Mathematics Problems Here are some problems designed to make you think. No answers are given because some problems require investigation and research. Do search the internet for help. 1. An area and volume problem Here is the graph of the curve for . (a) Find the shaded area (b) Rotate the area byabout the x-axis and find the volume of this solid of revolution. Now let and what happens? Explain what seems to be odd behaviour. 2. e and Without using a calculator show that. The fact that these numbers are so close is one of the many amazing coincidences that arise in mathematics. 3. Polite Numbers A polite number is a positive integer that is the sum of two or more consecutive integers. For example, 6 = 1 + 2 + 3, 18 = 3 + 4 + 5 + 6, 41= 20 + 21 are all polite. On the other hand 8 is not polite so is called an impolite number. Find all the impolite numbers. 4. Grads Grads are found on any scientific calculator. What is a grad and where is it used? 5. Number Systems Numbers are built up using natural numbers integers rational numbers real numbers complex numbers What comes next and what does it have to do with a bridge in Dublin? How far can this be taken? 6. An Easy(?) Derivative Differentiate log(log(sin(x))) 7. Littlewood's Problem Find the smallest positive integer N with the property that if you shift the first digit of N to the end the result is exactly the original number N multiplied by. Try not to use a computer to solve this problem. 8. Euler's Formula Euler discovered that the expression n2 + n + 41 is a prime number for many values of the positive integer n - nearly half of the first 10 million values are prime. What's the smallest value of n for which n2 + n + 41 is not prime? 9. Infinity of primes Euclid proved that there an infinite number of prime numbers. His proof is a masterpiece and ranks as one of the finest proofs in mathematics. There are other proofs which are also interesting in their own right, and Proofs From The Book gives six of them. How many proofs can you find that there are an infinite number of primes? 10. Maximum and Minimum Prove the following result, which gives an easy way to find (local) maxima and minima for some complicated derivatives: Now that your appetite had been whetted look for more on the internet Links

Steve Mayer

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