Review 11.2 Vectors in space

Basic Information: Component form: v = <v1, v2, v3> Unit vector form: v = v1i + v2j + v3k Standard unit vector notation uses these unit vectors: i = <1, 0, 0>, j = <0, 1, 0> and k = <0, 0, 1>. Zero vector: 0 = <0, 0, 0>

Vectors in Space: 1. Two vectors are equal if and only if their corresponding components are equal. 2. The length of vector u = <u1, u2, u3> is: ||u|| = √(u1 3. A unit vector u in the direction of v is given by: u = v / ||v||, v ≠ 0 4. The sum of u = <u1, u2, u3> and v = <v1, v2, v3> is: u + v = <u1 + v1, u2 + v2, u3 + v3> 5. The scalar multiple of the real number c and vector u is: cu = <cu1, cu2, cu3> 6. The dot product of vector u and vector v is: u • v = u1v1 + u2v2 + u3v3

Angle Between Two Nonzero Vectors If θ is the angle between two nonzero vectors u and v, then: cosθ = (u • v) / (||u|| ||v||) If the dot product of two nonzero vectors is 0, the vectors are orthogonal (perpendicular). In general, two nonzero vectors u and v are parallel if there is some scalar c such that u = cv.

Using Vectors to Determine Collinear Points The points P, Q, and R are collinear if and only if the vectors PQ and PR are parallel.

Example #1:

Find the dot product of the vectors: u = <5, 8, 2> and v = <7, 1, 6>.

u • v = (5)(7) + (8)(1) + (2)(6) = 35 + 8 + 12 = 55

Review 5.2 Verifying Trigonometric Identities

In 5.2, the purpose is to verify trigonometric identities. In order to verify trigonometric identities, fundamental identities and the rules of algebra are important to be memorized. There are 4 

rules for verifying trigonometric identities:
1. Work with only one side the equation.
2. Factor an expression, add fractions, square binomials, or create a monomaniacal denominator.
3. Look for opportunities to use the fundamental identities .
4. Convert all terms to sines and cosines if the preceding guidelines do not help.


Example#1: 
2sec^2x - 2sec^2xsin^2x - sin^2x - cos^2x
=2sec^2x (1-sin^2x)-sin^2x-cos^2x Factor
=2sec^2x(cos^2x)-sin^x-cos^2x Pythagorean identities
=2(1/cos^2x)(cos^2x)-sin^x-cos^2x Reciprocal identities
=2-sin^2x-(1-sin^2x) Pythagorean identities
=2-sin^2x-1+sin^x Simplify

Review 5.3 Solving Trigonometric Equations

Summary:
• To solve a trigonometric equation by using standard algebraic techniques.
• Solving trigonometric equations of quadratic type.
• Solving trigonometric equations involving multiple angles.
• Use inverse function to solve the trigonometric equations.


Rules for solving trigonometric equation:
• Algebra rule (factor, simplify, combine terms).
• Constantly be aware of to the interval for each problem.
• For inverse functions, try to use several different identities to work out the problem.
• Do not forget to add 2nπ for general solution of sine and cosine; add n π for general solution of tangent.


Example 1: Solve 2sin²x+3cosx-3=0
Try to rewriting the equation so that it has only cosine functions.
2sin²x+3cosx-3=0
2（1-cos²x）+3cosx-3=0 Pythagorean identity
Setting each factor equal to zero, to get the general solution:
2cos²x-3cosx+1=0 Multiply both sides by-1
(2cosx-1)(cos x-1)=0 Factor
cos x=1/2 and cos x=1
x=2nπ，x=π/3+2nπ，x=5π/3+2nπ

MATH GAME

This game is centuries old, Captain James Cook used to play it with his fellow officers on his long voyages, and so it has also been called "Captain's Mistress". Milton Bradley (now owned by Hasbro) published a version of this game called "Connect Four" in 1974.

Other names for this this game are "Four-in-a-Row" and "Plot Four".

http://www.mathsisfun.com/games/connect4.html

here is the link for the game. have fun guys

12.4 Limits at Infinity and Limits of Sequences

Example:

The numerator is always 100 and the denominator  approaches  as x approaches  , so that the resulting fraction approaches 0.

12.5 The area Problem

In order calculate the area of region, we need know the following summation formulas and properties, which are used to evaluate finite and infinite summations. 

finding math talents

Sara, who is 5 years old, listens as her 32-year-old father comments that today is her grandmother's 64th birthday. "Grandma's age is just twice my age," he observes.
Although outwardly Sara does not seem to react to this information, her mind is whirling. A few moments pass, and then the young girl excitedly replies, "You know Dad, you will only be 54 when your age is twice mine!"
Sara has been intrigued by numbers and numerical relationships since she was very small. At first this could be seen in the way she liked to count things and organize groups of objects. She showed a fascination for calendars, telephone numbers, dates, ages, measurements, and almost anything else dealing with numbers. Sara learned and remembered this information quickly and easily, but what was even more amazing was the way she played with and manipulated the information she was learning. She would carefully examine each idea and eagerly search to discover new, interesting, and unusual relationships and patterns. Although Sara has had little formal instruction in mathematics, at the age of 5 she has acquired an incredible amount of mathematical knowledge and is amazingly sophisticated in using this knowledge to discover new ideas and solve problems.
Sara is an example of a young child who is highly talented in the area of mathematics. Like most individuals with this unusual talent, Sara exhibits characteristics and behaviors that are clues to her ability. Some mathematically talented people radiate many or obvious clues, others offer only a few, or subtle ones. Recognizing these clues is often an important first step in discovering an individual's high ability in mathematics. It is difficult to believe, but many people with a high degree of mathematical talent have their talent underestimated or even unrecognized. Their clues have gone unnoticed or ignored, and the true nature of their ability remains unexplored. If Sara's talent in mathematics is to be discovered and appropriately nurtured, it is important that her parents and teachers recognize the clues.

  

Information fromhttp://www.gifted.uconn.edu/siegle/tag/Digests/e482.html