6.090, IAP 2005—Lecture 2 1MASSACHVSETTS INSTITVTE OF TECHNOLOGYDepartment of Electrical Engineering and Computer Science6.090—Building Programming ExperienceIAP 2005Lecture 2Scheme1. Basic Elements(a) self-evaluating - expressions whose value is the same as the expression.(b) names - Name is looked up in the symbol table to find the value associated with it.Names may be made of any collection of characters that doesn’t start with a number.2. Combinati on( procedure arguments-separated-by-spaces )Value is determined by evaluating the expression for the procedure and applying the resultingvalue to the value of the arguments.3. Special Forms(a) define - (define name value)The name is bound to the result of evaluating the the value. Return value is unspecified.(b) if - (if test consequent alternative)If the value of the test is not false (#f), evaluate the consequent, otherwise evaluate thealternative.(c) lambda - (lambda parameters body)Creates a procedure with the given parameters and body. Parameters is a list of namesof variables. Body is one or more scheme expressions. When the procedure is applied,the body expressions are evaluated in order and the value of the last one is returned.6.090, IAP 2005—Lecture 2 2Problems1. Evaluation - For each expression:(a) Write the type of the expression(b) Write your guess as to the expression’s return value. If the expression is erroneoussimply indicate “error” for the value. If the expression returns an unspecified value,write whatever you want! If the expression returns a procedure, indicate “procedure”for the value.(c) Evaluate the expression, and copy the response from the *scheme* buffer.(lambda (x) x)((lambda (x) x) 17)((lambda (x y) x) 42 17)((lambda (x y) y) (/ 1 0) 3)((lambda (x y) (x y 3)) (lambda (a b) (+ a b)) 14)2. Writing Procedures - Write the procedure indicated. Then test it in scheme to make sure itworks.(a) Write a procedure cube that returns the cube of it’s input.(define cube(b) Write a procedure the-answer?, which returns true (#t) if the input is the number 42.(define the-answer?(c) Write a procedure sign that returns 1 if it’s input is positive, -1 if it’s input is negative,and 0 if it’s input is 0.(define sign6.090, IAP 2005—Lecture 2 3(d) Given a margin width m, which is both the top, bottom, left, and right margin of thepage, write a procedure that computes the ”usable” (non margin) area of the 8.5in by11in sheet of paper.(usable-page 0);Value: 93.5(usable-page 1);Value: 58.5(define usable-page(e) Write a procedure that when given a width, returns the length of the mos t beautifulrectangle having that width. According to studies, the most beautiful rectangle is onewhose ratio of length to width is the golden ratio. The golden ratio can be most easilybe expressed as (sqrt(5)+1)/2.(beautiful-rectangle 1);Value: 1.618033988749895(beautiful-rectangle 34.5);Value: 55.82217261187137(define beautiful-rectangle(f) Write a procedure that c omputes the positive root of the quadratic polynomial using thequadratic formula. The positive root is the larger of the two roots. If the polynomialhas complex roots, your procedure should return the string ”complex roots”.(postive-root 1 -2 1);Value: 1(positive-root 3 1 3);Value: "complex roots"(define postive-root6.090, IAP 2005—Lecture 2 43. Biggie-Sizing!Suppose we’re designing an point-of-sale and order-tracking system for Wendy’s1. Luckilythe¨Uber-Qwuick drive through supports only 4 options: Classic Single Combo (hamburgerwith one patty), Classic Double With Cheese Combo (2 patties), and Classic Triple withCheese Combo (3 patties), Avant-Garde Quadruple with Guacamole Combo (4 patties). Weshall encode these combos as 1, 2, 3, and 4 resp ec tively. Each meal can be biggie-sized toacquire a larger box of fries and drink. A biggie-sized combo is represented by 5, 6, 7, and 8respectively.(a) Write a procedure named biggie-size which when given a regular combo returns abiggie-sized version.(b) Write a procedure named unbiggie-size which when given a biggie-sized combo returnsa non-biggie-sized version.(c) Write a procedure named biggie-size? which when given a combo, returns true if thecombo has been biggie-sized and false otherwise.(d) Write a procedure named combo-price which takes a combo and returns the price ofthe combo. Each patty costs $1.17, and a biggie-sized version costs $.50 extra overall.16.090 and MIT do not endorse and are not affiliated with Wendy’s in any way. They merely capitalize on thepleasant way “biggie-size” rolls off the tongue.6.090, IAP 2005—Lecture 2 5RecursionMore Problems4. Write expt, a procedure that raises x to the nth power. You may assume that n is non-negativeand integer.(expt 2 2);Value: 4(expt 2 3);Value: 8Plan:(define expt(lambda (x n)6.090, IAP 2005—Lecture 2 65. Write remainder, a procedure that computes the remainder of x divided by y.(remainder 2 5);Value: 2(remainder 7 5);Value: 2Plan:(define remainder(lambda (x y)6. Write fib, a procedure that computes the nth fibonacci number.(fib 0);Value: 1(fib 1);Value: 1(fib 2);Value: 2(fib 6);Value: 13Plan:(define fib(lambda
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