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

# # Problem 1

# Integrate the following equation using the Trapazoid Rule, Simpson's rule, and Gauss-Legendre quadrature from x=0 to x=1:
# 
# $$f(x) = x\cdot\tanh(d\cdot(x-0.5))+1.$$
# 
# Use d=10. Make a log-log plot of the relative error of the methods versus the number of grid points.  To get the "exact" integral, you can use the built-in integrator.  
# 
# In python, use the following:
# 
#     from scipy.integrate import quad
#     (Ie, abserr) = quad(f,0,1)
#     
# When varying the number of points, because we are plotting on a log scale, and because the error goes as a power, we want the number of points to increase as a power.  I did something like:
# 
# npoints = 2^1, 2^2, 2^3, 2^4, 2^5, 2^6, 2^7, 2^8, 2^9, 2^10,
#     
# for 10 runs.  Then, when you plot on a log scale you'll have an even spacing of points on the x-axis.
# 
# Use your results to verify that the trapazoid and Simpson methods have convergence rates $O(\Delta x^2)$ and $O(\Delta x^4)$, respectively.

# In[1]:


get_ipython().run_line_magic('matplotlib', 'inline')
import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import quad


# ## Problem 2

# This problem is from Hoffman's book: Chapter 4 problem 106 (b).
# 
# The following temperature and rate data were measured:
# 
# | T(K) | K$_f$ |
# |:----:|:-----:|
# |1000  | 7.5E15|
# |2000  | 3.8E15|
# |3000  | 2.5E15|
# |4000  | 1.9E15|
# |5000  | 1.5E15|
# 
# Find parameters $B$, $\alpha$, and $E/R$ for the model
# 
# $$K = BT^{\alpha}\exp\left(-\frac{E}{RT}\right).$$
# 
# Plot the model with the best coefficients with a smooth line (use lots of T points). Also plot the measured points using data markers (not lines).

# In[ ]:





# ## Problem 3

# Evaluate the function $f(x)=\exp(4x)$ at $x=0.55$ using a cubic spline fit to points $x=0,\,0.2,\,0.4,\,0.8,\,1.0.$  Also report the relative error at $x=0.55$. (Code the spline yourself.)

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