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

1 files changed, 77 insertions, 0 deletions

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)