Μ 2 exercises p670 1 14 103 normal distributions

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- μ 2 * Exercises: p.670 1-14 10.3 Normal Distributions * Terms: (standard) normal random variable, (standard) normal distribution, (standard) normal curve * The normal probability density function with mean μ and standard deviation σ is defined to be f ( x ) = e - ( 1 2 )[ ( x - μ ) σ ] 2 σ 2 π , ( -∞ < x < ) * Know how to read the chart to compute various probabilities with the standard normal distribution. * If X is a normal random variable with mean μ and standard deviation σ , then Z = X - μ σ and P ( a < X < b ) = P ( a - μ σ < Z < b - μ σ ) where Z is the standard normal random variable. * Exercises: p.682 1-20 Chapter 11 Taylor Polynomials and Infinite Series 11.1 Taylor Polynomials * The n th Taylor Polynomial: Suppose that the function f and its first n derivatives are defined at x = a . Then the n th Taylor polynomial of f at x = a is the polynomial P n ( x ) = f ( a ) + f 0 ( a )( x - a ) + f 00 ( a ) 2! ( x - a ) 2 + · · · + f ( n ) ( a ) n ! ( x - a ) n which coincides with f ( x ) , f 0 ( x ) , . . . , f ( n ) ( x ) at x = a ; that is, P n ( a ) = f ( a ) , P 0 ( a ) = f 0 ( a ) , . . . , P ( n ) ( a ) = f ( n ) ( a ) * Know how to use a Taylor Polynomial of a function at x = a to approximate its integral at x = a . * Exercises: pp.698-699 1-22, 27,28 11.2 Infinite Sequences * Terms: infinite sequence, terms, n th term, convergent, divergent * Limit of a Sequence: Let { a n } be a sequence. We say that the sequence { a n } converges and has the limit L , written lim n →∞ a n = L if the terms of the sequence, a n , can be made as close to L as we please by taking n sufficiently large. If a sequence is not convergent, it is said to be divergent. * Limit Properties of Sequences: Suppose lim n →∞ a n = A and lim n →∞ b n = B Then 10
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1. lim n →∞ ca n = c lim n →∞ a n = cA , ( c , a constant) 2. lim n →∞ a n ± b n = lim n →∞ a n ± lim n →∞ b n = A ± B 3. lim n →∞ a n b n = ( lim n →∞ a n )( lim n →∞ b n ) = AB 4. lim n →∞ a n b n = lim n →∞ a n lim n →∞ b n = A B provided B 6 = 0 and b n > 0 * Exercises: p.707 1-21,30-44 11.3 Infinite Series * Terms: infinite series, terms, n th term, general term, N th partial sum, telescoping series, geometric series * N th Partial Sum of an Infinite Series: Given an infinite series X n =1 a n = a 1 + · · · + a n + · · · , the N th partial sum of the series is S N = N X n =1 a n = a 1 + · · · + a N . If the sequence of partial sums { S N } converges to the number S – lim N →∞ S N = S –then the series a n converges and has sum S , written X n =1 a n = a 1 + · · · + a n · · · = S If { S N } diverges, then the series a n diverges. * Know how to find the N th partial sum of simple series (such as telescoping series) and use it to compute the sum of the series. * If | r | < 1, then the geometric series X n =0 ar n = a + ar + ar 2 + · · · + ar n + · · · coverges and its sum is a 1 - r ; that is, X n =0 ar n = a + ar + ar 2 + · · · + ar n + · · · = a 1 - r The series diverges if | r | ≥ 1. * Warning, the above fact talks about geometric series beginning at n = 0. If your geometric series does not begin at n = 0, you need to manipulate it so that it does. Here’s how: X n = k ar n = X n =0 ( ar k ) r n * Properties of Infinite Series: If X n =1 a n and X n =1 b n are convergent infinite series and c is a constant, then 1.
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