From 56ab459970db5ee7d05f3f30cdbd750f64f1800b Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 04:47:27 +0530 Subject: Rename FSF-2020/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py to FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py --- .../Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py | 29 ++++++++++++++++++++++ 1 file changed, 29 insertions(+) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py new file mode 100644 index 0000000..41c3b61 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py @@ -0,0 +1,29 @@ +from manimlib.imports import* + +#---- visualization of the function +class ExampleAnimation(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- f(x,y) = (y-x)(1-2x-3y) + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + (v-u)*(1-2*u-3*v) + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [PURPLE_B,PURPLE_C,PURPLE_D, PURPLE_E]).scale(1).fade(0.2).shift([0.2,0.2,0]) + + f_text = TextMobject("$f(x,y) = (y-x)(1-2x-3y)$").to_corner(UL) + + self.set_camera_orientation(phi = 60 * DEGREES, theta = 75 * DEGREES) + self.begin_ambient_camera_rotation(rate=0.1) + self.add_fixed_in_frame_mobjects(f_text) + self.wait(1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.wait(1) + self.play(Write(f)) + self.wait(4) -- cgit From 91f4e736577c6a4bf1f17b5eded2942905a0573d Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 04:49:41 +0530 Subject: Rename FSF-2020/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.py to FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.py --- .../file4_Types_of_critical_points.py | 134 +++++++++++++++++++++ 1 file changed, 134 insertions(+) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.py new file mode 100644 index 0000000..656fb68 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.py @@ -0,0 +1,134 @@ +from manimlib.imports import* + + +#---- Relative Maxima +class firstScene(ThreeDScene): + def construct(self): + + r_text = TextMobject("Relative Maximum at ORIGIN",color ='#87CEFA') + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #----graph of the function f(x,y) = -x**2-y**2 + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [YELLOW_B,YELLOW_C,YELLOW_D, YELLOW_E]).scale(1.5).shift([0,0,-0.51]).fade(0.3) + + f_text = TextMobject("$f(x,y) = -x^2-y^2$").to_corner(UL) + + d = Dot(color = "#800000").shift([0,0,0]) #---- critical point + + self.set_camera_orientation(phi = 75 * DEGREES, theta = 45 * DEGREES) + self.add_fixed_in_frame_mobjects(r_text) + self.wait(1) + self.play(FadeOut(r_text)) + self.begin_ambient_camera_rotation(rate = 0.1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(surface),Write(d)) + self.add_fixed_in_frame_mobjects(f_text) + self.wait(2) + self.play(FadeOut(axes),FadeOut(surface),FadeOut(f_text),FadeOut(d),FadeOut(label_x),FadeOut(label_y)) + + +#---- Relative Minima +class secondScene(ThreeDScene): + def construct(self): + + r2_text = TextMobject("Relative Minimum at ORIGIN",color ='#87CEFA') + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #----graph of the function g(x,y) = x**2+y**2 + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2+v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors =[TEAL_B,TEAL_C,TEAL_D,TEAL_E]).scale(1.5).shift([0,0,0.55]).fade(0.1) + + d = Dot(color = "#800000").shift([0,0,0]) #---- critical point + + g_text = TextMobject("$f(x,y) = x^2+y^2$").to_corner(UL) + + self.set_camera_orientation(phi = 75 * DEGREES, theta = 45 * DEGREES) + self.add_fixed_in_frame_mobjects(r2_text) + self.wait(1) + self.play(FadeOut(r2_text)) + self.begin_ambient_camera_rotation(rate = 0.1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(surface),Write(d)) + self.add_fixed_in_frame_mobjects(g_text) + self.wait(2) + self.play(FadeOut(axes),FadeOut(surface),FadeOut(g_text),FadeOut(d),FadeOut(label_x),FadeOut(label_y)) + + + +#---- Saddle Point +class thirdScene(ThreeDScene): + def construct(self): + + r3_text = TextMobject("Saddle Point", color = '#87CEFA') + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- graph of function h(x,y) = -x^2 + y^2 + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2+v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1,checkerboard_colors = [PURPLE_B,PURPLE_C,PURPLE_D,PURPLE_E]).scale(1.5).shift([0,0,0]) + + #---- curve(trace) along x axis + curve_x = ParametricSurface( + lambda u, v: np.array([ + u*0.4, + v, + v**2 + ]),v_min = -1, v_max = 1, u_min = -0.2, u_max = 0.2).shift([0,0,0.34]).scale(1.5).set_color("#800000") + + #---- curve(trace) along y axis + curve_y = ParametricSurface( + lambda u, v: np.array([ + u, + v*0.4, + -u**2 + ]),v_min = -0.2, v_max = 0.2, u_min = -1, u_max = 1).scale(1.6).shift([0,0,-0.1]).set_color("#800000") + + d = Dot(color = GREEN).shift([0,0,0.1]) #---- critical point + + h_text = TextMobject("$f(x,y) = -x^2+y^2$").to_corner(UL) + + self.add_fixed_in_frame_mobjects(r3_text) + self.wait(1) + self.set_camera_orientation(phi = 50 * DEGREES,theta = 45 * DEGREES) + self.play(FadeOut(r3_text)) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.begin_ambient_camera_rotation(rate = 0.3) + self.add_fixed_in_frame_mobjects(h_text) + self.play(Write(surface)) + self.wait(1) + self.add(curve_y) + self.add(d) + self.wait(1) + self.play(FadeOut(curve_y)) + self.wait(1) + self.add(curve_x) + self.wait(1) + self.add(d) + self.wait(1) -- cgit From 9c6d394101ad89681ef497cd0667fa23e6f3d168 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 04:50:58 +0530 Subject: Rename FSF-2020/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.py to FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.py --- ...file3_Tangent_plane_at_extrema_of_a_function.py | 73 ++++++++++++++++++++++ 1 file changed, 73 insertions(+) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.py new file mode 100644 index 0000000..e674113 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.py @@ -0,0 +1,73 @@ +from manimlib.imports import* + +#---- tangent plane to minima of the function +class firstScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- parabola: f(x,y) = x**2 + y**2 + parabola = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2+v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [GREEN_E,GREEN_D,GREEN_C,GREEN_B], resolution = (20, 20)).scale(1) + + d = Dot(np.array([0,0,0]), color = '#800000') # ---- critical point + + tangent_plane = Rectangle(fill_color = '#C0C0C0', fill_opacity = 0.3).move_to(ORIGIN).fade(0.7) # ----tangent plane + + parabola_text = TextMobject("Minimum with horizontal tangent plane").scale(0.7).to_corner(UL) + + self.set_camera_orientation(phi = 75 * DEGREES, theta = 45 * DEGREES) + self.begin_ambient_camera_rotation(rate = 0.2) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.add_fixed_in_frame_mobjects(parabola_text) + self.wait(1) + self.play(Write(parabola)) + self.play(ShowCreation(d)) + self.wait(1) + self.play(ShowCreation(tangent_plane)) + self.wait(2) + self.play(FadeOut(parabola_text),FadeOut(parabola),FadeOut(tangent_plane),FadeOut(d),FadeOut(label_x),FadeOut(label_y),FadeOut(axes)) + + +#---- tangent plane to maxima of the function +class secondScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #----parabola: g(x,y) = -x**2-y**2 + parabola = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [BLUE_E,BLUE_D,BLUE_C,BLUE_B], resolution = (20, 20)).scale(1) + + d = Dot(np.array([0,0,0]), color = '#800000') #---- critical point + + tangent_plane = Rectangle(fill_color = '#C0C0C0',fill_opacity = 0.3).move_to(ORIGIN).fade(0.7) #---- tangent plane + + parabola_text = TextMobject("Maximum with horizontal tangent plane").scale(0.7).to_corner(UL) + + self.set_camera_orientation(phi = 75 * DEGREES, theta = 45 * DEGREES) + self.begin_ambient_camera_rotation(rate = 0.2) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.add_fixed_in_frame_mobjects(parabola_text) + self.wait(1) + self.play(Write(parabola)) + self.play(ShowCreation(d)) + self.wait(1) + self.play(ShowCreation(tangent_plane)) + self.wait(2) -- cgit From a79446a128f65df2de58229ee5b48a12fe29cdd4 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 04:51:58 +0530 Subject: Rename FSF-2020/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.py to FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.py --- .../Critical-Points/file2_Traces_and_Tangent.py | 88 ++++++++++++++++++++++ 1 file changed, 88 insertions(+) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.py new file mode 100644 index 0000000..4b020e1 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.py @@ -0,0 +1,88 @@ +from manimlib.imports import* +import math as m + +#---- tangent to the trace with x constant +class firstScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes().scale(1) + label_x = TextMobject("$x$").shift([5.8,-0.5,0]) + label_y = TextMobject("$y$").shift([-0.5,-5.6,0]).rotate(-4.5) + + #---- graph of f(x,y) = -x^2-y^2 + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),u_min=-1,u_max=1, v_min=-1,v_max=1,checkerboard_colors=[PURPLE_C,PURPLE_D,PURPLE_E,PURPLE_B]).scale(1.5).shift([0,0,2]).rotate(0.2) + + #---- curve(trace) along y axis + curve = ParametricSurface( + lambda u, v: np.array([ + u*0.4, + v, + -v**2 + ]),v_min =-1 , v_max =1 , u_min = -0.1, u_max = 0.1).scale(1.6).shift([0.02,0.1,2.3]).set_color("#800000").rotate(0.1) + + d = Dot(color =YELLOW).shift([-0.05,-0.2,2.3]) #---- critical point + + x_text = TextMobject("Tangent to the trace with $x$ constant at critical point").shift(3*RIGHT+2*UP).scale(0.5).to_corner(UL) + + tangent_line = Line([-0.05,-1.5,2.3],[-0.05,1.5,2.3],color = '#228B22') + + self.add(axes) + self.set_camera_orientation(phi = 40 * DEGREES, theta = 55 * DEGREES) + self.begin_ambient_camera_rotation(rate = 0.1) + self.add(label_x) + self.add(label_y) + self.play(Write(surface)) + self.add_fixed_in_frame_mobjects(x_text) + self.add(curve) + self.wait(1) + self.play(Write(tangent_line),Write(d)) + self.wait(1) + + + +#---- tangent to the trace with y constant +class secondScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes().scale(1) + label_x = TextMobject("$x$").shift([5.8,-0.5,0]) + label_y = TextMobject("$y$").shift([-0.5,-5.6,0]).rotate(-4.5) + + #---- graph of f(x,y) = -x^2-y^2 + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + -u**2-v**2 + ]),u_min = -1, u_max = 1, v_min = -1, v_max = 1, checkerboard_colors = [PURPLE_B,PURPLE_C,PURPLE_D,PURPLE_E]).scale(1.5).shift([0,0,2]).rotate(0.2) + + #---- curve(trace) along x axis + curve = ParametricSurface( + lambda u, v: np.array([ + u, + v*0.4, + -u**2 + ]),v_min = -0.1, v_max = 0.1, u_min = -1, u_max = 1).scale(1.6).shift([0.07,0.1,2.3]).set_color("#800000") + + d = Dot(color = YELLOW).shift([0,-0.2,2.3]) #---- critical point + + tangent_line = Line(color = '#228B22').scale(1).shift([0,-0.2,2.3]).rotate(m.radians(190),LEFT) + + y_text = TextMobject("Tangent to the trace with $y$ constant at critical point").shift(3*RIGHT+2*UP).scale(0.5).to_corner(UL) + + self.add(axes) + self.set_camera_orientation(phi = 40 * DEGREES, theta = 55 * DEGREES) + self.add(label_x) + self.add(label_y) + self.begin_ambient_camera_rotation(rate = 0.1) + self.play(Write(surface)) + self.add_fixed_in_frame_mobjects(y_text) + self.add(curve) + self.wait(1.5) + self.play(Write(tangent_line),Write(d)) + self.wait(0.5) -- cgit From 9977b22482702c409517e391ba380f59c5cd557c Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 04:53:16 +0530 Subject: Rename FSF-2020/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.py to FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.py --- .../file1_Critical_Point_of_a_function.py | 77 ++++++++++++++++++++++ 1 file changed, 77 insertions(+) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.py new file mode 100644 index 0000000..e8cb08d --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.py @@ -0,0 +1,77 @@ +from manimlib.imports import* +import math as m + +#---- case 1: parial derivatives exist at critical point of the function +class firstScene(ThreeDScene): + def construct(self): + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- f(x,y) = e^(-10x^2-10y^2) + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + m.exp(-10*u**2-10*v**2) + ]),u_min = -1, u_max = 1, v_min = -1, v_max = 1, checkerboard_colors = [TEAL_E,TEAL_D,TEAL_C,TEAL_B]).fade(0.6).scale(3.5).shift([0,0,1.5]) + + l1 = Line([0,0,3.75],[0,0,0],color = '#800000') + + d = Dot([0,0,3.75],color = '#800000') #---- critical point + + d_text = TextMobject("$\\frac{\\partial f}{\\partial x}=\\frac{\\partial f}{\\partial y} = 0$").scale(0.8).to_corner(UL) + + f_text = TextMobject("Critical Point ",color = YELLOW).shift(3.4*UP).scale(0.5) + + self.set_camera_orientation(phi = 45*DEGREES, theta = 40*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.add_fixed_in_frame_mobjects(d_text) + self.begin_ambient_camera_rotation(rate = 0.2) + self.play(Write(surface)) + self.wait(1) + self.play(Write(l1)) + self.play(Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(f_text) + self.wait(3) + self.play(FadeOut(f_text),FadeOut(surface),FadeOut(axes),FadeOut(d_text),FadeOut(d),FadeOut(l1),FadeOut(label_x),FadeOut(label_y)) + + +#---- case 2: parial derivatives do not exist at critical point of the function +class secondScene(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- g(x,y)= |x|+|y| + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + abs(u)+abs(v) + ]),u_min = -1.5, u_max = 1.5, v_min = -1.5, v_max = 1.5, checkerboard_colors = [TEAL_E,TEAL_D,TEAL_C,TEAL_B]) + + d2 = Dot([0,0,0],color = '#800000') #---- critical point + + d2_text = TextMobject("$\\frac{\\partial f}{\\partial x}$ and/or $\\frac{\\partial f}{\\partial y}$ does not exist").scale(0.7).to_corner(UL) + + g_text = TextMobject("Critical Point",color = YELLOW).shift(1.2*RIGHT).scale(0.6) + + self.set_camera_orientation(phi = 60*DEGREES, theta = 40*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.add_fixed_in_frame_mobjects(d2_text) + self.begin_ambient_camera_rotation(rate = 0.2) + self.wait(1) + self.play(Write(surface2)) + self.wait(1) + self.play(Write(d2)) + self.wait(1) + self.add_fixed_in_frame_mobjects(g_text) + self.wait(2) -- cgit From b94975950b6d53d171cb15123a65c4d4abad75b4 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 04:54:27 +0530 Subject: Rename FSF-2020/approximations-and-optimizations/Critical-Points/README.md to FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md --- .../Critical-Points/README.md | 32 ++++++++++++++++++++++ 1 file changed, 32 insertions(+) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md new file mode 100644 index 0000000..f0747bb --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md @@ -0,0 +1,32 @@ +

SubTopic: Critical Points

+

+ +file1_Critical_Point_of_a_function + +![file1_Critical_Point_of_a_function](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif?raw=true) +

+

+ +file2_Traces_and_Tangent + +![file2_Traces_and_Tangent](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif?raw=true) +

+

+ +file3_Tangent_plane_at_extrema_of_a_function + +![file3_Tangent_plane_at_extrema_of_a_function](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif?raw=true) +

+

+ +file4_Types_of_critical_points + +![file4_Types_of_critical_points](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif?raw=true) +

+

+ +file5_f(x,y)=(y-x)(1-2x-3y) + +![file5_f(x,y)=(y-x)(1-2x-3y)](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file5_f(x%2Cy)%3D(y-x)(1-2x-3y).gif?raw=true) +

+

-- cgit From 6cb42c9f196cdc4c29671a4d5dbccaf1ebcc780e Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 05:10:35 +0530 Subject: Add files via upload --- .../file1_Critical_Point_of_a_function.gif | Bin 0 -> 8077401 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif new file mode 100644 index 0000000..ca3989c Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif differ -- cgit From dea86496011a0c1e907f04a0c998f67a120a95b6 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 05:11:15 +0530 Subject: Add files via upload --- .../Critical-Points/file2_Traces_and_Tangent.gif | Bin 0 -> 2552938 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif new file mode 100644 index 0000000..84acf2e Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif differ -- cgit From b902d20de6d51e29dddaaf8b7f2623f22ac2ef05 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 05:12:08 +0530 Subject: Add files via upload --- .../file3_Tangent_plane_at_extrema_of_a_function.gif | Bin 0 -> 2198637 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif new file mode 100644 index 0000000..14fb318 Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif differ -- cgit From 0ce6295d88d64bca989c79d6f92e2c12f8c02bc3 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 05:12:50 +0530 Subject: Add files via upload --- .../file4_Types_of_critical_points.gif | Bin 0 -> 3864765 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif new file mode 100644 index 0000000..91e7084 Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif differ -- cgit From 23d5769f486b79619bbef9da101fd08e299d59fe Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 05:13:41 +0530 Subject: Add files via upload --- .../Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).gif | Bin 0 -> 1522415 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).gif new file mode 100644 index 0000000..4bc92f8 Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).gif differ -- cgit From f2e7374ca9dcb5803490067da9cd937de2acc838 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Wed, 24 Jun 2020 05:15:06 +0530 Subject: Update README.md --- .../approximations-and-optimizations/Critical-Points/README.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md index f0747bb..5bd9cc5 100644 --- a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md @@ -3,30 +3,30 @@ file1_Critical_Point_of_a_function -![file1_Critical_Point_of_a_function](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif?raw=true) +![file1_Critical_Point_of_a_function](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file1_Critical_Point_of_a_function.gif?raw=true)



file2_Traces_and_Tangent -![file2_Traces_and_Tangent](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif?raw=true) +![file2_Traces_and_Tangent](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file2_Traces_and_Tangent.gif?raw=true)



file3_Tangent_plane_at_extrema_of_a_function -![file3_Tangent_plane_at_extrema_of_a_function](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif?raw=true) +![file3_Tangent_plane_at_extrema_of_a_function](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file3_Tangent_plane_at_extrema_of_a_function.gif?raw=true)



file4_Types_of_critical_points -![file4_Types_of_critical_points](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif?raw=true) +![file4_Types_of_critical_points](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif?raw=true)



file5_f(x,y)=(y-x)(1-2x-3y) -![file5_f(x,y)=(y-x)(1-2x-3y)](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/approximations-and-optimizations/Critical-Points/file5_f(x%2Cy)%3D(y-x)(1-2x-3y).gif?raw=true) +![file5_f(x,y)=(y-x)(1-2x-3y)](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x%2Cy)%3D(y-x)(1-2x-3y).gif?raw=true)



-- cgit From c6468ed920ee42e9b9b712065ffea02d1a426837 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 03:31:59 +0530 Subject: Rename file5_f(x,y)=(y-x)(1-2x-3y).py to file6_f(x,y)=(y-x)(1-2x-3y).py --- .../Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py | 29 ---------------------- .../Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).py | 29 ++++++++++++++++++++++ 2 files changed, 29 insertions(+), 29 deletions(-) delete mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py deleted file mode 100644 index 41c3b61..0000000 --- a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).py +++ /dev/null @@ -1,29 +0,0 @@ -from manimlib.imports import* - -#---- visualization of the function -class ExampleAnimation(ThreeDScene): - def construct(self): - axes = ThreeDAxes() - label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis - label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis - - #---- f(x,y) = (y-x)(1-2x-3y) - surface = ParametricSurface( - lambda u, v: np.array([ - u, - v, - (v-u)*(1-2*u-3*v) - ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [PURPLE_B,PURPLE_C,PURPLE_D, PURPLE_E]).scale(1).fade(0.2).shift([0.2,0.2,0]) - - f_text = TextMobject("$f(x,y) = (y-x)(1-2x-3y)$").to_corner(UL) - - self.set_camera_orientation(phi = 60 * DEGREES, theta = 75 * DEGREES) - self.begin_ambient_camera_rotation(rate=0.1) - self.add_fixed_in_frame_mobjects(f_text) - self.wait(1) - self.add(axes) - self.add(label_x) - self.add(label_y) - self.wait(1) - self.play(Write(f)) - self.wait(4) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).py new file mode 100644 index 0000000..41c3b61 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).py @@ -0,0 +1,29 @@ +from manimlib.imports import* + +#---- visualization of the function +class ExampleAnimation(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis + + #---- f(x,y) = (y-x)(1-2x-3y) + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + (v-u)*(1-2*u-3*v) + ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [PURPLE_B,PURPLE_C,PURPLE_D, PURPLE_E]).scale(1).fade(0.2).shift([0.2,0.2,0]) + + f_text = TextMobject("$f(x,y) = (y-x)(1-2x-3y)$").to_corner(UL) + + self.set_camera_orientation(phi = 60 * DEGREES, theta = 75 * DEGREES) + self.begin_ambient_camera_rotation(rate=0.1) + self.add_fixed_in_frame_mobjects(f_text) + self.wait(1) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.wait(1) + self.play(Write(f)) + self.wait(4) -- cgit From d26ae780a0b1f66c94ed8ffcf903cd23dc92846d Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 03:32:50 +0530 Subject: Delete file5_f(x,y)=(y-x)(1-2x-3y).gif --- .../Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).gif | Bin 1522415 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).gif deleted file mode 100644 index 4bc92f8..0000000 Binary files a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x,y)=(y-x)(1-2x-3y).gif and /dev/null differ -- cgit From 46c0d7092a97118b1268595236d74455e92bc550 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 03:35:36 +0530 Subject: Add files via upload --- .../Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).gif | Bin 0 -> 1522415 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).gif new file mode 100644 index 0000000..4bc92f8 Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x,y)=(y-x)(1-2x-3y).gif differ -- cgit From f1dfbd78a5fea187577cc648f0d251c897a9c907 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 03:36:38 +0530 Subject: Delete file4_Types_of_critical_points.gif --- .../file4_Types_of_critical_points.gif | Bin 3864765 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif deleted file mode 100644 index 91e7084..0000000 Binary files a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif and /dev/null differ -- cgit From cfba6959b00fab943e6d590a28bd57c24032d71e Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 03:37:33 +0530 Subject: Update and rename file4_Types_of_critical_points.py to file4_Relative_Maxima_and_Relative_Minima.py --- .../file4_Relative_Maxima_and_Relative_Minima.py | 51 ++++++++ .../file4_Types_of_critical_points.py | 134 --------------------- 2 files changed, 51 insertions(+), 134 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.py delete mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.py new file mode 100644 index 0000000..7999527 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.py @@ -0,0 +1,51 @@ +from manimlib.imports import* +import math as m + +#---- locating extrema of a funtion using critical points +class Extrema(ThreeDScene): + def construct(self): + + h_text = TextMobject("Relative Maxima and Relative Minima",color = GREEN) + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis + + #---- f(x,y) = 5(x+y)e^(-x^2-y^2) + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + 5*(u+v)*m.exp(-u**2-v**2) + ]),u_min = -PI, u_max = PI, v_min = -PI, v_max = PI).set_color(TEAL).shift([0,0,0]).fade(0.4) + + d1 = Dot(color = YELLOW).shift([0.5,0.5,3.02]) #---- critical point for maxima + l1 = Line([0.5,0.5,0.1],[0.5,0.5,3],color = YELLOW) + + d2 = Dot(color = YELLOW).shift([-1.15,0,-2.98]) #---- critical point for minima + l2 = Line([-1.15,0,0],[-1.15,0,-2.98],color = YELLOW) + + max_text = TextMobject("Relative Maxima").shift(3.1*UP+1.5*RIGHT).scale(0.5) + min_text = TextMobject("Relative Minima").shift(3.1*DOWN+1.5*LEFT).scale(0.5) + + self.add_fixed_in_frame_mobjects(h_text) + self.wait(1) + self.wait(1) + self.play(FadeOut(h_text)) + self.wait(1) + self.set_camera_orientation(phi = 100*DEGREES, theta = -40*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(surface)) + self.wait(1) + self.play(Write(l1),Write(d1)) + self.add_fixed_in_frame_mobjects(max_text) + self.wait(1) + self.play(Write(l2),Write(d2)) + self.add_fixed_in_frame_mobjects(min_text) + self.wait(1) + self.wait(1) + self.play(FadeOut(l1),FadeOut(d1),FadeOut(l2),FadeOut(d2),FadeOut(max_text),FadeOut(min_text)) + self.begin_ambient_camera_rotation(rate = 0.3) + self.wait(3) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.py deleted file mode 100644 index 656fb68..0000000 --- a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.py +++ /dev/null @@ -1,134 +0,0 @@ -from manimlib.imports import* - - -#---- Relative Maxima -class firstScene(ThreeDScene): - def construct(self): - - r_text = TextMobject("Relative Maximum at ORIGIN",color ='#87CEFA') - - axes = ThreeDAxes() - label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis - label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis - - #----graph of the function f(x,y) = -x**2-y**2 - surface = ParametricSurface( - lambda u, v: np.array([ - u, - v, - -u**2-v**2 - ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors = [YELLOW_B,YELLOW_C,YELLOW_D, YELLOW_E]).scale(1.5).shift([0,0,-0.51]).fade(0.3) - - f_text = TextMobject("$f(x,y) = -x^2-y^2$").to_corner(UL) - - d = Dot(color = "#800000").shift([0,0,0]) #---- critical point - - self.set_camera_orientation(phi = 75 * DEGREES, theta = 45 * DEGREES) - self.add_fixed_in_frame_mobjects(r_text) - self.wait(1) - self.play(FadeOut(r_text)) - self.begin_ambient_camera_rotation(rate = 0.1) - self.add(axes) - self.add(label_x) - self.add(label_y) - self.play(Write(surface),Write(d)) - self.add_fixed_in_frame_mobjects(f_text) - self.wait(2) - self.play(FadeOut(axes),FadeOut(surface),FadeOut(f_text),FadeOut(d),FadeOut(label_x),FadeOut(label_y)) - - -#---- Relative Minima -class secondScene(ThreeDScene): - def construct(self): - - r2_text = TextMobject("Relative Minimum at ORIGIN",color ='#87CEFA') - - axes = ThreeDAxes() - label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis - label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis - - #----graph of the function g(x,y) = x**2+y**2 - surface = ParametricSurface( - lambda u, v: np.array([ - u, - v, - u**2+v**2 - ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1, checkerboard_colors =[TEAL_B,TEAL_C,TEAL_D,TEAL_E]).scale(1.5).shift([0,0,0.55]).fade(0.1) - - d = Dot(color = "#800000").shift([0,0,0]) #---- critical point - - g_text = TextMobject("$f(x,y) = x^2+y^2$").to_corner(UL) - - self.set_camera_orientation(phi = 75 * DEGREES, theta = 45 * DEGREES) - self.add_fixed_in_frame_mobjects(r2_text) - self.wait(1) - self.play(FadeOut(r2_text)) - self.begin_ambient_camera_rotation(rate = 0.1) - self.add(axes) - self.add(label_x) - self.add(label_y) - self.play(Write(surface),Write(d)) - self.add_fixed_in_frame_mobjects(g_text) - self.wait(2) - self.play(FadeOut(axes),FadeOut(surface),FadeOut(g_text),FadeOut(d),FadeOut(label_x),FadeOut(label_y)) - - - -#---- Saddle Point -class thirdScene(ThreeDScene): - def construct(self): - - r3_text = TextMobject("Saddle Point", color = '#87CEFA') - - axes = ThreeDAxes() - label_x = TextMobject("$x$").shift([5.5,-0.5,0]) #---- x axis - label_y = TextMobject("$y$").shift([-0.5,5.5,0]).rotate(-4.5) #---- y axis - - #---- graph of function h(x,y) = -x^2 + y^2 - surface = ParametricSurface( - lambda u, v: np.array([ - u, - v, - -u**2+v**2 - ]),v_min = -1, v_max = 1, u_min = -1, u_max = 1,checkerboard_colors = [PURPLE_B,PURPLE_C,PURPLE_D,PURPLE_E]).scale(1.5).shift([0,0,0]) - - #---- curve(trace) along x axis - curve_x = ParametricSurface( - lambda u, v: np.array([ - u*0.4, - v, - v**2 - ]),v_min = -1, v_max = 1, u_min = -0.2, u_max = 0.2).shift([0,0,0.34]).scale(1.5).set_color("#800000") - - #---- curve(trace) along y axis - curve_y = ParametricSurface( - lambda u, v: np.array([ - u, - v*0.4, - -u**2 - ]),v_min = -0.2, v_max = 0.2, u_min = -1, u_max = 1).scale(1.6).shift([0,0,-0.1]).set_color("#800000") - - d = Dot(color = GREEN).shift([0,0,0.1]) #---- critical point - - h_text = TextMobject("$f(x,y) = -x^2+y^2$").to_corner(UL) - - self.add_fixed_in_frame_mobjects(r3_text) - self.wait(1) - self.set_camera_orientation(phi = 50 * DEGREES,theta = 45 * DEGREES) - self.play(FadeOut(r3_text)) - self.add(axes) - self.add(label_x) - self.add(label_y) - self.begin_ambient_camera_rotation(rate = 0.3) - self.add_fixed_in_frame_mobjects(h_text) - self.play(Write(surface)) - self.wait(1) - self.add(curve_y) - self.add(d) - self.wait(1) - self.play(FadeOut(curve_y)) - self.wait(1) - self.add(curve_x) - self.wait(1) - self.add(d) - self.wait(1) -- cgit From 90aeb7ca1eb59ee91a49852045e4e3b1614a515f Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 03:39:45 +0530 Subject: Create file5_Saddle_Point.py --- .../Critical-Points/file5_Saddle_Point.py | 71 ++++++++++++++++++++++ 1 file changed, 71 insertions(+) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.py new file mode 100644 index 0000000..67dbb18 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.py @@ -0,0 +1,71 @@ +from manimlib.imports import* +import math as m + +#---- saddle point of a function +class SaddlePoint(ThreeDScene): + def construct(self): + + h_text = TextMobject("Saddle Point",color = GREEN) + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis + + #---- f(x,y) = -x^2-y^2 + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + u**2-v**2 + ]),u_min = -1, u_max = 1, v_min = -1, v_max = 1,checkerboard_colors = [BLUE_B,BLUE_C,BLUE_D,BLUE_E]).shift([0,0,0]).scale(3) + + #---- curve(trace) along y axis + curve_x = ParametricSurface( + lambda u, v: np.array([ + u*0.1, + v, + v**2 + ]),v_min = -1, v_max = 1, u_min = -0.2, u_max = 0.2).shift([0,0,-2]).scale(3.1).set_color("#800000").rotate(m.radians(180),UP) + + x_text = TextMobject("A dip at critical point along x axis").scale(0.5).to_corner(UL) + + #---- curve(trace) along x axis + curve_y = ParametricSurface( + lambda u, v: np.array([ + u, + v*0.1, + -u**2 + ]),v_min = -0.2, v_max = 0.2, u_min = -1, u_max = 1).scale(3).shift([0.1,0,2.2]).set_color("#800000").rotate(m.radians(182),DOWN) + + y_text = TextMobject("A peak at critical point along y axis").scale(0.5).to_corner(UL) + + d = Dot(color = YELLOW).shift([0,-0.22,0]) #---- critical point(saddle point) + + self.add_fixed_in_frame_mobjects(h_text) + self.wait(1) + self.play(FadeOut(h_text)) + self.wait(1) + self.set_camera_orientation(phi = 75*DEGREES, theta = 40*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(surface)) + self.wait(1) + self.move_camera(phi = 45*DEGREES, theta = 70*DEGREES) + self.add(curve_y) + self.play(Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(x_text) + self.wait(1) + self.wait(1) + self.play(FadeOut(curve_y),FadeOut(d),FadeOut(x_text)) + self.wait(1) + self.move_camera(phi = 40*DEGREES, theta = 30*DEGREES) + self.add(curve_x) + self.play(Write(d)) + self.wait(1) + self.add_fixed_in_frame_mobjects(y_text) + self.begin_ambient_camera_rotation(rate = 0.3) + self.wait(3) + self.play(FadeOut(curve_x),FadeOut(d),FadeOut(y_text)) + self.wait(1) -- cgit From b8e7cacb5bc993e0e760486270c0314d85ec943d Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 03:42:38 +0530 Subject: Add files via upload --- .../file4_Relative_Maxima_and_Relative_Minima.gif | Bin 0 -> 1574608 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.gif new file mode 100644 index 0000000..30cea35 Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.gif differ -- cgit From 8e3006aadce5dfa840e0a45a94f569711daa8805 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 03:44:16 +0530 Subject: Add files via upload --- .../Critical-Points/file5_Saddle_Point.gif | Bin 0 -> 7136893 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.gif new file mode 100644 index 0000000..7300f3a Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.gif differ -- cgit From ca9d449b68c2393a58dcb090f664739a8282919f Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Fri, 26 Jun 2020 03:46:19 +0530 Subject: Update README.md --- .../Critical-Points/README.md | 12 +++++++++--- 1 file changed, 9 insertions(+), 3 deletions(-) (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md index 5bd9cc5..6cf8f40 100644 --- a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md @@ -21,12 +21,18 @@ file4_Types_of_critical_points -![file4_Types_of_critical_points](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Types_of_critical_points.gif?raw=true) +![file4_Relative_Maxima_and_Relative_Minima](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.gif?raw=true)



-file5_f(x,y)=(y-x)(1-2x-3y) +file5_Saddle_Point -![file5_f(x,y)=(y-x)(1-2x-3y)](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_f(x%2Cy)%3D(y-x)(1-2x-3y).gif?raw=true) +![file5_Saddle_Point](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file5_Saddle_Point.gif?raw=true) +

+

+ +file6_f(x,y)=(y-x)(1-2x-3y) + +![file6_f(x,y)=(y-x)(1-2x-3y)](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file6_f(x%2Cy)%3D(y-x)(1-2x-3y).gif?raw=true)



-- cgit From a68eb145ac849444d222c3eaea619e7dfca9f410 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Sat, 27 Jun 2020 02:50:19 +0530 Subject: Update and rename file4_Relative_Maxima_and_Relative_Minima.py to file4_Relative_Maximum_and_Relative_Minimum.py --- .../file4_Relative_Maxima_and_Relative_Minima.py | 51 ---------------------- .../file4_Relative_Maximum_and_Relative_Minimum.py | 51 ++++++++++++++++++++++ 2 files changed, 51 insertions(+), 51 deletions(-) delete mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.py create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.py (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.py deleted file mode 100644 index 7999527..0000000 --- a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.py +++ /dev/null @@ -1,51 +0,0 @@ -from manimlib.imports import* -import math as m - -#---- locating extrema of a funtion using critical points -class Extrema(ThreeDScene): - def construct(self): - - h_text = TextMobject("Relative Maxima and Relative Minima",color = GREEN) - - axes = ThreeDAxes() - label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis - label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis - - #---- f(x,y) = 5(x+y)e^(-x^2-y^2) - surface = ParametricSurface( - lambda u, v: np.array([ - u, - v, - 5*(u+v)*m.exp(-u**2-v**2) - ]),u_min = -PI, u_max = PI, v_min = -PI, v_max = PI).set_color(TEAL).shift([0,0,0]).fade(0.4) - - d1 = Dot(color = YELLOW).shift([0.5,0.5,3.02]) #---- critical point for maxima - l1 = Line([0.5,0.5,0.1],[0.5,0.5,3],color = YELLOW) - - d2 = Dot(color = YELLOW).shift([-1.15,0,-2.98]) #---- critical point for minima - l2 = Line([-1.15,0,0],[-1.15,0,-2.98],color = YELLOW) - - max_text = TextMobject("Relative Maxima").shift(3.1*UP+1.5*RIGHT).scale(0.5) - min_text = TextMobject("Relative Minima").shift(3.1*DOWN+1.5*LEFT).scale(0.5) - - self.add_fixed_in_frame_mobjects(h_text) - self.wait(1) - self.wait(1) - self.play(FadeOut(h_text)) - self.wait(1) - self.set_camera_orientation(phi = 100*DEGREES, theta = -40*DEGREES) - self.add(axes) - self.add(label_x) - self.add(label_y) - self.play(Write(surface)) - self.wait(1) - self.play(Write(l1),Write(d1)) - self.add_fixed_in_frame_mobjects(max_text) - self.wait(1) - self.play(Write(l2),Write(d2)) - self.add_fixed_in_frame_mobjects(min_text) - self.wait(1) - self.wait(1) - self.play(FadeOut(l1),FadeOut(d1),FadeOut(l2),FadeOut(d2),FadeOut(max_text),FadeOut(min_text)) - self.begin_ambient_camera_rotation(rate = 0.3) - self.wait(3) diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.py b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.py new file mode 100644 index 0000000..3bd810d --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.py @@ -0,0 +1,51 @@ +from manimlib.imports import* +import math as m + +#---- locating extrema of a funtion using critical points +class Extrema(ThreeDScene): + def construct(self): + + h_text = TextMobject("Relative Maximum and Relative Minimum",color = GREEN) + + axes = ThreeDAxes() + label_x = TextMobject("$x$").shift([5.5,-0.3,0]) #---- x axis + label_y = TextMobject("$y$").shift([-0.3,5.5,0]).rotate(-4.5) #---- y axis + + #---- f(x,y) = 5(x+y)e^(-x^2-y^2) + surface = ParametricSurface( + lambda u, v: np.array([ + u, + v, + 5*(u+v)*m.exp(-u**2-v**2) + ]),u_min = -PI, u_max = PI, v_min = -PI, v_max = PI).set_color(TEAL).shift([0,0,0]).fade(0.4) + + d1 = Dot(color = YELLOW).shift([0.5,0.5,3.02]) #---- critical point for maxima + l1 = Line([0.5,0.5,0.1],[0.5,0.5,3],color = YELLOW) + + d2 = Dot(color = YELLOW).shift([-1.15,0,-2.98]) #---- critical point for minima + l2 = Line([-1.15,0,0],[-1.15,0,-2.98],color = YELLOW) + + max_text = TextMobject("Relative Maximum").shift(3.1*UP+1.5*RIGHT).scale(0.5) + min_text = TextMobject("Relative Minimum").shift(3.1*DOWN+1.5*LEFT).scale(0.5) + + self.add_fixed_in_frame_mobjects(h_text) + self.wait(1) + self.wait(1) + self.play(FadeOut(h_text)) + self.wait(1) + self.set_camera_orientation(phi = 100*DEGREES, theta = -40*DEGREES) + self.add(axes) + self.add(label_x) + self.add(label_y) + self.play(Write(surface)) + self.wait(1) + self.play(Write(l1),Write(d1)) + self.add_fixed_in_frame_mobjects(max_text) + self.wait(1) + self.play(Write(l2),Write(d2)) + self.add_fixed_in_frame_mobjects(min_text) + self.wait(1) + self.wait(1) + self.play(FadeOut(l1),FadeOut(d1),FadeOut(l2),FadeOut(d2),FadeOut(max_text),FadeOut(min_text)) + self.begin_ambient_camera_rotation(rate = 0.3) + self.wait(3) -- cgit From ff66437567d0d7923d0e2291edba668aa05e7197 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Sat, 27 Jun 2020 02:50:43 +0530 Subject: Delete file4_Relative_Maxima_and_Relative_Minima.gif --- .../file4_Relative_Maxima_and_Relative_Minima.gif | Bin 1574608 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.gif deleted file mode 100644 index 30cea35..0000000 Binary files a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.gif and /dev/null differ -- cgit From 825d0d1b187d3d378869f8d77cfe1af5bd9c8a67 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Sat, 27 Jun 2020 02:51:20 +0530 Subject: Update README.md --- .../approximations-and-optimizations/Critical-Points/README.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md index 6cf8f40..83abfd3 100644 --- a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md @@ -19,9 +19,9 @@



-file4_Types_of_critical_points +file4_Relative_Maximum_and_Relative_Minimum -![file4_Relative_Maxima_and_Relative_Minima](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maxima_and_Relative_Minima.gif?raw=true) +![file4_Relative_Maxima_and_Relative_Minima](?raw=true)



-- cgit From 8ead1443c9c8f9eac9fa698e78c4dabfbac6c188 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Sat, 27 Jun 2020 02:52:12 +0530 Subject: Add files via upload --- .../file4_Relative_Maximum_and_Relative_Minimum.gif | Bin 0 -> 1587319 bytes 1 file changed, 0 insertions(+), 0 deletions(-) create mode 100644 FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.gif (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.gif b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.gif new file mode 100644 index 0000000..6b93359 Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.gif differ -- cgit From 9950f6e2906a69175c0474545ec489a952735920 Mon Sep 17 00:00:00 2001 From: Vaishnavi Date: Sat, 27 Jun 2020 02:54:14 +0530 Subject: Update README.md --- .../approximations-and-optimizations/Critical-Points/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) (limited to 'FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points') diff --git a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md index 83abfd3..857d298 100644 --- a/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md +++ b/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/README.md @@ -21,7 +21,7 @@ file4_Relative_Maximum_and_Relative_Minimum -![file4_Relative_Maxima_and_Relative_Minima](?raw=true) +![file4_Relative_Maxima_and_Relative_Minima](https://github.com/vnb09/FSF-mathematics-python-code-archive/blob/fsf_tasks/FSF-2020/calculus-of-several-variables/approximations-and-optimizations/Critical-Points/file4_Relative_Maximum_and_Relative_Minimum.gif?raw=true)



-- cgit