def loc_xtrm_1st_drv_test(expr):
#step 1 take derivative
myderiv = deriv(expr)
#step2 solve for x
zeros = find_poly_2_zeros(myderiv)
# for c in zeros:
# print c
pf = tof(expr)
extremas = []
for c in zeros:
yval = abs(pf(c.get_val()) - 0.0)
if (pf(c.get_val()) - 1 > 0 & pf(c.get_val()) + 1 < 0):
extremas.append("min", point2d(c.get_val(), yval))
else:
extremas.append("max", point2d(c.get_val(), yval))
return extremas
def max_profit(cost_fun, rev_fun):
profit = plus(cost_fun, prod(rev_fun,-1))
profit_deriv = deriv(profit)
zeroes = find_poly_2_zeros(profit_deriv)
return max(zeroes)
def test_02():
f0 = make_prod(make_const(0.5), make_pwr('x', 2.0))
f1 = make_prod(make_const(6.0), make_pwr('x', 1.0))
f2 = make_plus(f0, f1)
poly = make_plus(f2, make_const(0.0))
print '+++', poly
zeros = find_poly_2_zeros(poly)
for c in zeros:
print c
pf = tof(poly)
for c in zeros:
assert abs(pf(c.get_val()) - 0.0) <= 0.0001
print 'True zeros!'
def loc_xtrm_1st_drv_test(expr):
exprFn = tof(expr)
derivativeExpr = deriv(expr)
derivfn = tof(derivativeExpr)
degree = findDegree(derivativeExpr)
critical_points = []
if degree == 2:
xvalues = find_poly_2_zeros(derivativeExpr)
for x in xvalues:
y = exprFn(x.get_val())
critical_points.append(make_point2d(x.get_val(), y))
elif degree == 1:
x = find_poly_1_zeros(derivativeExpr)
y = exprFn(x.get_val())
critical_points.append(make_point2d(x.get_val(), y))
elif degree == 0:
# The derivative is just a constant so all values will be just the constant
# f` = 5
x = derivativeExpr.get_val()
else:
raise Exception("Not a first or second degree polynomial degree=",
degree)
maxima = None
minima = None
for p in critical_points:
x = p.get_x().get_val()
less = derivfn(x - 0.5)
more = derivfn(x + 0.5)
if less < 0 and more > 0:
y = exprFn(x)
minima = make_point2d(x, y)
elif less > 0 and more < 0:
y = exprFn(x)
maxima = make_point2d(x, y)
if not maxima and not minima:
return None
else:
results = []
if maxima:
results.append(("max", maxima))
if minima:
results.append(("min", minima))
return results
def find_infl_pnts(expr):
drv = deriv(expr)
drv_2 = deriv(drv)
# print 'f`(x) = ', drv
# print 'f``(x) = ', drv_2
zeros = None
try:
zeros = find_poly_1_zeros(drv_2)
# print '1st degree poly'
except Exception:
# print 'not a 1st degree poly'
pass
try:
zeros = find_poly_2_zeros(drv_2)
# print '2nd degree poly'
except Exception:
# print 'not a 2nd degree poly'
pass
if isinstance(zeros, const):
zeros = [zeros]
# print 'zeros:'
# for z in zeros:
# print ':',z.get_val()
f = tof(expr)
f_dp = tof(drv_2)
delta = 0.1
points = [] #will store array of points and if they are max/min
#zeros may be None if no inflection points
if not zeros == None:
for z in zeros:
z_val = f(z.get_val())
z_minus = f_dp(z.get_val() - delta)
z_plus = f_dp(z.get_val() + delta)
if z_minus < 0 and z_plus > 0:
points.append(point2d(z, const(z_val)))
if z_minus > 0 and z_plus < 0:
points.append(point2d(z, const(z_val)))
else:
return None
if points == []:
return None
else:
# print points
return points
def max_norman_window_area(p):
assert isinstance(p, const)
hexpr = quot(plus(plus(p,prod(const(-2.0),'r')), prod(prod(np.pi,'r'), const(-1.0)))/const(2.0))
area = plus(prod(const(0.5), prod(np.pi, pwr('r', const(2.0)))), prod(prod(const(2.0),'r'), hexpr))
area_deriv = deriv(area)
zeroes = find_poly_2_zeros(area_deriv)
ans = area(zeroes)
return max(ans)
def find_infl_pnts(expr):
#find the second derivative
second_drv = deriv(deriv(expr))
degree = findDegree(second_drv)
expr_tof = tof(expr)
inflection_points = []
if degree == 2:
zeros = find_poly_2_zeros(second_drv)
for x in zeros:
y = expr_tof(x.get_val())
inflection_points.append(make_point2d(x.get_val() , y))
else:
x = find_poly_1_zeros(second_drv)
y = expr_tof(x.get_val())
inflection_points.append(make_point2d(x.get_val(), y))
return inflection_points
def loc_xtrm_2nd_drv_test(expr):
drv = deriv(expr)
drv_2 = deriv(drv)
# print 'f`(x) = ', drv
# print 'f``(x) = ', drv_2
zeros = None
try:
zeros = find_poly_1_zeros(drv)
# print '1st degree poly'
except Exception:
print('not a 1st degree poly')
pass
try:
zeros = find_poly_2_zeros(drv)
# print '2nd degree poly'
except Exception:
print('not a 2nd degree poly')
pass
if isinstance(zeros, const):
zeros = [zeros]
print('zeros:')
for z in zeros:
print(':', z.get_val())
f = tof(expr)
f_dp = tof(drv_2)
points = [] #will store array of points and if they are max/min
for z in zeros:
z_val = f(z.get_val())
z_f_dp = f_dp(z.get_val())
if z_f_dp < 0:
points.append(('max', point2d(z, const(z_val))))
if z_f_dp > 0:
points.append(('min', point2d(z, const(z_val))))
if points == []:
return None
else:
# print points
return points
def loc_xtrm_1st_drv_test(expr):
drv = deriv(expr)
# print 'f`(x) = ', drv
zeros = None
try:
zeros = find_poly_1_zeros(drv)
# print '1st degree poly'
except Exception:
# print 'not a 1st degree poly'
pass
try:
zeros = find_poly_2_zeros(drv)
# print '2nd degree poly'
except Exception:
# print 'not a 2nd degree poly'
pass
if isinstance(zeros, const):
zeros = [zeros]
# print 'zeros:'
# for z in zeros:
# print ':',z.get_val()
f = tof(expr)
delta = 0.1
points = [] #will store array of points and if they are max/min
for z in zeros:
z_val = f(z.get_val())
z_minus = f(z.get_val() - delta)
z_plus = f(z.get_val() + delta)
if z_minus < z_val and z_plus < z_val:
points.append(('max', point2d(z, const(z_val))))
if z_minus > z_val and z_plus > z_val:
points.append(('min', point2d(z, const(z_val))))
if points == []:
return None
else:
# print points
return points