Approximating $\sqrt{2}$

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def sqrt2approx(n):
    if n == 0:
        return 1
    else:
        return sqrt2approx(n-1)-(sqrt2approx(n-1)**2-2)/(2*sqrt2approx(n-1))

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sqrt2approx(0)

1

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sqrt2approx(1)

1.5

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sqrt2approx(2)

1.4166666666666667

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sqrt2approx(3)

1.4142156862745099

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2**0.5

1.4142135623730951

Newton's Method (general version)

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def newton(f,df,x,n,err):
    i = 0
    while abs(f(x)) > err and i <= n:
        x = x - f(x)/df(x)
        i += 1
    if i > n:
        return False
    else:
        return x

Approximating $\sqrt[3]{4}$

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def myfun(x):
    return x**3-4

def myfunDeriv(x):
    return 3*x**2

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newton(myfun,myfunDeriv,1.5,5,0.0001)

1.587401052148227

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1.587401052148227**3

4.000000001360922

Line tangent to $e^x$ and $\ln{x}$

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import math

def f(x):
    return

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Approximating 2\sqrt{2}2​

Newton's Method (general version)

Approximating 43\sqrt[3]{4}34​

Line tangent to exe^xex and ln⁡x\ln{x}lnx

Approximating $\sqrt{2}$

Approximating $\sqrt[3]{4}$

Line tangent to $e^x$ and $\ln{x}$