导数的几何意义

1from manim import *2 3# 配置中文环境4class DerivativeDefinition(Scene):5    def construct(self):6        # 使用黑色背景7        self.camera.background_color = BLACK8        9        # 创建坐标系 - 不显示数值10        axes = Axes(11            x_range=[-1, 4, 1],12            y_range=[-1.5, 1.5, 0.5],13            axis_config={"color": BLUE, "include_numbers": False},  # 移除数值14            x_length=6,15            y_length=616        )17        18        # 创建坐标轴标签，并手动调整y标签位置19        x_label = axes.get_x_axis_label("x")20        y_label = axes.get_y_axis_label("y").shift(DOWN * 0.5)  # 将y标签下移21        axes_labels = VGroup(x_label, y_label)22        23        # 创建新函数 f(x) = sin(x)，但只用通用表达式显示24        func = lambda x: np.sin(x)25        graph = axes.plot(func, color=BLUE, stroke_width=3)26        27        # 修改为通用表达式 y = f(x)28        func_label = MathTex("y = f(x)").next_to(graph, UP).shift(RIGHT)29        30        # 选择点 x₀ = π/4 (原来的a)31        x0 = np.pi/432        point_x0 = axes.coords_to_point(x0, func(x0))33        dot_x0 = Dot(point_x0, color=YELLOW, radius=0.08)  # 稍微增大点的半径34        35        # 添加点x₀的标签36        x0_label = MathTex("x_0").next_to(dot_x0, DOWN+RIGHT, buff=0.1).scale(0.9)37        38        # 创建切线 - 提前创建，一直显示39        exact_derivative = np.cos(x0)  # sin(x)的导数是cos(x)40        tangent_slope = exact_derivative41        tangent_line = Line(42            start=axes.coords_to_point(x0 - 2, func(x0) - 2*tangent_slope),  # 延长切线43            end=axes.coords_to_point(x0 + 2, func(x0) + 2*tangent_slope),    # 延长切线44            color=RED,45            stroke_width=2.546        )47        48        # 创建切线标签49        tangent_label = Text("切线", font="SimSun", color=RED).scale(0.7)50        # 将标签定位到切线的左下角51        tangent_label.next_to(52            axes.coords_to_point(x0 - 1.5, func(x0) - 1.5*tangent_slope),  # 切线左下方的点53            DOWN + LEFT,  # 放在这个点的左下方54            buff=0.255        )56        57        # 最终结论: 切线斜率 = 计算公式 = f'(x0)58        final_conclusion_1 = Text("切线斜率 =", font="SimSun").scale(0.7).to_corner(UL, buff=0.3)59        final_conclusion_2 = MathTex("\\lim_{\\Delta x \\to 0} \\frac{f(x_0+\\Delta x) - f(x_0)}{\\Delta x} = f'(x_0)").scale(0.7).next_to(final_conclusion_1, RIGHT, buff=0.1)60        61        # 在右侧显示割线斜率计算公式 - 调整为两行显示，并向右移动62        secant_slope_text = Text("割线斜率 =", font="SimSun").scale(0.7).to_edge(RIGHT, buff=2.0).shift(RIGHT * 1.0)  # 减小右边距，并额外右移63        secant_slope_formula_1 = MathTex("\\frac{f(x_0+\\Delta x) - f(x_0)}{\\Delta x}").scale(0.7).next_to(secant_slope_text, DOWN, buff=0.3).align_to(secant_slope_text, RIGHT)  # 对齐右侧64        65        # 初始设置66        self.play(67            Create(axes),68            Write(axes_labels),69            Create(graph),70            Write(func_label)71        )72        self.play(Create(dot_x0), Write(x0_label))73        self.wait()74        75        # 先显示切线，然后显示切线标签76        self.play(Create(tangent_line))77        self.play(Write(tangent_label))  # 添加切线标签78        self.add(tangent_line)  # 确保切线始终可见79        80        # 使用ValueTracker实现连续动画 - 变量h改为delta_x81        delta_x_tracker = ValueTracker(2.0)  # 起始Δx值更大，更明显82        83        # 创建依赖于delta_x_tracker的动态对象84        def get_secant_point():85            delta_x = delta_x_tracker.get_value()86            return axes.coords_to_point(x0 + delta_x, func(x0 + delta_x))87        88        dot_x0_plus_delta = Dot(color=GREEN, radius=0.08)89        dot_x0_plus_delta.add_updater(lambda d: d.move_to(get_secant_point()))90        91        # 添加x₀+Δx标签92        x0_plus_delta_label = MathTex("x_0 + \\Delta x").scale(0.8)93        x0_plus_delta_label.add_updater(94            lambda l: l.next_to(dot_x0_plus_delta, UP+RIGHT, buff=0.1)95        )96        97        # 创建延长的割线98        secant_line = Line(color=GREEN, stroke_width=4.0)99        100        # 更新函数，使割线延长101        def update_secant_line(line):102            delta_x = delta_x_tracker.get_value()103            end_point = axes.coords_to_point(x0 + delta_x, func(x0 + delta_x))104            105            # 计算延长线的向量方向106            direction = end_point - point_x0107            direction = direction / np.linalg.norm(direction)108            109            # 延长线的起点和终点110            extended_start = point_x0 - direction * 3  # 向后延长111            extended_end = end_point + direction * 3  # 向前延长112            113            line.put_start_and_end_on(extended_start, extended_end)114            return line115        116        secant_line.add_updater(update_secant_line)117        118        # 动态更新Δx值显示119        delta_x_label = Text("", font="SimSun")120        delta_x_label.add_updater(121            lambda t: t.become(122                Text(f"Δx = {delta_x_tracker.get_value():.2f}", font="SimSun")123            ).next_to(secant_slope_formula_1, DOWN, buff=0.3)124        )125        126        # 1. 首先显示割线和点 - 修改显示顺序127        self.play(Create(dot_x0_plus_delta), Write(x0_plus_delta_label), Create(secant_line))128        self.wait()129        130        # 2. 然后显示割线斜率计算公式 - 修改为两行显示131        self.play(Write(secant_slope_text), Write(secant_slope_formula_1), Write(delta_x_label))132        133        # 突出显示当前公式，表示极限过程134        limit_process_text = Text("极限过程", font="SimSun", color=YELLOW).scale(0.7).next_to(delta_x_label, DOWN, buff=0.5)135        limit_arrow = Arrow(limit_process_text.get_top(), delta_x_label.get_bottom(), buff=0.1, color=YELLOW)136        self.play(Write(limit_process_text), Create(limit_arrow))137        138        # 3. 执行Δx从大到小变化的动画，展示割线趋近于切线139        self.play(delta_x_tracker.animate.set_value(0.05), rate_func=lambda t: 1 - (1-t)**4, run_time=8)140        self.wait()141        142        # 4. 显示结论：割线趋近于切线143        conclusion = Text("当 Δx → 0 时，割线趋近于切线", font="SimSun").to_edge(DOWN)144        self.play(Write(conclusion))145        self.wait()146        147        # 5. 最后显示总结性公式：切线斜率 = 计算公式 = f'(x0)148        self.play(Write(final_conclusion_1), Write(final_conclusion_2))149        self.wait(2)