Opsimath

2021/09/17阅读：80主题：默认主题

# 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)?

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