Logarithms. For example:

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1 Math Review Summation Formulas Let >, let A, B, and C e constants, and let f and g e any functions. Then: f C Cf ) ) S: factor out constant ± ± g f g f ) ) )) ) S: separate summed terms C C ) 6 ) ) Computer Science Dept Va Tech July McQuain WD S3: sum of constant S4: sum of 6 S5: sum of squared S6: sum of ^ ) S7: sum of ^-)

2 Logarithms Math Review Let e a real numer, > and. Then, for any real numer >, the arithm of to ase is the power to which must e raised to yield. That is: y ) y if and only if For eample: 64) 6 ecause 6 64 / 8) 3 ecause 3 / 8 ) ecause If the ase is omitted, the standard convention in mathematics is that ase is intended; in computer science the standard convention is that ase is intended. Computer Science Dept Va Tech July McQuain WD

3 Logarithms Math Review 3 Let a and e real numers, oth positive and neither equal to. Let > and y > e real numers. L: ) L7: y) ) y) L: L3: L4: ) ) < for all < < ) > for all > L8: L9: y y ) y ) ) y) L5: L6: y ) ) y L: ) a a ) ) Computer Science Dept Va Tech July McQuain WD

4 Limit of a Function Math Review 4 Definition: Let f) e a function with domain a, ) and let a < c <. The it of f) as approaches c is L if, for every positive real numer ε, there is a positive real numer δ such that whenever -c < δ then f) L < ε. The definition eing cumersome, the following theorems on its are useful. We assume f) is a function with domain as descried aove and that K is a constant. C: K K c C: r r C3: c for all r> Computer Science Dept Va Tech July McQuain WD

5 Limit of a Function Math Review 5 Here assume f) and g) are functions with domain as descried aove and that K is a constant, and that oth the following its eist and are finite): f ) A g ) B Then: C4: Kf ) K f ) C5: C6: C7: f ) ± g ) ) f ) ± g ) f )* g ) ) f )* g ) f ) / g ) ) f ) / g ) provided B Computer Science Dept Va Tech July McQuain WD

6 Limit as Approaches Infinity Math Review 6 Definition: Let f) e a function with domain [, ). The it of f) as approaches is L if, for every positive real numer ε, there is a positive real numer such that whenever > then f) L < ε. The definition eing cumersome, the following theorems on its are useful. We assume f) is a function with domain [, ) and that K is a constant. C8: K K C9: C: for all r> r Computer Science Dept Va Tech July McQuain WD

7 Limit of a Rational Function Math Review 7 Given a rational function the last two rules are sufficient if a little algera is employed: Divide y highest power of from the denominator. Tae its term y term. Apply theorem C3. Computer Science Dept Va Tech July McQuain WD

8 Infinite Limits Math Review 8 In some cases, the it may e infinite. Mathematically, this means that the it does not eist. C: r for all r> C3: e ) C: ) Eample: Computer Science Dept Va Tech July McQuain WD

9 l'hôpital's Rule Math Review 9 In some cases, the reduction tric shown for rational functions does not apply: 7 5 ) 5?? In such cases, l'hôpital's Rule is often useful. If f) and g) are differentiale functions such that then: f ) g ) c f ) f ) g ) g ) This also applies if the it is. Computer Science Dept Va Tech July McQuain WD

10 l'hôpital's Rule Eamples Math Review Applying l'hôpital's Rule: ) 5 7 Another eample: 3 e 3 e 6 e 6 e [ ] f ) f ) Recall that: D e e D[ f ) ] Computer Science Dept Va Tech July McQuain WD

11 Mathematical Induction Math Review Mathematical induction is a technique for proving that a statement is true for all integers in the range from to, where is typically or. First Principle of Mathematical Induction Let P) e a proposition regarding the integer, and let S e the set of all integers for which P) is true. If ) is in S, and ) whenever is in S then is also in S, then S contains all integers in the range [, ). To apply the PMI, we must first estalish that a specific integer,, is in S estalishing the asis) and then we must estalish that if a aritrary integer,, is in S then its successor,, is also in S. Computer Science Dept Va Tech July McQuain WD

12 Induction Eample Math Review Theorem: For all integers n, n n is a multiple of. proof: Let S e the set of all integers for which n n is a multiple of. If n, then n n, which is oviously a multiple of. This estalishes the asis, that is in S. ow suppose that some integer is an element of S. Then is a multiple of. We need to show that is an element of S; in other words, we must show that ) ) is a multiple of. Performing simple algera: ) ) ) ) 3 ow we now is a multiple of, and the epression aove can e grouped to show: ) ) ) ) ) ) The last epression is the sum of two multiples of, so it's also a multiple of. Therefore, is an element of S. Therefore, y PMI, S contains all integers [, ). QED Computer Science Dept Va Tech July McQuain WD

13 Inadequacy of the First Form of Induction Math Review 3 Theorem: Every integer greater than 3 can e written as a sum of 's and 5's. That is, if > 3, then there are nonnegative integers and y such that 5y.) This is not easily) provale using the First Principle of Induction. The prolem is that the way to write in terms of 's and 5's has little to do with the way is written in terms of 's and 5's. For eample, if we now that we can say that 5y 5y 5y ) 5 3) 5y ) ut we have no reason to elieve that y is nonnegative. Suppose for eample that is 9.) Computer Science Dept Va Tech July McQuain WD

14 "Strong" Form of Induction Math Review 4 There is a second statement of induction, sometimes called the "strong" form, that is adequate to prove the result on the preceding slide: Second Principle of Mathematical Induction Let P) e a proposition regarding the integer, and let S e the set of all integers for which P) is true. If ) is in S, and ) whenever through are in S then is also in S, then S contains all integers in the range [, ). Interestingly, the "strong" form of induction is ically equivalent to the "wea" form stated earlier; so in principle, anything that can e proved using the "strong" form can also e proved using the "wea" form. Computer Science Dept Va Tech July McQuain WD

15 Using the Second Form of Induction Math Review 5 Theorem: Every integer greater than 3 can e written as a sum of 's and 5's. proof: Let S e the set of all integers n > 3 for which n 5y for some nonnegative integers and y. If n 4, then n * 5*. If n 5, then n * 5*. This estalishes the asis, that 4 and 5 are in S. ow suppose that all integers from 4 through are elements of S, where 5. We need to show that is an element of S; in other words, we must show that r 5s for some nonnegative integers r and s. ow 6, so - 4. Therefore y our assumption, - 5y for some nonnegative integers and y. Then, simple algera yields that: - 5y ) 5y, whence is an element of S. Therefore, y the Second PMI, S contains all integers [4, ). QED Computer Science Dept Va Tech July McQuain WD