3110.sagews

Tutorial 19 August

Here are some examples of plotting functions:

var('x')
f(x) = log(x)
plot(f(x),(0,10)) # plot f(x) over x in [0,10]

g(x)=x^2
# Combining two plots with "+"
plot(f(x),(0,5)) + plot(g(x),(0,5),color="red")

Problem 1

Use plotting of the functions $f(x)=x^2+x+1$ and $g(x)=(\log x)^{10}$ to find out roughly for which $x>3$ the function $f$ becomes larger than $g$.

Problem 2

The following is from Question 4 of the final exam of MTH110 last year (now we do this in Python):

The Goldbach Conjecture asserts that every even integer greater than 2 can be written as the sum of two primes.

a) Write and test a Python function PrimeSum(n) which returns true if n is the sum of two odd primes and returns false otherwise. You can assume that n is a positive integer.

Hints: Use a for-loop with x running from 3 to n-3 and test if x and n-x are both primes. The built-in function is_prime(x) can be used to test if x is prime.

b) Write and test Python function Goldbach(a,b) which returns true if the Goldbach Conjecture is correct for all even n with 2a<=n<=2b and false otherwise.