Mathematics Interview Questions (XIV)
71. Integrate .
Solution. Consider integration by parts.
72. Draw on the same axes. What does that show you? As tends to infinity, what does tend to?
Solution. You can view here for a sample. Note that and are symmetric about , which is a property of inverse functions. For a function , suppose lies on it; then for its inverse , is on its graph, and these two points are symmetric about . Using L'Hôpital's rule, , which is also an obvious result from the graph.
73. Define the term 'prime number.'
Solution. An integer that is not a product of two smaller natural numbers.
74. Find a method to determine if a number is prime.
Solution. The easiest way is to check whether the number is divisble by primes smaller than it. One thing one may consider is that integers larger than the square root do not need to be checked because, whenever , one of the two factors and is less than or equal to . This method is therefore . There exists, however, better ways (in terms of time complexity), and the best algorithm discovered now bears time complexity, though I shall not introduce them here.
75. Prove that, if , and can't both be odd.
Solution. Suppose and are both odd, then (taking as an example) we can write them in the form , i.e., , and . As a result, . However, can only be , leading to a contradiction.