Eksamensopgaver

import numpy as np
import matplotlib.pyplot as plt

%matplotlib inline
plt.rc('font', size=16)

def part_over_y_0(a,c):
    # assume a < 0 and c > 0
    x0 = np.sqrt(-c/a)
    N = 100
    xs = np.linspace(-x0,x0,N)
    ys = a*xs**2 + c
    return xs,ys
    
a1 = -3
c1 = 5
a2 = -0.5
c2 = 2

# spmg a
x,y = part_over_y_0(a1,c1)
plt.plot(x,y)
x,y = part_over_y_0(a2,c2)
plt.plot(x,y)

plt.grid()
plt.axis('equal')
plt.xlabel('$x$')
plt.ylabel('$y$')

from scipy.optimize import fsolve

y1 = lambda x: a1*x**2 + c1
y2 = lambda x: a2*x**2 + c2
y3 = lambda x: y1(x) - y2(x)
x0 = fsolve(y3,1)

xs = np.array([-x0,x0])
plt.scatter(xs,[y1(xs)],c='r',s=100)

fig

import numpy as np

N = 100
xs = np.linspace(-np.pi/2,np.pi/2,N)
ys = np.cos(xs)
L = np.sum(np.sqrt(np.diff(xs)**2+np.diff(ys)**2))
print('L',L)

from scipy.integrate import quad

f = lambda x: np.sqrt(1 + np.sin(x)**2)
L2 = quad(f, -np.pi/2,np.pi/2)[0]
print('L2',L2)

L 3.820147518333091
L2 3.8201977890277115

from scipy.integrate import quad
from scipy.optimize import fsolve

La = lambda a: quad(lambda x: np.sqrt(1 + (a*np.sin(x))**2), -np.pi/2,np.pi/2)[0]

a = fsolve(lambda a: La(a)-5,2)
print('A',a)

A [1.83146772]

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors as colors
colors.colorConverter.colors['b'] = [0, 0.4470, 0.7410]
colors.colorConverter.colors['g'] = [0.4660, 0.6740, 0.1880]

plt.rc('font', size=16)
fig,ax = plt.subplots(1,1)

x0 = np.sqrt(3/4)

f1 = lambda x: np.sqrt(1-x**2)
for xinit,xfinal,sign,linestyle in [(-x0,x0,1,'-'),
                         (-1,1,1,'--'),
                         (-1,1,-1,'--')]:
    xs = np.linspace(xinit,xfinal)
    ax.plot(xs,sign*f1(xs),linestyle,color='g')

f2 = lambda x,a: a*x**2 + 0.5 -0.75*a
for xinit,xfinal,linestyle in [(-x0,x0,'-'),
                    (-1.4,-x0,'--'),
                    (x0,1.4,'--')]:
    xs = np.linspace(xinit,xfinal)
    ax.plot(xs,f2(xs,1),linestyle=linestyle,color='b')

ax.grid()
ax.axis('equal')
ax.set_xlabel('$x$')
ax.set_ylabel('$y$')
ax.set_yticks([-1,0,1])
ax.text(0.3,1.1,'$f_1$',color='g')
ax.text(0.3,0.2,'$f_2$',color='b')

from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt

plt.rc('font', size=16)
fig,ax = plt.subplots(1,1)

dydx = lambda x, y: -0.5 * (x - 2) * y

def plot_sol(xrange,yinit):
    mysol = solve_ivp(dydx, xrange, yinit)
    xs = mysol.t
    ys = mysol.y[0]
    ax.plot(xs,ys,'o',linestyle='--')

xinit = 2
yinit = [1]

for xfinal in [6,-2]:
    xrange = [xinit,xfinal]
    plot_sol(xrange,yinit)

ax.grid()
ax.set_ylim([0,1.2])
ax.set_xlabel('$x$')
ax.set_ylabel('$y$')