Questions and Answers
What is the general form of a parabolic function?
What maintains the termination of a recursive function?
How can iterative structures typically be replaced in functional programming?
What issue occurs when evaluating (factorial 1000)?
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Which component is NOT essential in defining a recursive function?
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What does each recursive call in the factorial function do?
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Which of the following statements about recursion is true?
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What is the factorial of 0 according to the defined factorial function?
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What characterizes a tail-recursive subroutine?
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Why does the factorial function not exhibit tail-recursion?
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What is the potential drawback of using helper functions in recursion?
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In the provided factorial function structure, what does 'fact-helper' aim to achieve?
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What does the 'let' construct accomplish in the context of local binding?
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What happens if a tail-recursive function is called with a large value?
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Which statement about the factorial function and recursion is true?
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What type of variables can be bound using the 'let' construct?
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What does the named-let allow in Scheme?
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In the Fibonacci function provided, what does the expression (if (< n 2) n ...) return when n is less than 2?
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What values are initialized for fm and fm-1 in the Fibonacci named-let?
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What is the base case for the Padovan sequence P(n)?
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How does the factorial function utilize named-let according to the pattern described?
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What is the purpose of the parameters (m 2), (fm 1), and (fm-1 1) in the Fibonacci named-let?
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What will the expression (if (= m n) fm ...) in the Fibonacci named-let do when m equals n?
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Study Notes
Parabola Function
- Parabola is defined by the quadratic function: ( y = ax^2 + bx + c ).
- Scheme function structure: ( (define parabola (lambda (x)...)) ) needs completion.
Functional Programming and Recursion
- Iterations often involve mutable variables, leading to side effects, which are undesirable in functional programming.
- Recursion can replace loops, maintaining immutability.
Recursion Fundamentals
- Establish a function with base cases to terminate and recursive cases to move closer to base.
- Recursive functions call themselves with modified parameters to progress toward base case.
Factorial Function
- Factorial defined as:
- ( n! = 1 ) if ( n == 0 )
- ( n! = n \times (n-1)! ) if ( n > 0 ).
- Needs completion for the Scheme definition: ( (define factorial (lambda (n)...)) ).
Stack Overflow and Recursion
- Evaluating ( (factorial 1000) ) causes StackOverflowError due to recursion depth.
- Each recursive call produces a unique stack frame, consuming stack space.
Tail Recursion
- Recursive calls that perform their last action as another recursive call are called tail-recursive.
- Tail recursion allows stack frames to be reused, simulating loops in recursion.
Improving the Factorial Function
- The traditional factorial function lacks tail recursion.
- Proposed solution involves a helper function to implement tail recursion.
- Example structure:
- ( (define fact-helper (lambda (n m fm)...)) )
- ( (define factorial (lambda (n) (if (< n 0) 1 (fact-helper n 1 1)))) ).
Local Binding with let
-
let
allows variable bindings for expressions, isolating names within the expression context. - Variables must remain independent.
Fibonacci Function with let
- Structure for Fibonacci can be completed using
let
:- ( (define fibonacci (lambda (n) (if (< n 2) n (let ((fn-1...) (fn-2...))...)))) ).
Named let
- Named
let
allows recursive evaluations without cluttering the variable namespace. - Format:
- ( (let name ((name1 expr1) (name2 expr2)...) expression) )
- Inside the expression,
name
can refer to the recursive call.
Final Fibonacci Implementation
- An optimized Fibonacci function using named
let
:- If ( n < 2 ), return ( n ); otherwise, define
f
to compute based on the recurrence relation.
- If ( n < 2 ), return ( n ); otherwise, define
Exercise on Factorial with Named let
- A different approach using named
let
for factorial is suggested.
Padovan Sequence
- Defined by:
- ( P(0) = P(1) = P(2) = 1 )
- Recurrence relation: ( P(n) = P(n-2) + P(n-3) ).
- Complete the definition: ( (define padovan (lambda (n)...)) ).
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Description
This quiz covers the concepts of recursion and functional programming in Scheme, particularly focusing on defining parabolic functions. You will work on completing a function definition and understand the challenges of iterative structures in functional programming.