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Math Reviews

Lab 1. Math Reviews. Math Reviews. Mathematical Induction Logarithm Mod Series Derivative Limit. Mathematical Induction. What is Mathematical induction? A method of proving statements defined for the class of positive integers (or some subset of this class) Proof of a predicate

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Math Reviews

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  1. Lab 1 Math Reviews

  2. Math Reviews • Mathematical Induction • Logarithm • Mod • Series • Derivative • Limit

  3. Mathematical Induction • What is Mathematical induction? • A method of proving statements defined for the class of positive integers (or some subset of this class) • Proof of a predicate • Let P(n) be a predicate defined for integers n, and let a be a fixed integer. • Motivation

  4. Mathematical Induction • Principle of mathematical induction: • 1.Basis Step. Show that P(a) is true. • 2.Inductive Step. First, the inductive hypothesis is made - we assume P(k) is true, k > a. Then, using this assumption, we show that P(k+1) must also be true. More formally, we prove the implication, P(k) → P(k+1) for all k > a. • If we prove both parts of the definition, we have shown that P(n) is true for all n > a.

  5. Mathematical Induction • Example: • Prove: 1+4+7…+(3n-2)=(3n2-n)/2

  6. Generalized Mathematical Induction • Let P(n) be a predicate defined for integers n, and let a and b be fixed integers with a < b. A proof by generalized mathematical induction has two steps, just like regular induction: • Basis Step. Show that P(a), P(a+1), P(a+2), ... , and P(b) are all true. • Inductive Step. Show that for any integer k, k > b, if we assume P(i) is true for all integers i with a <i < k (this is the inductive hypothesis), P(k) must also be true. In logical notation the inductive step is: P(a) & P(a+1) & P(a+2) & ... & P(k-1) -> P(k). • If a = b, this will be changed to regular induction.

  7. Generalized Mathematical Induction • Example: • Consider the Fibonacci number: • 0,1,1,2,3,5,8,13,21,... • Fn=Fn-1+Fn-2 with seed values F0=0, F1=1 • Prove:

  8. Logarithm • logax = y ↔ ay = x • Log x to base a is y means • a raised to the y-th power is x. • Examples: • log28 = 3, log39 = 2, log1010 = 1, log51 = 0 • Properties

  9. Mod • m mod n = m – lowerbound(m/n) • Lowerbound(1.6) = 1 • Lowerbound(-1.3) = -2 • 9 mod 2 = 1 • 8 mod 3 = 2

  10. Series • Arithmetic series • 1,3,5,7,… • 2,5,8,11,… • Geometric series • 2,6,18,54,… • 1,4,16,64,… • Harmonic series • 1,1+1/2,1+1/2+1/3,1+1/2+1/3+1/4,…

  11. Derivative • The rate of change… • Consider a uniform acceleration: • a – acceleration. • t – time. • D – displacement. D = ½ at2 • V – velocity. V = at • Properties • ( c * f(x) )’ = c * f’(x) • …

  12. Limit • Definition: • Properties • L’Hopital’s Rule

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