MIT6_042JS10_lec19_prob - Massachusetts Institute of...

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Massachusetts Institute of Technology 6.042J/18.062J, Spring ’10 : Mathematics for Computer Science March 17 Prof. Albert R. Meyer revised March 2, 2010, 733 minutes In-Class Problems Week 7, Wed. Problem 1. The Elementary 18.01 Functions (F18’s) are the set of functions of one real variable defined recur- sively as follows: Base cases: • The identity function, id ( x ) ::= x is an F18, • any constant function is an F18, • the sine function is an F18, Constructor cases: If f,g are F18’s, then so are 1. f + g , fg , e g (the constant e ), 2. the inverse function f ( 1) , 3. the composition f g . (a) Prove that the function 1 /x is an F18. Warning: Don’t confuse 1 /x = x 1 with the inverse, id ( 1) of the identity function id ( x ) . The inverse id ( 1) is equal to id. (b) Prove by Structural Induction on this definition that the Elementary 18.01 Functions are closed under taking derivatives . That is, show that if f ( x ) is an F18, then so is f ::= df/dx . (Just work out 2 or 3 of the most interesting constructor cases; you may skip the less interesting ones.) Problem 2. Let
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MIT6_042JS10_lec19_prob - Massachusetts Institute of...

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