# Using the information proivded in the 'Differentiation' tutorial on 
# http://sage.ccjohnson.org, answer the problems in the attached PDF using 
# Sage in this worksheet.  (The PDF is also available on Sakai.)



# 1
diff(cos(x^2) + sin(x^2) - cos(sin(x^2)) + sin(cos(x^2)))

# 2
graph = f(x) = x^2 * sqrt(sin(x) + 2)
Df(x) = diff(f)
curve = plot(f(x), (x, 0, 10))
var('y')
tanline = implicit_plot(y - f(3) == Df(3) * (x-3), (x, 0, 10), (y, 0, 15), \
                       color='red')

curve += plot(tanline)

show(curve)

# 3
f(x) = (arctan(sin(x * ln(x^2 + 1))))
curve = plot(f(x), (x, 0, 10))

Df = diff(f)
D2f = diff(Df)
curve += plot(Df, (x, 0, 10), color='red')
curve += plot(D2f, (x, 0 ,10), color='green')
show(curve)

# 4
s(x) = f(x)
s(x) = g(x)
s(x) = h(x)
Ds(x) = diff(s)
def newton(s, x0, n):
    counter = 1

    while counter <= n:
        x0 = x0 - s(x0) / Ds(x0)
        counter += 1

    return x0


# 5
P(x) = 6 * x^12 - 12 * x^6 + 7 * x^4
Q(x) = 12 * x^9 + x^7 + 3 * x
solve(P(x) == Q(x), x)
graph = plot(P(x), (x, 1, 2))
graph += plot(Q(x), (x, 1,2), color='red')
show(graph)

DP(x) = diff(P)
def newton(P, x0, n):
    counter = 1

    while counter <= n:
        x0 = x0 - s(x0) / Ds(x0)
        counter += 1

    return x0
print newton(P, 1.38366, 50)

DQ(x) = diff(Q)
def newton(Q, x0, n):
    counter = 1

    while counter <= n:
        x0 = x0 - s(x0) / Ds(x0)
        counter += 1

    return x0
print newton(Q, 1.38366, 50)

# t == 1.38366
# pont where P(x) and Q(x) cross is (1.38366, 0.933104439922602)
︠52c3df6e-4604-4d8a-883f-0652176dd7eb︠