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

# # Table of Contents
#  <p><div class="lev1 toc-item"><a href="#1" data-toc-modified-id="1-1"><span class="toc-item-num">1&nbsp;&nbsp;</span>1</a></div><div class="lev2 toc-item"><a href="#1(b)" data-toc-modified-id="1(b)-11"><span class="toc-item-num">1.1&nbsp;&nbsp;</span>1(b)</a></div><div class="lev1 toc-item"><a href="#2" data-toc-modified-id="2-2"><span class="toc-item-num">2&nbsp;&nbsp;</span>2</a></div><div class="lev2 toc-item"><a href="#2(b)" data-toc-modified-id="2(b)-21"><span class="toc-item-num">2.1&nbsp;&nbsp;</span>2(b)</a></div><div class="lev1 toc-item"><a href="#3(a)" data-toc-modified-id="3(a)-3"><span class="toc-item-num">3&nbsp;&nbsp;</span>3(a)</a></div><div class="lev2 toc-item"><a href="#3(b)" data-toc-modified-id="3(b)-31"><span class="toc-item-num">3.1&nbsp;&nbsp;</span>3(b)</a></div><div class="lev1 toc-item"><a href="#4" data-toc-modified-id="4-4"><span class="toc-item-num">4&nbsp;&nbsp;</span>4</a></div><div class="lev2 toc-item"><a href="#3(b)" data-toc-modified-id="3(b)-41"><span class="toc-item-num">4.1&nbsp;&nbsp;</span>3(b)</a></div>

# # 1

# In[5]:


from sympy import *
init_printing()
x = symbols('x')
f = sin(x)*cos(x)**3
f


# In[12]:


fs = f.series(x,0,15)
f0 = fs.removeO()


# In[13]:


f0


# In[20]:


plot(f,f0,(x,0,pi/2))


# In[26]:


g = integrate(f,(x,0,x))


# In[28]:


plot(f,g,(x,0,2*pi))


# ## 1(b)

# In[9]:


from sympy import *
init_printing()
y = symbols('y')
x = symbols('x',positive = True)
f = sqrt(x**2-Rational(1/2)*y**2)
f


# In[12]:


int1 = integrate(f,(y,0,x))
int1


# In[13]:


integrate(int1, (x,0,1))


# # 2
# 

# In[16]:


A=Matrix([[1,1,1],[1,-1,-3],[0,1,2]])
A


# In[17]:


A.rank()


# ## 2(b)

# In[19]:


x1=Matrix([1,1,0])
x2=Matrix([1,0,-1])
x3=Matrix([0,-1,1])


# In[32]:


y1=x1
a1=y1/y1.norm()
a1


# In[39]:


y2=x2-(x2.T*a1)[0]*a1


# In[40]:


x2


# In[42]:


x2a = (x2.T*a1)[0]*a1
x2a


# In[45]:


y2 = x2-x2a
y2


# In[48]:


a2=y2/y2.norm()
a2


# In[55]:


x3a1 = (x3.T*a1)[0]*a1
x3a1
x3a2 = (x3.T*a2)[0]*a2
x3a2
y3=x3-x3a1-x3a2


# In[56]:


y3


# In[59]:


a3 = y3/y3.norm()
a3


# In[60]:


a3.T*a1


# # 3(a)

# In[1]:


from sympy import *
init_printing()

a, x = symbols('a x')


# In[2]:


C1 = Rational(1/2)*x**2 + Rational(1/2)
C1


# In[3]:


C2 = Rational(1/4)*x**2
C2


# In[4]:


S = expand(integrate(C1-C2,(x,a,a+1)))
S


# In[5]:


S0 = solve(diff(S,a),a)[0]
S0


# In[6]:


S.subs({a:S0})


# ## 3(b)

# In[7]:


solve(C1-1,x)


# In[8]:


solve(C2-1,x)


# In[9]:


#from sympy.plotting import plot
import matplotlib.pyplot as plt
import numpy as np

plot(C1,C2,1,(x,0,3))


# ![](plot1.png)

# In[14]:


Su = expand(integrate(C1-1,(x,1,a+1)))
Su


# In[15]:


S-Su


# In[16]:


plot(S-Su,(a,0,2))


# In[19]:


solve(diff(S-Su,a),a)


# # 4

# In[20]:


from sympy import *
init_printing()

a, x = symbols('a x')


# In[21]:


C1 = Rational(1/2)*x**2 + Rational(1/2)
C1


# In[22]:


C2 = 0.1*x**2
C2


# In[23]:


S = expand(integrate(C1-C2,(x,a,a+1)))
S


# In[24]:


S0 = solve(diff(S,a),a)[0]
S0


# In[25]:


S.subs({a:S0})


# ## 3(b)

# In[26]:


solve(C1-1,x)


# In[28]:


s1 = solve(C2-1,x)
s1[1]


# In[30]:


#from sympy.plotting import plot
import matplotlib.pyplot as plt
import numpy as np

plot(C1,C2,1,(x,0,4))


# In[31]:


Su = expand(integrate(C1-1,(x,1,a+1)))
Su


# In[32]:


S-Su


# In[33]:


plot(S-Su,(a,0,4))


# In[34]:


solve(diff(S-Su,a),a)