Every language is also a special way of looking at the world and interpreting experience. Concealed in the structure of language are a whole set of unconscious assumptions about the world and the life in it.
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Nov. 27, 2019, 5:01 p.m.

The Trapezoid Geometric Method of Finding Area

By Maurice Ticas

Tags:

Continuity

Geometry

Integrals

Consder a real-valued continuous function from points a, b of its domain where a < b. We will use the formula for the area of a trapezoid to find an approximation of the area \(\int_{a}^{b} f(x) \, dx\). Let's begin.

The formula for the area of a trapezoid is \[(c + d)/2 \times h \] where \(c, d\) are the lengths of the trapezoid basis, respectively, and \(h\) is the trapezoid height. After dividing our closed interval \([a,b]\) into n smaller subintervals of width \(\frac{b-a}{n}\), we then have that \(\int_{a}^{b} f(x) \, dx \) is approximately equal to \[ \frac{f(x_0) + f(x_1)}{2} \times \frac{b-a}{n} + \frac{f(x_1) + f(x_2)}{2} \times \frac{b-a}{n} + \dots + \frac{f(x_n-1) + f(x_n)}{2} \times \frac{b-a}{n} \\ = \frac{b-a}{2n} ( f(x_0) + 2 \, f(x_1) + \cdots + 2 \, f(x_{n-1}) + f(x_n)) \]

Our geometric intuition would allow us to be convinced that when \(f\) is concave up on \([a, b]\), then the trapezoid method is an over-approximation. When \(f\) is concave down on \([a, b]\), the method is an under-approximation.

Now consider applying the method for \(\int_{-\pi}^{\pi} \sin(x) \, dx\) for when \(n = 3\) and allow your imagination to give you the geometric intuition to see that the approximation gives the correct answer of \(0\). In this case, the trapezoids become triangles and what your are left with are two pairs of triangles with areas that cancel each other out to zero.

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Nov. 4, 2019, 11:35 p.m.

When To Expect Good Work

By Maurice Ticas

Tags:

Motivational

Personal

Statistics

There is data to support an upbeat outlook in life. Reading from Zuckerman's excerpt about Mr. Simon encouraged me to share meaningful data, albeit small in size.

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May 28, 2019, 7:42 a.m.

Linear Algebra for the Motivated

By Maurice Ticas

Tags:

Algebra

Linear Algebra

Matroids

Mastering the concepts from Axler's book opens doors

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