Upload
amos-chandler
View
222
Download
3
Embed Size (px)
Citation preview
MATH 2160 1st Exam Review
Problem Solving; Venn Diagrams; Patterns; Pascal’s
Triangle; Sequences
Problem Solving
Polya’s 4 Steps Understand the problem Devise a plan Carry out the plan Look back
Problem Solving Strategies for Problem Solving
Make a chart or table Draw a picture or diagram Guess, test, and revise Form an algebraic model Look for a pattern Try a simpler version of the problem Work backward Restate the problem Eliminate impossible situations Use reasoning
Problem Solving
How many hand shakes? Playing darts Tetrominos Who am I? Triangle puzzle
Venn Diagrams Vocabulary
Universe Element Set Subset Disjoint Mutually Exclusive Finite
Intersection Union Compliment Empty Set Infinite
What can you say about A and B?
A B = A B = {A, B} A and B are mutually exclusive or disjoint
Venn Diagrams
A B
Venn Diagrams What can you say about A and B?
A B = A B = A’ B = A’ B = A B’ = A
B’ = A’ B’ = A’ B’ =
BA
Venn Diagrams What can you say
about A, B, and C? A B C =? A B C =? (A C) B =? A (C B) =? (A B) C =? C (A B) =? (B C) A =? B (C A) =? (A’ B) C =? (A’ B) C =? A’ B’ C’ =? A’ B’ C’ =?
Etc.
BA
C
Patterns Triangular Numbers
Etc.
T1 T2 T3 T4
Tn = Tn-1 + n
n
1xn xT
2
)1n(nTn
Patterns
Square Numbers
Etc.
S1 S2 S3 S4
Sn = n2
Patterns
Rectangular Numbers
Etc.
R1 R2 R3 R4
Rn = n (n + 1)
Rn = n2 + n
Pascal’s Triangle
Expanding a binomial expression:(a + b)0 = 1(a + b)1 = a + b(a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3
+ b4
Pascal’s Triangle
Vocabulary Expansion – the sum of all of the
terms Coefficient – the number in front of
the variable(s) for a particular term Variable(s) – the letters AND their
exponents for a particular term Term – the coefficient AND the
variable(s)
Pascal’s Triangle
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 1
Pascal’s Triangle
Magic 11’s110 1111 1 1112 1 2 1113 1 3 3 1114 1 4 6 4 1 Fails to work after this…
Arithmetic Sequences
The difference between any two consecutive terms is always the same. Examples:
1, 2, 3, … 1, 3, 5, 7, … 5, 10, 15, 20, …
Non-Examples 1, 4, 9, 16, … 2, 6, 12, 20, …
Arithmetic Sequences
The nth number in a series: an = a1 + (n – 1) d
Example Given 2, 5, 8, …; find the 100th term
n = 100; a1 = 2; d = 3 a100 = 2 + (100 – 1) 3 a100 = 2 + (99) 3 a100 = 2 + 297 a100 = 299
Arithmetic Sequences
Summing or adding up n terms in a sequence: Example:
Given 2, 5, 8, …; add the first 50 terms n = 50; a1 = 2; a50 = 2 + (50 – 1) 3 = 149 S50 = (50/2) (2 + 149) S50 = 25 (151) S50 = 3775
n1n aa2
nS
Arithmetic Sequences
Summing or adding up n terms in a sequence: Example:
Given 2, 5, 8, …; add the first 51 terms n = 51; a1 = 2; a2 = 5; a51 = 2 + (51 – 1) 3 = 152 S51 = 2 + ((51-1)/2) (5 + 152) S51 = 2 + (50/2) (5 + 152) S51 = 2 + 25 (157) S51 = 2 + 3925 S51 = 3927
n21n aa2
1-naS
Geometric Sequences
The ratio between any two consecutive terms is always the same. Examples:
1, 2, 4, 8, … 1, 3, 9, 27, … 5, 20, 80, 320, …
Non-Examples 1, 4, 9, 16, … 2, 6, 12, 20, …
Geometric Sequences
The nth number in a series: an = a1 r(n-1)
Example Given 5, 20, 80, 320, …; find the 10th term
n = 10; a1 = 5; r = 20/5 = 4 a10 = 5 (4(10-1)) a10 = 5 (49) a10 = 5 (262144) a10 = 1310720
Geometric Sequences
Summing or adding up n terms in a sequence: Example:
Given 5, 20, 80, 320, …; add the first 7 terms n = 7; a1 = 5; r= 20/5 = 4 S7 = 5(1 – 47)/(1 – 4) S7 = 5(1 – 16384)/(– 3) = 5(– 16383)/(– 3) S7 = (– 81915)/(– 3) = (81915)/(3) S7 = 27305
r1
r1aS
n1
n
Fibonacci Sequences
1, 1, 2, 3, … Seen in nature
Pine cone Sunflower Snails Star fish
Golden ratio (n + 1) term / n term of Fibonacci Golden ratio ≈ 1.618
Test Taking Tips Get a good nights rest before the exam Prepare materials for exam in advance
(scratch paper, pencil, and calculator) Read questions carefully and ask if you
have a question DURING the exam Remember: If you are prepared, you
need not fear