```python
import sympy

import adaptive_trapzint as p1
import sinesum1 as p2
import centered_diff as p3
import find_primes as p4
```

# Homework 1

### Taylor Patti
### 2/8/2016





## Exercises Completed:

1. Exercise 3.8 (```adaptive_trapzint.py```)
1. Exercise 3.15 (```sinesum1.py```)
1. Exercise 3.18 (```centered_diff.py```)
1. Exercise 3.20 (```find_primes.py```)




## adaptive_trapzint.py

Uses algorithm to calculate the number of series terms required to obtain integral value within a given error, then computes this quantity. A select sampling of functions, intervals,  and function performance is output below.


```python
p1.performance_table()
```

    
    Preformance Table of Various Functions
    
      Function        Interval           Error         N-value
    
       cos(x)       0.00 to 3.14      3.122502E-17       509
    
       sin(x)       0.00 to 3.14     -6.349115E-06       509
    
       sin(x)       0.00 to 1.57     -6.346204E-06       180
    


## sinesum1.py

Constructs a trigonometric series with a known point of conversion and ultimately produces a table which details the effectiveness of truncating at various indexes for a sampling of arguments.


```python
p2.table(1)
```

    
    Difference Between the Limit and the Sum of Sines
    
             n     a=0.01     a=0.25     a=0.49
    
             1       0.99       0.68       0.40
             3       0.96       0.15      -0.18
             5       0.94      -0.14       0.00
            10       0.87       0.01      -0.06
            30       0.63      -0.03      -0.01
           100      -0.02       0.01      -0.01


## centered_diff.py

Estimates the first derivative of a function by a truncated Taylor Series approximation. Ultimately, a table was produced in which the estimations were compared to the analytic derivatives, calculated using
sympy, and the error was deduced.


```python
p3.application()
```

    
                Function                Error
    
                  exp(x)              1.67E-05
    
            exp(-2*x**2)              0.00E+00
    
                  cos(x)             -2.45E-16
    
                  log(x)              3.33E-05
    





    [1.6666749992122476e-05, 0.0, -2.4492935982947064e-16, 3.3335333477158e-05]



## find_primes.py

This program identifies prime factors by iterating through a list from 2 to the number given as the function argument and eliminating numbers which are multiples of smaller numbers from the iteration list.
The iteration list is then returned and consists of only prime numbers lesser or equal to the function argument. Here, the prime numbers of 100 and less are shown.


```python
print p4.find_primes(100)
```

    [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]


420db775fada89d2abfacfc688266efaa314f88e
ipynb
Jupyter Notebook
hw1.ipynb
chapman-phys227-2016s/hw-1-patti102
352c0afdc6947b2f1da56d4cd66d690a33cfd3f7
MIT
2016-02-05T23:49:20.000Z
2016-02-05T23:49:20.000Z
hw1.ipynb
chapman-phys227-2016s/hw-1-patti102
352c0afdc6947b2f1da56d4cd66d690a33cfd3f7
MIT
hw1.ipynb
chapman-phys227-2016s/hw-1-patti102
352c0afdc6947b2f1da56d4cd66d690a33cfd3f7
MIT
Qwen/Qwen-72B
1. YES
2. YES