#!/usr/bin/env python
# coding: utf-8

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#from lab 1
#centered finite difference approximation of first derivative
#at a with step size h
def fd(f, a, h):
    return (f(a+h) - f(a-h)) / (2*h)


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#example in slides
f = lambda x: x ** 1.8
x = 0.5
h = 0.1
dxe = fd(f, x, h) #approximation


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df = lambda x: 1.8 * (x ** 0.8)
dx = df(x) #actual value


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#absolute error
abs(dxe - dx)


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#fractional error
abs(dxe - dx) / abs(dx)


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dxe, dx


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#first method from slide
from math import factorial, isinf
for n in range(1000):
    try:
        (n ** n) / factorial(n)
    except OverflowError:
        print(n)
        break


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#second method from slide
for n in range(1000):
    try:
        r = 1
        for i in range(n):
            r = r * (n / (n-i))
        if isinf(r):
            print(n)
            break
    except OverflowError:
        print(n)
        break


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0.3/0.1


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#compute the nth-order term of the Taylor series of exp(x) about 0 
def texp(x, n):
    return (x**n / factorial(n))    


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va = 0
for i in range(50):
    va = va + texp(-7, i)


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va


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vb = 0
for i in range(50):
    vb = vb + texp(7, i)
vb = 1 / vb


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vb


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from math import exp
v = exp(-7)


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#relative errors of the two estimates
ra = abs(va - v) / abs(v)
rb = abs(vb - v) / abs(v)
ra, rb


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