Discrete Math

Cycle B4 Calendar


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Date

Topics

Assignment

Thu, Apr 29

B4 - 24 days - ends June 2

Review
  • None
Fri, Apr 30
Review
  • None

Mon, May 3

Review

Review Assignment:

  • Scan each section in Ch 12. In particular, refresh your memory on the material in the colored boxes, and read through the example problems. Then work the exercises below:
  • 12-1 (p. 423) Written Exercises #1, 3
  • 12-2 (p. 429) WE #9, 11
  • 12-3 (p.435) WE #1, 3, 5
  • 12-4 (p. 444) WE #1, 3, 5
  • 12-5 (p. 450) WE #7, 9
  • 12-6 (p. 455) WE #1, 9

Tue, May 4

12-7 Determinants

  • What is a determinant?
    • rows and columns
    • size of a determinant
  • Evaluating a 2x2 determinant
    • eval a 2x2 determinant
  • Evaluating larger determinants
      • minors
        • the minor of an element is the determinant formed after you cross off the row and column containing that element
        • minor
        • sign of a minor
          • alternates, starting with "+"
          • sign of minors
        • evaluating a determinant using minors
          • expanding by minors
      • You can add or subtract any multiple of any row/column to any other row/column in a determinant without changing the determinant's value
  • Read:
    • 12-7 and study the examples
  • Work:
    • 12-7 Written Exercises #1-13

Wed, May 5

12-8 Applications of Determinants

  • Cramer's Rule
    • Solving 2 equations with 2 unknowns. If
      ax + by = c
      dx + ey = f, then
      cramer's rule
    • Solving 3 equations with 3 unknowns
  • Geometry
    • Area of parallelogram determined by A = (xa, ya) and B = (xb, yb) is:
      Area = absolute value of area of parallelogram
    • Area of parallelepiped determined by A = (xa, ya, za), B = (xb, yb, zb) and C = (xc, yc, zc)is:
      Area = absolute value of volume of parallelpiped
  • Read:
    • 12-8 and study the examples
  • Work:
    • 12-8 Written Exercises #1-17odd, 21
Thu, May 6

12-9 Determinants and Vectors in Three Dimensions

  • Unit Vectors
    • a unit vector is a vector whose length is 1
    • Special unit vectors
      • i is a unit vector in the +x direction
      • j is a unit vector in the +y direction
      • k is a unit vector in the +z direction
    • Unit vector notation
      • (3, 4, 5) = 3i + 4j + 5k
  • Cross Product
    • The cross product of two vectors A x B is a vector which is perpendicular to both A and B.
      • If A = (xa, ya, za) and B = (xb, yb, zb) then
        cross product
    • Geometric cross product
      • If thetais the angle between vectors A and B, then
        |A x B| = |A||B|sintheta
      • The direction of A x B is given by the right hand rule:
        "Point the fingers of your right hand in the direction of A, then curl them toward the direction of B. Your thumb points in the direction of A x B."
    • Properties of the Cross Product
      • A x B is perpendicular to both A and B .
      • A x B = -(B x A) . This means that the vectors A x B and B x A are opposite in direction - The cross product is NOT a commutative operation!
      • A x (B + C) = (A x B) + (A x C). The cross product is distributive across (vector) addition.
      • |A x B| is the area of the parallelogram formed by vectors A and B.
      • A x B = 0 means that A and B are parallel (and vice versa).
    • Cross product and dot product on the TI-89
  • Read:
    • 12-9 and study the examples
  • Work:
    • 12-9 Written Exercises (p. 467) #1-9, 11-15odd
Fri, May 7
12-9 Determinants and Vectors in Three Dimensions
  • Finish the 12-9 assignment

Mon, May 10

Cellular Automata - 1 Assignment
Tue, May 11
Cellular Automata - 2
Wed May 12
Cellular Automata - 3  

Thu, May 13

Cellular Automata - 4  

Fri, May 14

(Mr. S absent)

Review
  • Read:
    • Chapter Summary on P. 468-469
  • Answer:
    • Chapter Test (p. 469) #1-13
    • [Hint for #10: The center of the sphere is the point (0, -1, 3), so a vector normal (perpendicular) to the plane goes through the points (0, -1, 3) and (2, -3, -2).]

Mon, May 17

Reward Day
  • None

Tue, May 18

Review
  • None

Wed, May 19

Test - Ch 12 & Cellular Automata

  • None
Thu, May 20

13-1 Arithmetic and Geometric Sequences

  • Sequence
    • function whose domain is usually the positive integers (natural numbers)
    • first term, t1, second term t2, nth term tn
    • can be specified by a formula:
      tn = n2 - 1
  • Arithmetic Sequences
    • common difference, d = difference between any two adjacent terms
    • the nth term:
      • tn = t1 + d(n - 1)
    • Example:
      • 1, 4, 7, 10, 13, ... is an arithmetic sequence with common difference, d = 3.
      • t1 = 1, t2 = 4, etc.
      • the seventh term, t7 = 1 + 3(7-1) = 1 + 18 = 19
  • Geometric Sequences
    • common ratio, r = quotient of any two adjacent terms
    • the nth term:
      • tn = t1rn-1
    • Example:
    • 1, 4, 16, 64, 256... is a geometric sequence with common ratio, r = 4.
    • the seventh term, t7 = (1)(47-1) = 46 = 4096

 

  • Read:
    • 13-1 and study the examples
  • Work:
    • 13-1 Written Exercises (p. 476) #1-41odd
Fri, May 21

13-2 Recursive Definitions

  • The formulas we used in 13-1 are called explicit definitions
    • An explicit definition lets you calculate the value of any term
  • A recursive definition has two parts:
    • initial condition tells where the sequence starts
    • recursion formula tells how to get the next term from the current term
  • Finding:
    • an explicit definition from a recursive definition
      • Example: What is an explicit definition for the sequence t1 = 1, tn = tn-1 + 4?
        • Generate some terms: 1, 5, 9, 13, 17, ...
        • Recognize: This is an arithmetic sequence with t1 = 1 and d = 4.
        • Since tn = t1 + d(n - 1) for an arithmetic sequence, tn = 1 + 4(n - 1), which simplifies to tn = 4n - 3.
    • a recursive definition from an explicit definition
    • a recursive definition from a description
  • Some recursive functions
  • Read:
    • 13-2 and study the examples
  • Work:
    • 13-2 Written Exercises (p. 481) #1-5odd, 11-19odd, 21-25

Mon, May 24

13-3 Sums of Arithmetic and Geometric Series

  • Read:
    • 13-3 and study the examples
  • Work:
    • 13-3 Written Exercises (p. 489) #1-9odd, 17-23odd, 29, 31
Tue, May 25

13-4 Limits of Infinite Sequences

  • In mathematics, "limit" does not mean "limitation" or "handicap" - it means "a number that the terms of the sequence get closer and closer to, or homes in on" - it is a "target."
  • lim as n->inf Sn = L means "As the number of terms gets larger and larger, the value of the terms of sequence S get closer and closer to (or homes in on) L"
  • lim of rational function
  • limits that "do not exist"
    • infinite limits
  • Read:
    • 13-4 and study the examples
  • Work:
    • 13-4 Written Exercises (p. 496) #1-6, 9 (ans: 0), 10, 13, 15, 19, 20, 26
Wed, May 26
Quiz - Chapter 13
  • Seniors - turn in books

Thu, May 27

(Last day for seniors with privileges)

 
  • Seniors
    • turn in books
    • last day to turn in assignments

Fri, May 28

(Senior grades due 8 A.M.)

No Seniors (who still have senior privileges)

Goodbye Seniors and Best Wishes

Mon, May 31

 

Memorial Day - No School

  • None
Tue, June 1
   

Wed, June 2

End of B4 - 23 days

 

Have a GREAT summer break!


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Last update by JL Stanbrough