Mathematics Interview Questions (XVII)
86. Find the values of all the derivatives of at .
Solution. Its derivative is . The limit as can be calculated using first a change of variable , which implies . As a result, and . Accordingly, only one-sided derivatives exist, which are from the right and from the left respectively.
Extension. Draw its graph. Draw also the graph of the derivative on the same axes. Lastly, draw the graph of .
87. Show that is divisble by 12.
Solution. so it must be divisible by 6 (one of , , and must be divisble by 2 and one must be divisible by 3). If is even then is divisible by 4 as well. If is odd then and are both even, so it is also divisible by 4.
Extension. Do it again, by modular arithmetic this time.
88. Explain what integration is.
Solution. In the Riemann integral framework, there are two kinds of (single variable) integrations. One is called the indefinite integration. The indefinite integral of a function is a family of functions that is unique up to a constant satisfying . The other is called the definite integration. The definite integral of a function on the closed interval is defined as the Riemann sum
These two kinds of integrals are related by the fundamental theorem of calculus (a.k.a. the Newton-Leibniz formula)
Extension. What about the Lebesgue integral? If you haven't heard of it, look it up!
89. If is a perfect square and its second last digit is 7, what are the possibilities for the last digit of ?
Solution. Any number that is a square mod 100 is necessarily a square both mod 4 and mod 5, which is to say 0 or 1 mod 4 and 0, 1, or 4 mod 5. The only number in the 70s that satisfies both criteria is 76. The answer is therefore 6.
Extension. Prove a more general case: the last two decimal digits of a perfect square must be one of the following pairs: 00, e1, e4, 25, o6, e9 where e stand for an even digit and o for odd.
90. If where , show that .
Solution. If we let , this gives which implies or . However, if , so .
Extension. Solve that functional equation. What assumption(s) are you making? What if we drop this/these assumption(s)?