def is_odd_function(func, i=range(1, 10)):
"""
If the graph of f is unchanged by rotation through the angle π about
the origin, as in Figure 3.2(b), then f is said to be an odd function.
Thus f is odd if
f (−x) = −f (x), for all x in the domain of f.
"""
is_odd = all([_eval(func, -_x) == -_eval(func, _x) for _x in i])
return is_odd
def is_even_function(func, i=range(1, 10)):
"""
If the graph of a function f is unchanged under reflection in the
y-axis, as in Figure 3.2(a), then f is said to be an even function.
Thus f is even if
f (−x) = f (x), for all x in the domain of f.
"""
is_even = all([_eval(func, _x) == _eval(func, -_x) for _x in i])
return is_even
def graph_sketching_strategy(func):
"""
Module C1, Page 36
Strategy: To sketch the graph of a given function f, determine the
following features of f (where possible) and then show these features
in your sketch.
1. The domain of f .
2. Whether f is even or odd.
3. The x- and y-intercepts of f .
4. The intervals on which f is positive or negative.
5. The intervals on which f is increasing or decreasing and any
stationary points, local maxima and local minima.
6. The asymptotic behaviour of f .
"""
# Step 2 - even or odd or neither? This isn't really that necessary
# for this automated plot of the function but is nice to annotate
# the graph with and will have to be shown in any TMA/exam working
ftype = function_type(func)
# Step 3 - intercepts are easy to work out with a bit of algebra
# or evalutating it at x = 0
y_intercepts = _eval(func, 0) # y-intercept is solution(s) to f(0)
x_intercepts = filter_reals(solve(func)) # x-intercepts is solution(s) to f(x) = 0
# Step 4 - we need to find any poles (discontinuties). We can do that
# symbolically by solving the equation 1/f(x) = 0. If f has a pole then
# it will be a root of this equation
poles = solve(1/func, x)
# Now we'll check between these groups what the value of the function
# is to determine if postive or negative in that region. Again this would
# be obviously visible on the generated plot regardless but this is for
# _sketching_ manually, so we want to summarise and work all this out
# as if by hand too.
points = [-oo, ] + poles + [oo, ]
print """
The function %(func)s is of type: %(ftype)s. The y-intercept
are %(yints)r and the x-intercepts are %(xints)r.
""" % {
'func': func,
'ftype': ftype,
'yints': y_intercepts,
'xints': x_intercepts,
}
return
def classify_fixed_points(func):
"""
Module B1, Page 23, 24 - Behaviour near a fixed point
Let a be a fixed point of a smooth function f , and let xn be an
iteration sequence generated by f .
(a) If |f (a)| < 1, then there is an open interval I containing a with
the property that if x0 is in I, then xn → a as n → ∞.
(b) If |f (a)| > 1, then no iteration sequence generated by f
converges to a, unless xn = a for some value of n.
Takes a function as input, gets the fixed points and then classifies
them returning a list of tuples of (point, gradient value, classification)
"""
classified_points = list()
points = fixed_points(func)
# We'll need to use the first derivative of the function to classify the
# points according to the above criteria.
gradient_func = diff(func)
for point in points:
val = abs(_eval(gradient_func, point))
if val > 1:
point_type = 'repelling'
if val < 1:
point_type = 'attracting'
if val == 1:
point_type = 'indifferent'
if val == 0:
point_type = 'super-attracting'
classified_points.append((point, val, point_type))
print "Classified fixed points for %s:\n%r\n:" % (func, classified_points)
return classified_points