Derivatives of Trig Functions
Trigonometric functions are useful in our practical lives in diverse areas such as astronomy, physics, surveying, carpentry etc. How can we find the derivatives of the trigonometric functions? Our starting point is the following limit: Using the derivative language, this limit means that . To see why, it is enough to rewrite the expression involving the cosine as But , so we have This limit equals and thus In fact, we may use these limits to find the derivative of and at any point x=a. So which implies So we have proved that exists and Similarly, we obtain that exists and that Since , and are all quotients of the functions , we can compute their derivatives with the help of the quotient rule: It is quite interesting to see the close relationship between (and also between From the above results we get These two results are very useful in solving some differential equations. Example 1. . So using the product rule, we get which implies, using trigonometric identities, Exercise 1. at the point Answer. Exercise 2.
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Pi is Wrong: Tau
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Table of Trigonometric Identities
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Geometry
Geometry is all about shapes and their properties. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into: Point, Line, Plane and Solid A Point has no dimensions, only position A Line is one-dimensional A Plane is two dimensional (2D) A Solid is three-dimensional (3D) Why? Why do we do Geometry? Plane Geometry Plane Geometry is all about shapes on a flat surface (like on an endless piece of paper). Perimeter General Drawing Tool Polygons A Polygon is a 2-dimensional shape made of straight lines. Here are some more: The Circle Circle Theorems (Advanced Topic) Symbols There are many special symbols used in Geometry. Geometric Symbols Congruent and Similar Angles Types of Angles Transformations and Symmetry Transformations: Symmetry: Symmetry Artist Coordinates More Advanced Topics in Plane Geometry Pythagoras Conic Sections Trigonometry Trigonometry is a special subject of its own, so you might like to visit: Solid Geometry Common 3D Shapes
Solving Equations – MONSTER MATH!
Solving equations is one of the MOST important things that I do with middle school students (other than fractions, of course). I want them to learn the “process” of solving equations with simple, one-step equations. Hopefully when we get to multiple step equations they will have a process to fall back on. I always start with Hands On Equations. This year, I also had students do the Khan Academy’s equation solving intuition problem set before we even started the Hands On Equations. I used the Hands On Equation manipulates (which I called TOYS so they loved them before they even saw them) and my own worksheet (below) which they LOVED because I let them draw MONSTERS on it. OH MY! Day 2 we did not get out the “toys” but did do a pass-it equation solving unit where each student got to make up their own one-step equation (and thus their own monster) and then pass it to a friend for the second step. I still can’t believe what the right “spin” can do to a lesson with 6th graders! Like this:
Why Is American Teaching So Bad? by Jonathan Zimmerman
The Teacher Wars: A History of America’s Most Embattled Profession by Dana Goldstein Doubleday, 349 pp., $26.95 Building a Better Teacher: How Teaching Works (and How to Teach It to Everyone) by Elizabeth Green Norton, 372 pp., $27.95 Getting Schooled: The Reeducation of an American Teacher by Garret Keizer Metropolitan, 302 pp., $27.00 In 1853, the most important man in nineteenth-century American education gave a speech praising female teachers. her head encircled with a halo of heavenly light, her feet sweetening the earth on which she treads, and the celestial radiance of her benignity making vice begin its work of repentance through very envy of the beauty of virtue! In a rapidly industrializing nation, in which there were many perils of poverty and violence, as well as opportunity, schools needed to inculcate thrift, civility, and self-control in the young. The shift in goals has unfortunately done nothing to alter the tedious, anti-intellectual practices of American teaching.
Muhammad ibn Musa al-Khwarizmi
Dari Wikipedia, the free encyclopedia Muhammad ibn Mūsā al-Khwārizmī (Arabic: محمد بن موسى الخوارزمي) was a Persian[1] mathematician, astronomer, astrologer and geographer. He was born around 780 in Khwārizm[2] (now Khiva, Uzbekistan) and died around 850. He worked most of his life as a scholar in the House of Wisdom in Baghdad. His Algebra was the first book on the systematic solution of linear and quadratic equations. Consequently he is considered to be the father of algebra,[3] a title he shares with Diophantus. His contributions not only made a great impact on mathematics, but on language as well. Biography Few details about al-Khwārizmī’s life are known; it is not even certain where he was born. His kunya is given as either Abū ʿAbd Allāh (Arabic: أبو عبد الله) or Abū Jaʿfar (أبو جعفر in Arabic).[7] The historian al-Tabari gave his name as Muhammad ibn Musa al-Khwārizmī al-Majousi al-Katarbali (Arabic: محمد بن موسى الخوارزميّ المجوسيّ القطربّليّ). Algebra Arithmetic
