pbSetCh7.sagews

reset

<function reset at 0x7f591d340ed8>

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# Exercise 1
#Exercise 1. Make a CAS function riemann_sum(f, a, b, n, c) that takes five inputs:
#•end points a,b, a<b,of interval of integration
#•function f continuous on the interval
#•positive integer n, the number of subintervals of equal length in partition of interval [a,b]
#•and parameter p.
#The function returns
#• left Riemann sum if c = 0;
#• middle Riemann sum if c = 1;
#• right Riemann sum if c = 2
#Test your function on example of your choice.

def riemann_sum(f,a,b,n,p):#p takes only one of the values 0, 1, 2
    #Define the partition of the interval [a,b] into n subintervals of equal length:
    delt=(b-a)/n #encode the length of a subinterval #delt=(b-a)/n
    L=[a+delt*j for j in range(n+1)] # L is a list of end points of subintervals   #a+n*delt=b
    # For each case below S is the sum of the given function values at appropriate x-values
    if p==0:
        S=sum(a+k*delt for k in range(n))
    elif p==1:
        S=sum(((a+b)/2)+k*delt for k in range(n))
    else:
        S=sum(b-k*delt for k in range(n))
    return S*delt

#My example:
riemann_sum(x^3-3*x^2,0,2,30,0).n()

1.93333333333333

#Your example:
riemann_sum(2*x^3-3*x^2+x,0,3,40,0).n()

4.38750000000000

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#Exercise 2
#Make a CAS function integral_MVT(f, a, b) that takes the end points of a closed interval and a continuous function defined on this interval and returns a figure with the graph of the function and the line y=f__ave. Here f_ave is the average value of the function f on the interval [a,b].
def integral_MVT(f,a,b):
    G=Graphics()
    #Make a plot of f on [a,b]:
    fcn_plot=plot(f,a,b,legend_label="f(x)",figsize=[4,2],thickness=2)
    G+=fcn_plot
    #Find the average value of f on [a,b]:
    f_ave=f.integral(x,a,b)/(b-a)
    #Find location of the point c s.t. f(c)=f_ave
    c=find_root(f-f_ave,a,b)
    #Define the constant function equal the f_ave on [a,b] and make a plot of the function
    f_c(x)=f(c)
    f_ave_plot=plot(f_ave,thickness=2,legend_label="line y=f_ave",linestyle="dashed")
    G+=f_ave_plot
    # Make a plot of two points, c on the x-axis and corresponding point on the graph of f
    pt_plot=point([[c,0],[c,f(c)]],size=40,color='red')
    G+=pt_plot
    return G.show()
#My example:
f(x)=3*sin(x)
integral_MVT(f,0,pi)# Include your own example

# Your example:
f(x)=2*cos(x)
integral_MVT(f,0,2*pi)

############################

# Exercise 3
#Consider the parametric curve x=t^2-4,  y=t-t^3, -2<=t<=2.
#Plot the curve. You will see two parts of the area bounded by the curve, a curvilinear triangle and a loop.
#Consider only the area bounded by the curvilinear triangle. Use CAS assistance to find the vertical line that halves the area bounded by this curve. Make a figure with the curve plot and the vertical line.  Show your work.
var('t')
x(t)=t^2-4
y(t)=t-t^3

t

f_plot=plot(x(t),-3,t,3)
x_plot=plot(y(t),-3,t,3)

(f_plot+x_plot).show()

########################

#Mini project: Area of the tilted ellipse

#Part 1. Find the general formula for the area of ellipse that is defined by the formula
#A*x^2+2*B*x*y+C*y^2 = 1  with B^2-A*C < 0.
#Hint: Apply the completing the square technique to the first two terms in the lhs of the given equation. This allows you to rewrite the given equation in the form ((x+k)^(2))/(a^(2))+(y^(2))/(b^(2))=1 with k, a, b depending on parameters A, B, C. Then think about relation between the area of standard ellipse and the area of its image under shear transformations

#Part 2. Apply the formula derived in Part 1 to the ellipse with parameters A = 1/4, B = 2/5, C = 1.
#Plot the given (sheared) ellipse and the corresponding standard ellipse in one figure.

reset
var('A B C')

<function reset at 0x7f1483e590c8> (A, B, C)

a=1/sqrt(A);b=sqrt(A/A*C-B^2)

pi/sqrt(1/4-4/25)

10/3*pi

A=0.25;B=0;C=1
a=1/sqrt(A);b=sqrt(A/A*C-B^2)
a

2.00000000000000

b

1.00000000000000

x(t)=2*cos(t)-(8/3)*sin(t)
y(t)=(5/3)*sin(t)
p=parametric_plot(((x(t),y(t)),(t,0,2*pi)),figsize=[4,3],thickness=2);p

Error in lines 3-3 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1191, in execute flags=compile_flags), namespace, locals) File "", line 1, in <module> File "/ext/sage/sage-8.9_1804/local/lib/python2.7/site-packages/sage/misc/decorators.py", line 493, in wrapper return func(*args, **options) File "/ext/sage/sage-8.9_1804/local/lib/python2.7/site-packages/sage/plot/plot.py", line 2625, in parametric_plot raise ValueError("the number of functions and the number of variable ranges is not a supported combination for a 2d or 3d parametric plots") ValueError: the number of functions and the number of variable ranges is not a supported combination for a 2d or 3d parametric plots

︠d00e226f-d027-4acb-a62d-154d302b97fa︠
#################################################################################
︠406aed5f-f1dc-4210-9a9b-0a9bae8c3543︠
# Mini project: Lorenz curve and the Gini index
# Fill in the gaps following the steps suggestes in Lecture 7 or do the project your own way.
#Step 1
#(a)

#(b)

#(c)

#Step 2

#Step 3

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