CSE 428: Home Assignment 5

Spring 2001, CSE 428: Additional notes about Assignment 5

Exercise 2

Please represent the operators on simple expressions as the string containing the conventional mathematical symbol. More precisely:

   + (plus)            is represented by  "+"
   - (minus)           is represented by  "-"
   * (product)         is represented by  "*"
   / (division)        is represented by  "/"
   ^ (exponentiation)  is represented by  "^"

The last equation for differentiation, namely
```
    
   D(u^n) = n*u^(n-1)*Du  
```
is actually valid also for n=0 (originally the case 0 was excluded). So you should not test that n is different from 0 when you define this case. However, it is valid only when n is a number (not a function of x). This means that when you define this case, you have to write a line like
```
   | deriv (Opr("^",u,Num n), x) = ...
```
This will give you a "match nonexhaustive" warning, but that's ok. On the contrary, allowing the third argument of Opr to be a generic expression, in this definition, will be considered an error.
Please try to follow closely the definition of the table
```
   Dz = 0  if z is a number or a variable different from x
   Dx = 1
   D(u+v) = Du + Dv
   D(u-v) = Du - Dv
   D(u*v) = v*Du + u*Dv
   D(u/v) = (v*Du - u*Dv) / (v^2)
   D(u^n) = n*u^(n-1)*Du  where n is a number 
```
Namely, do not change the order in which the argument of an operator are written, like for instance Dv*u instead of u*Dv. Although these expressions are theoretically equivalent, they produce a different symbolic output, and it would make the correction difficult for the TA.
Please do not load any package in your code. You don't need any extra operators for solving any of the exercises of this assignment. Loading a package may redefine operators and mess up things when the TA tests other exercises. If you really think you need to use some other operations, like those available in the "Real" package, then you can use the name extension, like for instance Real.rem (remainder on reals). However, once more, you can write the solutions just by using those operators that are loaded authomatically when you start sml. In fact, the only operator on reals you need to use is the minus, and that is simply represented by the symbol -, which is overloaded (e.g. 5-1 is on int and 5.0-1.0 is on reals).

Exercise 3

You can assume that in (numbered n), the integer n ranges from 2 to 10. In other words, the ordering is the following (in increasing order):

  1   ace
  2   numbered 2
  3   numbered 3
  4   numbered 4
  5   numbered 5
  6   numbered 6
  7   numbered 7
  8   numbered 8
  9   numbered 9
  10  numbered 10
  11  jack
  12  queen
  13  king

you can assume that the function smaller is never applied to "non-cards", like for instance (numbered 11)