex( r"\forall\,\varepsilon>0\,,\ \exists\,\delta>0\text{ such that }", r"\quad 0<|x-a|<\delta\ \Rightarrow\ |f(x)-L|<\varepsilon." ).scale(0.9) formal.next_to(title, DOWN) self.play(Write(formal)) self.wait(1) # Introduce the concrete example example = MathTex(r"\text{Example: } f(x)=x^2,\ a=1,\ L=1").scale(0.9) example.next_to(formal, DOWN) self.play(FadeIn(example, shift=DOWN)) self.wait(0.5) # Set up axes and the graph of f(x)=x^2 axes = Axes( x_range=[-1, 3, 1], y_range=[-0.5, 4.5, 1], x_length=10, y_length=6, tips=False, ).to_edge(DOWN) x_label = axes.get_x_axis_label(MathTex("x")) y_label = axes.get_y_axis_label(MathTex("y")) graph = axes.plot(lambda x: x**2, x_range=[-1, 3], color=BLUE) graph_label = MathTex(r"f(x)=x^2").set_color(BLUE) graph_label.next_to(axes.c2p(2, 4), UP+RIGHT, buff=0.2) self.play(Create(axes), FadeIn(x_label), FadeIn(y_label)) self.play(Create(graph), FadeIn(graph_label, shift=UP)) self.wait(0.5) # Mark a=1 and L=1 a = 1 L = 1 pt_a = axes.c2p(a, L) a_vline = DashedLine(axes.c2p(a, axes.y_range[0]), axes.c2p(a, axes.y_range[1]), color=GREY) L_hline = DashedLine(axes.c2p(axes.x_range[0], L), axes.c2p(axes.x_range[1], L), color=GREY) a_label = MathTex(r"a=1").scale(0.8).next_to(axes.c2p(a, axes.y_range[0]), DOWN) L_label = MathTex(r"L=1").scale(0.8).next_to(axes.c2p(axes.x_range[0], L), LEFT) a_dot = Dot(axes.c2p(a, a**2), color=BLUE) self.play(Create(a_vline), Create(L_hline)) self.play(FadeIn(a_label), FadeIn(L_label), FadeIn(a_dot)) self.wait(0.5) # Put the formal implication for the example implication = MathTex(r"0<|x-1|<\delta\ \Rightarrow\ |x^2-1|<\varepsilon").scale(0.9) implication.next_to(example, DOWN) self.play(Write(implication)) self.wait(0.5) # Derivation to choose delta in terms of epsilon step1 = MathTex(r"|x^2-1|=|x-1|\,|x+1|").scale(0.9) step2 = MathTex(r"\text{If }|x-1|<1\ \Rightarrow\ |x+1|<3").scale(0.9) step3 = MathTex(r"\Rightarrow\ |x^2-1|<3\,|x-1|").scale(0.9) step4 = MathTex(r"\text{So if }|x-1|<\varepsilon/3,\ \ |x^2-1|<\varepsilon").scale(0.9) choice = MathTex(r"\boxed{\ \delta=\min\{1,\ \varepsilon/3\}\ }").set_color(GREEN) steps = VGroup(step1, step2, step3, step4, choice).arrange(DOWN, aligned_edge=LEFT, buff=0.2) steps.to_corner(UR).shift(LEFT*0.5 + DOWN*0.3) self.play(Write(step1)) self.play(Write(step2)) self.play(Write(step3)) self.play(Write(step4)) self.play(FadeIn(choice, scale=1.1)) self.wait(0.5) # Epsilon and delta visual bands with trackers eps_tracker = ValueTracker(0.6) def delta_value(): return min(1.0, eps_tracker.get_value()/3.0) x_min, x_max = axes.x_range[0], axes.x_range[1] y_min, y_max = axes.y_range[0], axes.y_range[1] # Epsilon strip (horizontal) and its dashed boundaries def make_epsilon_rect(): y_plus = L + eps_tracker.get_value() y_minus = L - eps_tracker.get_value() height = abs(axes.c2p(0, y_plus)[1] - axes.c2p(0, y_minus)[1]) width = axes.x_axis.get_length() rect = Rectangle(height=height, width=width, fill_color=YELLOW, fill_opacity=0.15, stroke_width=0) rect.move_to(axes.c2p((x_min + x_max)/2, L)) return rect epsilon_rect = always_redraw(make_epsilon_rect) eps_top = always_redraw(lambda: DashedLine( axes.c2p(x_min, L + eps_tracker.get_value()), axes.c2p(x_max, L + eps_tracker.get_value()), color=YELLOW, )) eps_bottom = always_redraw(lambda: DashedLine( axes.c2p(x_min, L - eps_tracker.get_value()), axes.c2p(x_max, L - eps_tracker.get_value()), color=YELLOW, )) # Delta strip (vertical) and its dashed boundaries def make_delta_rect(): d = delta_value() x_left = a - d x_right = a + d width = abs(axes.c2p(x_right, 0)[0] - axes.c2p(x_left, 0)[0]) height = axes.y_axis.get_length() rect = Rectangle(width=width, height=height, fill_color=GREEN, fill_opacity=0.15, stroke_width=0) rect.move_to(axes.c2p(a, (y_min + y_max)/2)) return rect delta_rect = always_redraw(make_delta_rect) delta_left = always_redraw(lambda: DashedLine( axes.c2p(a - delta_value(), y_min), axes.c2p(a - delta_value(), y_max), color=GREEN, )) delta_right = always_redraw(lambda: DashedLine( axes.c2p(a + delta_value(), y_min), axes.c2p(a + delta_value(), y_max), color=GREEN, )) # Arrows and labels for epsilon and delta sizes eps_arrow = always_redraw(lambda: DoubleArrow( axes.c2p(x_min + 0.15*(x_max-x_min), L - eps_tracker.get_value()), axes.c2p(x_min + 0.15*(x_max-x_min), L + eps_tracker.get_value()), color=YELLOW, buff=0 )) eps_label = always_redraw(lambda: MathTex(r"\varepsilon").set_color(YELLOW).scale(0.9).next_to(eps_arrow, LEFT, buff=0.1)) delta_arrow = always_redraw(lambda: DoubleArrow( axes.c2p(a - delta_value(), y_min + 0.1*(y_max-y_min)), axes.c2p(a + delta_value(), y_min + 0.1*(y_max-y_min)), color=GREEN, buff=0 )) delta_label = always_redraw(lambda: MathTex(r"\delta").set_color(GREEN).scale(0.9).next_to(delta_arrow, DOWN, buff=0.1)) # Example points within the delta-interval and their images left_point = always_redraw(lambda: Dot(axes.c2p(a - 0.8*delta_value(), (a - 0.8*delta_value())**2), color=MAROON)) right_point = always_redraw(lambda: Dot(axes.c2p(a + 0.8*delta_value(), (a + 0.8*delta_value())**2), color=MAROON)) # Dotted guides from these points to show they fall inside the epsilon band left_h = always_redraw(lambda: DashedLine( axes.c2p(x_min, (a - 0.8*delta_value())**2), axes.c2p(x_max, (a - 0.8*delta_value())**2), color=MAROON, stroke_opacity=0.7 )) right_h = always_redraw(lambda: DashedLine( axes.c2p(x_min, (a + 0.8*delta_value())**2), axes.c2p(x_max, (a + 0.8*delta_value())**2), color=MAROON, stroke_opacity=0.7 )) # Bring in the geometric picture self.play( FadeIn(epsilon_rect), Create(eps_top), Create(eps_bottom), FadeIn(delta_rect), Create(delta_left), Create(delta_right), ) self.play(Create(eps_arrow), FadeIn(eps_label), Create(delta_arrow), FadeIn(delta_label)) self.wait(0.5) self.play(FadeIn(left_point), FadeIn(right_point)) self.play(Create(left_h), Create(right_h)) self.wait(0.5) # Animate changing epsilon and see delta update via our choice self.play(eps_tracker.animate.set_value(0.9), run_time=1.5) self.wait(0.3) self.play(eps_tracker.animate.set_value(0.3), run_time=1.5) self.wait(0.3) self.play(eps_tracker.animate.set_value(0.6), run_time=1.2) self.wait(0.5) # Highlight the logical flow one more time box = SurroundingRectangle(implication, color=WHITE) self.play(Create(box)) self.wait(1) # Clean focus on the bands and statement self.play( FadeOut(step1), FadeOut(step2), FadeOut(step3), FadeOut(step4), FadeOut(choice), FadeOut(graph_label), FadeOut(a_dot), ) self.wait(0.5) final_remark = Tex("For every $\\varepsilon$-band around $L$, we can find a $\\delta$-window around $a$ so that $f(x)$ stays inside.").scale(0.9) final_remark.to_edge(DOWN) self.play(Write(final_remark)) self.wait(2)

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Reinforcement Learning Dynamics

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