Callahan
    7.2: 2, 4, 5, 7, 9, 10, 18, 26, 27
    7.3: 10, 18 (Use Euler's Method)

2)  Determine the period and frequency of the following functions:

    a)  :

    To find the period we set the term inside the sine or cosine function equal to (the natural period these functions) and solve for the variable:

    We know that the frequency equals , so:

    And:

    If we graph sums of periodic functions we see that if the function with the longer period has period that is a integer multiple of the function with the shorter period then the longer period is the period of the overall function:

    b)  :

    And:

    In this case we can't just use the period of the function with the longest period, rather we need to find out what the smallest integer multiple of one period is that equals an integer multiple of the other period:

    So our period is:

    c)  :

    And:

    And:

*4)

    a)  What are the amplitude and frequency of ?

    b)  What are the amplitude and frequency of ?

5)

    a)  Is the antiderivative periodic?

    So it is periodic:

    b)  If so what are its amplitude and frequency?

7)

    a)  What is the average value of the function over the interval ?

    From p.351 we have the average value of an arbitrary function is given by:

    So:

    So the average value of the function over the interval is .

    b)  What is the average value of the function over the interval ?

    c)  What is the average value of the function over the interval ?

    d)  What is the average value of the function over the interval for any real number ?

    e)  Your work should demonstrate that the average value of over the full period does not depend on the point where you begin the period.  Is the same true for the average value over the half period?

    So yes the average value of over the full period does not depend on the point .

    However the average value over a half period varies over the range because the function lacks symmetry over the half period.-

*9)  The function and have the same amplitude and frequency; they differ only in phase.  In other words,

    for an appropriately chosen phase difference .  What is ?

    Since the phase difference is a constant we can choose any convenient value of and find :

    But since both functions are periodic:

10)  The function and also differ only in phase.  What is ?

    Since the phase difference is a constant we can choose any convenient value of and find :

    But since both functions are periodic:

18)

    a)  Let .  Using a graphical utility, sketch the graph of .

    a)  The function is periodic.  What is its period?  Estimate the amplitude from the graph:

    c)  In fact, can be viewed as a "phase-shifted" sine function:

    From your graph, estimate the phase difference and the amplitude .

    To find the phase difference we solve for a value of where :

    So we have:

*26)  The function whose graph is sketched at the left has the form:

    Determine the values of , , , and .

    First lets find using the peak-to-peak amplitude:

    Second lets find using the :

    Third lets find using the period given in the graph:

    So because when we have:

    Last, lets find using the fact that normally passes through the origin.  We can see that if the graph were not vertically shifted , it would just be a sine shifted of its period to the right.  So in order to be shifted to the right the expression inside the sine term must equal zero at the point .  This fact allows us to determine the value of :

    So our overall equation is:

27)  Write equations for three different functions all having amplitude 4 and period 5, whose graphs pass through the point :

    To make the amplitude and period correct we set and , and because when :

    To make a point on this curve pass through the point we adjust and appropriately.  To start with, lets just translate the point on the sine curve to the point :

    So we have:

    And the general form is:

    7.3: 10, 18  (use Euler's method to estimate the answers)

10)  Nonlinear Spring:    (p386)

    a)  Suppose the acceleration of the weight on a hard spring depends on the displacement of the weight according to the formula .  If you pull the weight down , hold it motionless, and then release it, what will its frequency be?

    b)  How far must you pull the weight so that its frequency will be double the frequency in part (a)?  

from visual.graph import *
graph = gdisplay()
    
tinitial = 0
tfinal = 6
t = tinitial
x = 2
v = 0

numberofsteps = 10000
deltat = (tfinal - tinitial) / float(numberofsteps)

vcurve = gcurve()
xcurve = gcurve()

for k in range(numberofsteps):
    vprime = -16*x-(x**3)
    xprime = v
    
    deltav = vprime * deltat
    deltax = xprime * deltat
    
    t = t + deltat
    v = v + deltav
    x = x + deltax
    
    vcurve.plot(pos=(t,v))
    xcurve.plot(pos=(t,x))

*18)  The Pendulum:    (p392)

    Take and give the pendulum three different initial impulses:

    Using Euler's method to determine the period of the motion in each case.  Are the periods noticeably different?

    The periods are not noticeably different, however the amplitude is:

v	Black	Solid
x	Red

from visual.graph import *
graph = gdisplay()

for i in [0.05, 0.1, 0.2]:
    tinitial = 0
    tfinal = 7
    t = tinitial
    x = 0
    v = i

    numberofsteps = 10000
    deltat = (tfinal - tinitial) / float(numberofsteps)

    xcurve = gcurve()
    xcurve.plot(pos=(t,x))

    for k in range(numberofsteps):
        vprime = -sin(x)
        xprime = v
        
        deltav = vprime * deltat
        deltax = xprime * deltat
        
        t = t + deltat
        v = v + deltav
        x = x + deltax

        xcurve.plot(pos=(t,x))