Math

# python 2
#
# Homework 3, Problem 3
# List comprehensions!
#
# Name:Shivam jindal
#

# this gives us functions like sin and cos
from math import *

# two more functions (not in the math library above)
def dbl(x):
    """ doubler!  input: x, a number """
    return float(2*x)

def sq(x):
    """ squarer!  input: x, a number """
    return float(x**2)

# examples for getting used to list comprehensions
def lc_mult( N ):
    """ this example takes in an int N
        and returns a list of integers
        from 0 to N-1, **each multiplied by 2**
    """
    return [ db(x) for x in range(N) ]

def lc_idiv( N ):
    """ this example takes in an int N
        and returns a list of integers
        from 0 to N-1, **each divided by 2**
        WARNING: this is INTEGER division...!
    """
    return [(x/2) for x in range(N) ]

def lc_fdiv( N ):
    """ this example takes in an int N
        and returns a list of integers
        from 0 to N-1, **each divided by 2**
        NOTE: this is floating-point division...!
    """
    return [ float(x)/2 for x in range(N) ]

def interp(low,hi,frac):
    """ returns a value frac of the way from low to hi """
    return low + (hi-low)*frac

# Here is where your functions start for the homework:

# Step 1, part 1
def unitfracs( N ):
    """ this example takes in an int N
        and returs a list of N number between 0 and 1
        which have same difference
        NOTE: this is floating-point division...!
    """
    
    return [float(x)/N for x in range(N)]


def scaledfracs(low,hi,N):
    """ this example takes in three value low,hi,and N
        and returs a list of N number between low and hi
        which have same difference
        NOTE: this is floating-point division...!
    """
    return [float(low+x*(hi-low)) for x in unitfracs(N)]

def sqfracs(low,hi,N):
    """ this example takes in three value low,hi and N
        and returs a list of N number which is the
        square of its scaled fraction
        NOTE: this is floating-point division...!
    """
    return [sq(x) for x in scaledfracs(low,hi,N)]


def f_of_fracs(f,low,hi,N):
    """ this example takes in four value f as function,
        low and hi is limit and N is number of division
        and returs a list of N number which is the
        result of f of its scaled fraction
        NOTE: this is floating-point division...!
    """
    return [f(x) for x in scaledfracs(low,hi,N)]

def integrate(f,low,hi,N):
    """ integrate returns an estimate of the definite integral
     of the function f (the first input)
     with lower limit low (the second input)
     and upper limit hi (the third input)
     where N steps are taken (the fourth input)

    integrate simply returns the sum of the areas of rectangles
    under f, drawn at the left endpoints of N uniform steps
    from low to hi
  """
    diff = (hi-low)/float(N)
    return sum(f_of_fracs(f,low,hi,N))*diff

def c(x):
    """ c is a semicircular function of radius two """
    return (4-x**2)**0.5

"""
    Q.2
    integrate(c,0,2,200) is 3.15117
    integrate(c,0,2,2000) is 3.14258
"""
"""
    Q.2
    if N goes to infinity then integral become zero
    because if N zero then diff = (hi-low)/float(N) is zero then
    sum(f_of_fracs(f,low,hi,N))*diff become zero so integral become zero.
"""