avg_rate_change.ipynb

Kernel: SageMath (stable)

Suppose that the height $s$ of a ball at time $t$ (in seconds) is described by $s(t)=64-16(t-1)^2$

import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt

t = np.arange(-3,10,0.1)
s = 64-16*(t-1)**2
plt.xlabel("time")
plt.ylabel("height")
plt.title(r"Height as a function of time from $t=-3$ to $t=10$")
plt.plot(t,s)
plt.show()

What is the average velocity of the ball between $t=0$ and $t=5$?

We can answer this question using the formula

$$rate = \frac{distance}{time}$$

This formula actually gives the average rate over a time interval.

T = np.array([0,5])
S = 64-16*(T-1)**2
plt.scatter(T,S)

plt.plot([T[0],T[0],T[1],T[0]],[S[0],S[1],S[1],S[0]])

t = np.arange(-3,10,0.1)
s = 64-16*(t-1)**2
plt.xlabel("time")
plt.ylabel("height")
plt.title(r"Height as a function of time from $t=-3$ to $t=10$")
plt.plot(t,s)



plt.show()

The average velocity between $t=0$ and $t=5$ is the slope of the hypotenuse of the blue triange.

We call this the secant line.

The slope of the secant line between $(t_1,s_1)$ and $(t_2,s_2)$ is given by the formula

$$\frac{s_2-s_1}{t_2-t_1}$$

In the case of this particular problem we can compute this value like so:

t1,s1 = 0,64-16*(0-1)**2
t2,s2 = 5,64-16*(5-1)**2

(s2-s1)/(t2-t1)

-48

This means that over this time interval the average velocity of the ball was $48$ ft per second downward.

import matplotlib.lines as mlines

def newline(p1, p2,lab="",axis=plt):
    ax = axis.gca()
    xmin, xmax = ax.get_xbound()

    if(p2[0] == p1[0]):
        xmin = xmax = p1[0]
        ymin, ymax = ax.get_ybound()
    else:
        ymax = p1[1]+(p2[1]-p1[1])/(p2[0]-p1[0])*(xmax-p1[0])
        ymin = p1[1]+(p2[1]-p1[1])/(p2[0]-p1[0])*(xmin-p1[0])

    l = mlines.Line2D([xmin,xmax], [ymin,ymax],label=lab,color='purple')
    ax.add_line(l)
    return l


t = np.arange(-3,10,0.1)
s = 64-16*(t-1)**2
plt.xlabel("time")
plt.ylabel("height")
plt.title(r"Height as a function of time from $t=-3$ to $t=10$")
plt.plot(t,s)

newline([T[0],S[0]],[T[1],S[1]],"secant line")

tangent = 32*t+48
plt.plot(t,tangent,label="tangent line")
plt.legend()
plt.show()

It's interesting to compare the secant line to the tangent line.

Let's look at the secant that comes from computing the average velocity over the time period $t=0$ to $t=3$

import matplotlib.lines as mlines

t = np.arange(-3,10,0.1)
s = 64-16*(t-1)**2
plt.xlabel("time")
plt.ylabel("height")
plt.title(r"Height as a function of time from $t=-3$ to $t=10$")
plt.plot(t,s)

T=np.array([0,.1])
S= 64-16*(T-1)**2

newline([T[0],S[0]],[T[1],S[1]],"secant line")

tangent = 32*t+48
plt.plot(t,tangent,label="tangent line")
plt.legend()
plt.show()

It seems that on this shorter interval, the secant line is more like the tangent line.

Remember that the meaning of the slope of the secant line is average velocity.

What's the average velocity given by the new secant line?

You can see from the calculation below that it's -16 ft/s.

t1,s1 = 0,64-16*(0-1)**2
t2,s2 = 3,64-16*(3-1)**2

(s2-s1)/(t2-t1)

-16.0

plt.plot(t,s)

[<matplotlib.lines.Line2D object at 0x7fd965943790>]

We can look at the average velocity on small intervals near each point.

np.diff([1,2,3,4])

array([1, 1, 1])

dt = np.diff(t)
ds = np.diff(s)

dt

array([ 0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,
        0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1,  0.1])

ds

array([ 12.64,  12.32,  12.  ,  11.68,  11.36,  11.04,  10.72,  10.4 ,
        10.08,   9.76,   9.44,   9.12,   8.8 ,   8.48,   8.16,   7.84,
         7.52,   7.2 ,   6.88,   6.56,   6.24,   5.92,   5.6 ,   5.28,
         4.96,   4.64,   4.32,   4.  ,   3.68,   3.36,   3.04,   2.72,
         2.4 ,   2.08,   1.76,   1.44,   1.12,   0.8 ,   0.48,   0.16,
        -0.16,  -0.48,  -0.8 ,  -1.12,  -1.44,  -1.76,  -2.08,  -2.4 ,
        -2.72,  -3.04,  -3.36,  -3.68,  -4.  ,  -4.32,  -4.64,  -4.96,
        -5.28,  -5.6 ,  -5.92,  -6.24,  -6.56,  -6.88,  -7.2 ,  -7.52,
        -7.84,  -8.16,  -8.48,  -8.8 ,  -9.12,  -9.44,  -9.76, -10.08,
       -10.4 , -10.72, -11.04, -11.36, -11.68, -12.  , -12.32, -12.64,
       -12.96, -13.28, -13.6 , -13.92, -14.24, -14.56, -14.88, -15.2 ,
       -15.52, -15.84, -16.16, -16.48, -16.8 , -17.12, -17.44, -17.76,
       -18.08, -18.4 , -18.72, -19.04, -19.36, -19.68, -20.  , -20.32,
       -20.64, -20.96, -21.28, -21.6 , -21.92, -22.24, -22.56, -22.88,
       -23.2 , -23.52, -23.84, -24.16, -24.48, -24.8 , -25.12, -25.44,
       -25.76, -26.08, -26.4 , -26.72, -27.04, -27.36, -27.68, -28.  ,
       -28.32])

ds/dt

array([ 126.4,  123.2,  120. ,  116.8,  113.6,  110.4,  107.2,  104. ,
        100.8,   97.6,   94.4,   91.2,   88. ,   84.8,   81.6,   78.4,
         75.2,   72. ,   68.8,   65.6,   62.4,   59.2,   56. ,   52.8,
         49.6,   46.4,   43.2,   40. ,   36.8,   33.6,   30.4,   27.2,
         24. ,   20.8,   17.6,   14.4,   11.2,    8. ,    4.8,    1.6,
         -1.6,   -4.8,   -8. ,  -11.2,  -14.4,  -17.6,  -20.8,  -24. ,
        -27.2,  -30.4,  -33.6,  -36.8,  -40. ,  -43.2,  -46.4,  -49.6,
        -52.8,  -56. ,  -59.2,  -62.4,  -65.6,  -68.8,  -72. ,  -75.2,
        -78.4,  -81.6,  -84.8,  -88. ,  -91.2,  -94.4,  -97.6, -100.8,
       -104. , -107.2, -110.4, -113.6, -116.8, -120. , -123.2, -126.4,
       -129.6, -132.8, -136. , -139.2, -142.4, -145.6, -148.8, -152. ,
       -155.2, -158.4, -161.6, -164.8, -168. , -171.2, -174.4, -177.6,
       -180.8, -184. , -187.2, -190.4, -193.6, -196.8, -200. , -203.2,
       -206.4, -209.6, -212.8, -216. , -219.2, -222.4, -225.6, -228.8,
       -232. , -235.2, -238.4, -241.6, -244.8, -248. , -251.2, -254.4,
       -257.6, -260.8, -264. , -267.2, -270.4, -273.6, -276.8, -280. ,
       -283.2])

plt.plot(t[1:],ds/dt)
plt.title("The average velocity very near each point.")
plt.xlabel("time")
plt.ylabel("feet per second")
plt.show()

When the time intervals $dt$ are extremely small, we can think of the above graph as giving the instantaneous velocity of the ball at each time.