sage math exam 2 practice .sagews

var('Y')                     ##the main variable is 'Y'
f(Y)=(1-(3/2)*Y)*Y*(1-Y)     ##the given equation
df(Y)=expand(diff(f(Y),Y))   ##find derivative of output and input. 
show(df(Y))                  ##derivative of the input
show(expand(f(Y)))           ##
df(0),df(1),df(2/3)          ##given input values for the derivative equation

Y

$\displaystyle \frac{9}{2} \, Y^{2} - 5 \, Y + 1$

$\displaystyle \frac{3}{2} \, Y^{3} - \frac{5}{2} \, Y^{2} + Y$

(1, 1/2, -1/3)

dflist=[]                        ##open derivative list 
dfvals=[]                        ##plugs 
flist=[x^2+2*x, 8*x^3, (1-x)/(1+x), x*e^(2*x^2)] ##
for f in flist:                  ##
    df(x)=diff(f(x),x)           ##
    dflist.append(df(x))         ##
    dfval=df(1.0)                ##
    dfvals.append(dfval)         ##
dflist                           ##
dfvals                           ##

[2*x + 2, 24*x^2, -1/(x + 1) + (x - 1)/(x + 1)^2, 4*x^2*e^(2*x^2) + e^(2*x^2)] [4.00000000000000, 24.0000000000000, -0.500000000000000, 36.9452804946533]

_=var("T,S")                     ##
vector_field(T,S)=[0.5*T-0.01*S*T,0.005*S*T-0.2*S]  ##
                                 ##
p=plot_vector_field(vector_field,##
          (T,0,120),(S,0,120),   ##
          frame=False,           ##
          color="green",         ##
          axes_labels=("T","S")) ##
show(p,figsize=5,aspect_ratio=1) ##

var("N")                     ##
f(N)=0.2*N*(1-N/100)         ##
dt=0.1                       ##
N=2.                         ##
Nvals=[N]                    ##
tvals=srange(0,40,dt,include_endpoint=True) ##
for t in tvals:              ##
   Np=f(N)                   ##
   dN=dt*Np                  ##
   N=N+dN                    ##
   Nvals.append(N)           ##
p=list_plot(zip(tvals,Nvals),##
            plotjoined=True, ##
            axes_labels=["t","N"])##
show(p,ymin=0,figsize=5)     ##

N

var("J,R")
vector_field(J,R)=[R,-J]    ##
                            ##
pvf=plot_vector_field(vector_field,     ##
                      (J,-2,2),(R,-2,2),##
                      plot_points=10,   ##
                      color="green")    ##
dt=0.001                    ##
J=1                         ##
R=0                         ##
Jvals=[J]                   ##
Rvals=[R]                   ##
tvals=srange(0.,7,dt, include_endpoint=True) ##
for t in tvals:              ##
    [Jp,Rp]=vector_field(J,R)##
    dJ=dt*Jp                 ##
    dR=dt*Rp                 ##
    J=J+dJ                   ##
    R=R+dR                   ##
    Jvals.append(J)          ##
    Rvals.append(R)          ##
ptr=list_plot(zip(Jvals,Rvals),plotjoined=True,axes_labels=["J","R"],aspect_ratio=1)##
pts1=list_plot(zip(tvals,Jvals),plotjoined=True,color='black',legend_label="Juliet")##
pts2=list_plot(zip(tvals,Rvals),plotjoined=True,color='red',legend_label="Romeo")   ##
show(pvf+ptr,frame=False,axes_labels=["J","R"],aspect_ratio=1)                      ##
show(pts1+pts2,axes_labels=["t","Emotion"])                                         ##

(J, R)

var('t')                   ##
g(t)=(t^3-2*t^2+1)*e^(t/3) ##
t0=-1                      ##
t1=0.25                    ##
g0=g(t0)                   ##
g1=g(t1)                   ##
m=(g1-g0)/(t1-t0)          ##
L(t)=m*(t-t0)+g0           ##
p=plot(g(t),(t,-2,2),color='red')    ##
p=p+plot(L(t),(t,-2,2),color='green')##
p=p+point([t0,g0],size=40)           ##
p=p+point([t1,g1],size=40)           ##
p=p+text('secant line',(1.2,3.5),color='green') ##
p=p+text('y=g(t)',(0.8,0.9),color='red') ##
show(p,axes_labels=['t',''])         ##

t

h(x)=8*x^2
dh(x)=diff(h(x),x)
@interact
def plot_tangent_line(x0=(-2,2,.25)):
    y0=h(x0)
    m=dh(x0)
    L(x)=y0+m*(x-x0)
    p1=plot(h(x),(x,-3,3),axes_labels=["x","y"])
    p2=point([x0,y0],color="red",size=40)
    p3=plot(L(x),(x,x0-1,x0+1),color="green")
    show(p1+p2+p3,ymin=-4, ymax=40,figsize=[7,4])

Interact: please open in CoCalc

h(x)=8*x^2
dh(x)=diff(h(x),x)
p1=plot(h(x),(x,-3,3),axes_labels=["x","y"],ymin=-4, ymax=40)
frames=[]
for x0 in srange(-2,2,0.1):
    y0=h(x0)
    m=dh(x0)
    L(x)=y0+m*(x-x0)
    p2=point([x0,y0],color="red",size=40)
    p3=plot(L(x),(x,x0-1,x0+1),color="green")
    frames.append(p1+p2+p3)
ani=animate(frames)
show(ani)