Solution to Math problem #278:
Let $x$ be the first term, then $18-x$ is the second term. Accordingly, we have to find the minimum value of the following quadratic polynomial:
$$f(x) = x^2 + (18-x)^2 = 2x^2 - 36x + 18^2.$$
Taking the derivative and equating the result to zero, we get
$$f'(x) = 4x - 36 = 0.$$
Therefore $x=9$ is a critical value. When $x<9$, $f'(x)<0$, while $x>9$, $f'(x)>0$. Therefore $x=9$ is the maximum value.
by Leila Schneps and Coralie Colmez
Mathematics is fast becoming one of the most important techniques in crime detection. Where once a Sherlock Holmes would have had to be content with a magnifying glass, or a jury with gut instinct and rational discussion, now a range of methods from probability and statistics are available to help. Today, mathematics lies behind expert conclusions on a hundred forensic matters from fingerprints to DNA.
Statistics can be a precious tool when identifying the patterns behind confusing or misleading phenomena.
The University of California at Berkeley was sued for gender bias when it was observed that just 35 percent of female applicants to graduate school were being accepted, versus 44 percent of males. The investigators began by narrowing the problem down to six major departments for which, combined, the inequality shifted to an even more incriminating 46 percent of males versus just 30 percent of females. But then, a department-by-department analysis showed the exact contrary of a bias against women: in four of the six departments, they were actually accepted at a higher percentage rate than males, and in the other two, the male-female ratio was 37-34 percent and 28-24 percent, discrepancies too small to have caused the overall appearance of inequality. Continue
Math problem: Express the number 18 as a sum of two numbers $a$ and $b$: $18 = a+b$ so that the sum $a^2 + b^2$ is minimal.
Discussions and hints: Let $x = a$, then $b = 18- x$, which means we have to minimize the following function:
$$f(x) = x^2 + (18-x)^2,$$
which is a quadratic polynomial.
Solution
Below please find February's top ten most popular posts published on Virtual Math Tutor blog - as per total # of pageviews (in that order):
Solution to Math problem #277: According to the statement of the problem, we have $$a_1 + a_2 + \cdots + a_{10} > 0,$$ $$a_2 + a_3 + \cdots + a_{11} > 0,$$ $$a_3 + a_4 + \cdots + a_{12} > 0,$$ $$\cdots$$ $$a_{2013} + a_1 + \cdots + a_9 > 0.$$ Adding up these inequalities, we arrive at $$10(a_1 + a_2 + \cdots + a_{2013} > 0$$ and the result follows.
Below please find January's top ten most popular posts published on Virtual Math Tutor blog - as per total # of pageviews (in that order):
I am home from the Joint Mathematics Meetings, and I’m still trying
to process everything I learned and got fired up about there. Most of my
time was spent in formal, serious lectures, many of them quite
technical, but on Friday night, I went to something completely
different: a mathematical poetry reading, facilitated by the Journal of Humanistic Mathematics. Some of the featured poets are mathematicians or math teachers, while some are full-time writers with an interest in math.
Mathematics is often portrayed as a sterile, robotic discipline, but continue reading
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