3508.sagews

#8a)

s(i,n) = -5 + (10*i)/n
f(x) = 1/(1+x^2)
var('x')

def L(i,n):
    prod = 1
    for j in range(n):
        if j != i:
            prod = prod*((x - s(j,n)) / (s(i,n) - s(j,n)))
    return prod;
        
def p(n):
    sum = 0;
    for i in range(n):
        sum = sum + f(s(i,n))*L(i,n)
    return sum

n=20
plot(p(n),(-4,4),color="green") + plot(f(x),(-4,4),color="red")
#show(p(n))

n=30
plot(p(n),(-4,4),color="green") + plot(f(x),(-4,4),color="red")
#show(p(n))

#8b)

T(i,n) = cos(((2*i + 1)/(n+1))*(pi/2))
s(i,n) = (1/2)*((5 + 5)*T(i,n) -5 + 5)
f(x) = 1/(1+x^2)
var('x')

def L(i,n):
    prod = 1
    for j in range(n):
        if j != i:
            prod = prod*((x - s(j,n)) / (s(i,n) - s(j,n)))
    return prod;
        
def p(n):
    sum = 0;
    for i in range(n):
        sum = sum + f(s(i,n))*L(i,n)
    return sum

n=20
plot(p(n),(-4,4),color="green") + plot(f(x),(-4,4),color="red")

n=30
plot(p(n),(-4,4),color="green") + plot(f(x),(-4,4),color="red")

#9b)

s(i,n) = (pi*i)/n
f(x) = cos(x)

def L(i,n):
    prod = 1
    for j in range(n):
        if j != i:
            prod = prod*((x - s(j,n)) / (s(i,n) - s(j,n)))
    return prod;
        
def p(n):
    sum = 0;
    for i in range(n):
        sum = sum + f(s(i,n))*L(i,n)
    return sum

eps = 10^-6
for n in range (3,10):
    diff(x) = f(x) - p(n)
    max = diff.find_maximum_on_interval(0,pi)
    if(max[0] < eps):
        print n

8 9

#9c)

T(i,n) = cos(((2*i + 1)/(n+1))*(pi/2))
s(i,n) = (1/2)*(pi*T(i,n) + pi)
f(x) = cos(x)
var('x')

def L(i,n):
    prod = 1
    for j in range(n):
        if j != i:
            prod = prod*((x - s(j,n)) / (s(i,n) - s(j,n)))
    return prod;
        
def p(n):
    sum = 0;
    for i in range(n):
        sum = sum + f(s(i,n))*L(i,n)
    return sum

eps = 10^-6
for n in range (3,10):
    diff(x) = f(x) - p(n)
    max = diff.find_maximum_on_interval(0,pi)
    if(max[0] < eps):
        print n

8 9