4148.sagews

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#
# G.2.a - f(x) = x^4 - 10000000*x^2;
#
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f(x) = x^4 - 10000000*(x^2) # the function f(x)
fig1 = plot(f,(x,-1000,1000),rgbcolor='red')       # fig1 ... (red)
show(fig1,figsize=[8,4],fontsize=8) # show() with some argument.

#############################
# Since the second element of the equation reduces lower values of x well below the x axis, a better global 
# representation of this function is created by setting the range to about 10,000 or greater, as displayed
# below.
#############################
f(x) = x^4 - 10000000*(x^2) # the function f(x)
fig1 = plot(f,(x,-10000,10000),rgbcolor='green')       # fig1 ... (green)
show(fig1,figsize=[8,4],fontsize=8) # show() with some argument.

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#
# G.2.b - f(x) = (2*x^6 + 50*x^2)/(x^6+3*x^2+1)
#
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f(x) = (2*x^6 + 50*x^2)/(x^6+3*x^2+1) # the function f(x)
fig1 = plot(f,(x,-10,10),rgbcolor='red')       # fig1 ... (red)
show(fig1,figsize=[8,4],fontsize=8) # show() with some argument.

###############################
# By plotting this with really low values, we can see a tendency emerge. Specifically, the global behavior
# tends toward a constant value of 2.
# Now, let's take the equation analysis, and we see that the dominant terms for the numerator and denominator
# are 2(x^6)/x^6, which results in a constant value of 2. See plot below.
###############################
f(x) = (2*x^6 + 50*x^2)/(x^6+3*x^2+1) # the function f(x)
fig1 = plot(f,(x,-1000,1000),rgbcolor='green')       
show(fig1,figsize=[8,4],fontsize=8) # show() with some argument.
##############################
# Therefore, the limit in both the negative and positive infinite direction is equal to 2.
##############################

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#
#G.2.c f(x)=(x^9+4*e^(0.6*x))/(3*x^12+2*e^(0.6*x))
#
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f(x) = (x^9+4*e^(0.6*x))/(3*x^12+2*e^(0.6*x)) # the function f(x)
fig1 = plot(f,(x,-100,1000),rgbcolor='red')       # fig1 ... (red)
show(fig1,figsize=[8,4],fontsize=8) # show() with some argument.

##########################
# Plotting this function, we can see that the global behavior of the function is a constant 2 in the positive
# infinite direction.
# Now, looking at the equation, we can see that the dominant elements are the exponentional elements because
# over a global range, exponential always dominates power elements or functions. 
# So, (4*e^(0.6*x))/(2*e^(0.6*x)= 2, which results in a positive infinity limit of 2.
##########################

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#
#G.2.d f(x)=(3*x^8 - 123*cos(x)-6*x^2)/(e^(0.4*x))
#
#######################################

f(x) = (3*x^8 - 123*cos(x)-6*x^2)/(e^(0.4*x)) # the function f(x)
fig1 = plot(f,(x,-200,150),xmin=1500,xmax=1500,ymin=200,ymax=1500,rgbcolor='red')       # fig1 ... (red)
show(fig1,figsize=[8,4],fontsize=8) # show() with some argument.

######################
# Ignoring all but the dominant terms results in the plot below, which is almost identical to the original
# formula. Exponential growth dominates power growth so the limit as x approaches infinity of 
# (3*x^8)/(e^(0.4*x))=0.
# 
f(x) = (3*x^8)/(e^(0.4*x)) # the function f(x)
fig1 = plot(f,(x,-2000,15000),xmin=1500,xmax=1500,ymin=200,ymax=1500,rgbcolor='red')       # fig1 ... (red)
show(fig1,figsize=[8,4],fontsize=8) # show() with some argument.
# Some large values appear as x is near 0, but no line is displayed (zero) for large x values.

#######################################
#
#G.2.e f(x)=(e^(0.8*x))(1 + 5*x^6)
#
#######################################

f(x) = (e^(0.8*x))*(1 + 5*(x^6)) # the function f(x)
fig1 = plot(f,(x,-2,20),rgbcolor='red')       # fig1 ... (red)
show(fig1,figsize=[8,4],fontsize=8) # show() with some argument.

########################
# Analysis and adjustments to the plot below indicate that there is no positive infinite limit
# value for this function. No matter what positive value of x is input for the function, a great
# exponential value is output.
# Analysis of the original equation and expanding the equation results in (e^(0.8*x))+ (e^(0.8*x))*(5*(x^6)),
# which has no limiting value toward positive infinity.
########################

#######################################
#
#G.2.f f(x)=(3*e^(-x) - e^(-3*x)) / (e^(-3*x) + e^(-x))
#
#######################################
# Multiplying the top and bottom by e^(3*x) results in (3*e^(-x))/(e^(-x)) = 3. Therefore, the positive
# infinity limiting value is 3. Now, let's look at a plot.
#######################################
f(x) = (3*e^(-x) - e^(-3*x)) / (e^(-3*x) + e^(-x)) # the function f(x)
fig1 = plot(f,(x,-2,20),rgbcolor='red')       # fig1 ... (red)
show(fig1,figsize=[8,4],fontsize=8) # show() with some argument.

#######################################
#
# G.2.g --> See http://springnote.com
# G.2.h f(x) = (2*x^4 - 40*x + 1)/(x^2 + x + 12)
#
#######################################
# 2*x^4 is the dominant growth element in the numerator and x^2 is the dominant element in the
# denominator. (2*x^4)/x^2 simplifies to 2*x^2, so this mimmicks this function, so let's see below.
#######################################
f(x) = (2*x^4 - 40*x + 1)/(x^2 + x + 12) # the function f(x)
g(x) = 2*x^2
fig1 = plot(f,(x,-100,100),rgbcolor='red')       # fig1 ... (red)
fig2 = plot(g,(x,-100,100),rgbcolor='blue')
figs = fig1 + fig2
show(figs,figsize=[8,4],fontsize=8) # show() with some argument.

#################################
#
# Give a number b so that f[x] is in its global scale behavior for |x| > b.
# f[x] seems to be in its global scale behavior at |x|=100, so b = 100.
# 
#################################
# Confirm this with a mentor!
#################################