JNB LAB: Calculus Animations

8.9. JNB LAB: Calculus Animations#

In this JNB, we will show how to create animations that illustrate basic calculus concepts. To create an animation, we use a for loop to create single frames of the animation ‘0.png’, ‘1.png’, ‘2.png’,…. and then use a block of code to combine the individual frames into a .gif animation.

8.9.1. Tangent Lines#

Let’s begin by showing how secants approach a tangent line. We will use the function \(f(x)=x^2\) and create a sequence of secants that approach the tangent line at point \(x=4\).

First, let’s define our function and specify the point of tangency.

f = lambda x : x**2 # Define the function f(x)
a=4   #x coordinate of point of tangency

Now let’s create the first frame (\(n=0\)) of the sequence of images \(\{n.png\}\) used to create the animation.

To create our animation, we will:

i) import a couple of functions used to create the animation

ii) specify the number of frames

iii) specify our function and point of tangency (assuming we haven’t done so previously)

iv) use a for loop to create the individual frames in the animation

v) combine the frames into an animation file “secant.gif” which is found in the same folder as this lab.

Exercise#

Exercise

1.1 Create an animation with 3 frames which shows secants approaching the tangent line to \(y=\frac{1}{x}\) at \(x=1\).

8.9.2. Riemann Sums#

Just as the tangent line is obtained by a limiting process, the area under curve is obtainable as a limit of a Riemann Sum. We can create an animation which shows the approximating rectangles filling in the area under a curve. For example, we consider the area under \(y=1/x\) in the interval \(1\le x \le 5\).

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.animation as animation
from matplotlib.animation import FuncAnimation
%matplotlib notebook
f = lambda x : 1/x
x=np.arange(1,6,.0001)
y=f(x)
frames=10
for n in np.arange(0,frames,1):
    plt.figure(figsize=(10,7))
    plt.gca().set_xticks(np.arange(1,7,1))
    plt.gca().set_yticks([.1,.2,.3,.4,.5,.6,.7,.8,.9,1])
    plt.plot(x,y,'k',markersize=1)
    x_left = np.linspace(1,6,n+2)
    x_left=x_left[:-1]
    plt.bar(x_left,1/x_left,width=5/(n+1),alpha=1-.9*(1/(n+1)),align='edge',color='r',edgecolor='k')
    plt.title('Riemann Sums')
    plt.xlabel('x')
    plt.ylabel('y')
    plt.ylim((0,1.1)) 
    plt.text(3,1,"Area Under Curve = ln(5) = 1.609...")
    Rn=0
    for k in np.arange(0,n+1,1):
        Rn=Rn+(5/(n+1))*(1/x_left[k])      
    plt.text(4,.9, "R"+str(n+1)+"="+str(Rn))
    #plt.grid()
    plt.savefig(str(n)+'.png')
    plt.show()
#----- Save the Animation as riemannsums.gif
from PIL import Image
images = []
for n in range(frames):
    exec('a'+str(n)+'=Image.open("'+str(n)+'.png")')
    images.append(eval('a'+str(n)))
images[0].save('riemannsums.gif',
               save_all=True,
               append_images=images[1:],
               duration=800,
               loop=0)

Exercise#

Exercise

2.1 Create an animation for a Riemann sum approximation of \(\int_0^{\pi/2}\cos(x)\,dx\).

8.9.3. Parametric Curves#

JNBs can be used to create .gif animations of parametric curves. For example, the code below is used to animate a cycloid curve traced by a point on the rim of a wheel.

Exercise#

Exercise

3.1 A hypocycloid curve is formed by a point on a circle rotating along the inside rim of another circle. The parametric equations are

\(x(t) = (a-b) \cos t + b \cos(\frac{a-b}{b}t)\)

\(y(t) = (a-b) \sin t - b \sin(\frac{a-b}{b}t)\)

Create an animation of a hypocycloid curve for the case \(a=1\), \(b=1/4\). Your animation should resemble the one shown below.

JNB LAB: Calculus Animations

Contents

8.9. JNB LAB: Calculus Animations#

8.9.1. Tangent Lines#

Exercise#

8.9.2. Riemann Sums#

Exercise#

8.9.3. Parametric Curves#

Exercise#