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Published byMeredith Norma Harrington Modified over 6 years ago
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The beginning We are going to write a RECURRANCE RELATIONSHIP for the following sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 … RECURRANCE RELATIONSHIP is an equation describing which number comes next. By Tom Bolan
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1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 … First we must understand what is happening: 1+1=2 Each number is the sum of the two before it 1+2=3 2+3=5 3+5=8 5+8=13
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1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 … The first term is 1 The second term is 1 The third term is 2 The seventh term is 13
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1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 The nth term is the sum of the 2 terms before it
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1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 This is a famous sequence (probably the only famous one) called the “Fibonacci Sequence” Believe it or not, this actually shows up in the real world.
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Today we are going to learn about 2 things
SIGMA NOTATION Shorthand for writing series “Sigma”: Σ Example: LIMITS The “target” of a sequence or series What a function gets infinitely close to.
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Sigma (Σ) notation 3 + 8 + 13 +…
Find the sum of the first 10 terms of the series …
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Sigma (Σ) notation 3 + 8 + 13 +… The long way to do it:
Find the sum of the first 10 terms of the series … The long way to do it:
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Sigma (Σ) notation 3 + 8 + 13 +… The long way to do it:
Find the sum of the first 10 terms of the series … The long way to do it: But we know better, if n was 100, this would take forever, so we take a shortcut . . .
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3 + 8 + 13 +… + t10 Find the Explicit definition: tn = 3+5(n-1)
How much each term goes up by The first term
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3 + 8 + 13 +… + t10 Find the Explicit definition: tn = 3+5(n-1)
Find the first and last terms t1 = t10 = 48 We get this by putting in 10 for n
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3 + 8 + 13 +… + t10 Find the Explicit definition: tn = 3+5(n-1)
Find the first and last terms t1 = t10 = 48 Use this formula To find the sum of the first “n” terms. “n”, in this case, is 10
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3 + 8 + 13 +… + t10 Find the Explicit definition: tn = 3+5(n-1)
Find the first and last terms t1 = t10 = 48 Use the formula 4. Plug n’ chug Put in 10 for n!
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Term #1 Term #10
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Along with a shorter way to solve it, there is a shorter way to write the question:
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Instead of writing: Find the sum of the first 10 terms of the series 3 + 8 + 13 +…
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Find the sum of the first 10 terms of the series 3 + 8 + 13 +…
Read: “the sum of 3+5(n-1) as n goes from 1 to 10 is 255.”
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Stop here: Start here: Put 1 in for n
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We put in 1-10 We got out this sequence 3,8,13,18,23,28,33,38,43,48 But this symbol means add ‘em up!
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Write each of the following in sigma notation
D
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B
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C
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D
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NOW FOR CALCULUS:
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LIMITS Find a partner Find a location 8 squares away from a wall
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LIMITS 1 partner will do the walking, the other will record
X=number of steps Y = number of squares to the wall At each step go half the distance to the wall Record until y = 0 X Y 8 1 4 2
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START WALKIN’
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So how many steps does it take to reach the wall?
4? ∞ 8? 64?
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So you never actually reach the wall…
…but you get infinitely close! As the number of steps you take approaches infinity… Your distance from the wall approaches 0!
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The sequence describing your distance to the wall…
8, 4, 2, 1, ½, … Is described by the function: Where “n” is the number of steps you take.
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Distance = And as n (the number of steps) approaches infinity
Distance to the wall approaches 0.
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“The limit, as n approaches infinity,
of is 0.”
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Limits are the basic math behind calculus
The 2 basic things calculus is used to find are AREA and SLOPE! But for really ugly functions
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Pretend this is our function f(x)
And we want to know the slope right here You can’t use rise/run because its curved! But…
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IF WE ZOOM IN…
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A gazillion times
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A gazillion times Then it looks like a straight line!
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A gazillion times Then it looks like a straight line!
When you do it an infinite number of times it is a straight line!
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A gazillion times Then it looks like a straight line!
When you do it an infinite number of times it is a straight line! So you can find the slope!
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What if you want to find the area under the line?
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You estimate, by finding the area of the rectangle.
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If that estimate isn’t good enough, use more rectangles!
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You guessed it… An INFINITE number of rectangles!
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CALCULUS IS . . . Using the mathematics of infinity (limits), to make otherwise impossible calculations. O.K., so I left a bunch of stuff out, but you get the basic idea.
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