A Visual Introduction to Differentiation (using vpython)

Posted October 15, 2015

by Lensyl Urbano

Screen capture: Enter an x value and the program calculates the slope for the function and draws the tangent line.

Screen capture: Enter an x value and the program calculates the slope for the function and draws the tangent line.

This quick program is intended to introduce differentiation as a way of finding the slope of a line. Students know how to find the slope of a tangent line at least conceptually (by drawing). We pick a curve: in this case:

f(x) = x^2

then enter values of x in the program to see how x, the function value and the differential compare to each other.

x f(x) f'(x)
0.5 0.25 1
1 1 2
2 2 4
3 9 6

Because it’s quick you have to change the function in the code, and enter the values for x in the python shell.

With a sin curve.

With a sin curve.

differentiation_intro_numeric.py

from visual import *

class tangent_line:
    def __init__(self):
        self.dx = 0.1
        self.line = curve()
        self.tangent_line = curve()
        self.point = sphere(radius=.25,color=color.yellow)
        self.point.visible = False
        self.label = label(pos=(-5,-8))

    '''CHANGE FUNCTION (y) HERE'''
    # the original function
    def f(self, x):
        #y = sin(x)
        y = x**2
        return y
    '''END CHANGE FUNCTION HERE'''

    def find_slope(self, x):
        sdx = .00001
        m = (self.f(x+sdx)-self.f(x))/sdx
        return round(m,3)
        
    def draw(self):
        for x in arange(xmin, xmax+self.dx, self.dx):
            self.line.append(pos=(x, self.f(x)))

    def draw_tangent(self, x):
        m = self.find_slope(x)
        y = self.f(x)
        b = y - m * x
        print "When x = ", x, " slope = ", m
        self.label.text = "point: (%1.2f, %1.2f)\nSlope: %1.2f" % (x,y,m)
        self.plot_point(x)

        #draw tangent
        self.tangent_line.visible = False
        self.tangent_line = curve(pos=[(xmin,m*xmin+b),(xmax,m*xmax+b)], color=color.yellow)
              
    def plot_point(self, x):
        self.point.visible = True
        self.point.pos = (x, self.f(x)) 

#axes
xmin = -10.
xmax = 10.
ymin = -10.
ymax = 10.
xaxis = curve(pos=[(xmin,0),(xmax,0)])
yaxis = curve(pos=[(0,ymin),(0,ymax)])

#tick marks
tic_dx = 1.0
tic_h = .5
for i in arange(xmin,xmax+tic_dx,tic_dx):
    tic = curve(pos=[(i,-0.5*tic_h),(i,0.5*tic_h)])
for i in arange(ymin,ymax+tic_dx,tic_dx):
    tic = curve(pos=[(-0.5*tic_h,i),(0.5*tic_h,i)])

#stop scene from zooming out too far when the curve is drawn
scene.autoscale = False

# draw curve
func = tangent_line()
func.draw()

# get input
while 1:
    xin = raw_input("Enter x value: ")
    func.draw_tangent(float(xin))


Citing this post: Urbano, L., 2015. A Visual Introduction to Differentiation (using vpython), Retrieved March 30th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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