W01: From function inputs to executable contracts

Prerequisites and observable goals#

The central question: what exactly is being replaced?#

The notation $f(x)$ does not instruct us to multiply an object $f$ by $x$. It names the output obtained by applying the function to an input. A useful mental model is an expression with an input slot. If $f(u)=u^2-u-3$, the letter $u$ is replaceable; the rule, including its parentheses and signs, is not. Therefore

By contrast, $f(x)+h=x^2-x-3+h$ changes an already computed output. These are two different operations: move the input, or move the output. A graph makes the distinction visible. Moving the input travels to another point of the curve; adding to the output shifts vertically at the same input.

A program has an analogous boundary. A function receiving two coordinates is not automatically receiving a velocity, a direction, or an already transformed position. Write the input representation and units before arithmetic. In a game, confusing world position with screen position is the same category of error as confusing an input with its output: the numbers may look plausible while representing different objects.

A complete chain: value, average rate, derivative, line#

Let $f(x)=x^2-x-3$ and focus on input $a=2$. The point’s height is

For the nearby input, substitute the whole expression before simplifying:

The output change is the new output minus the entire old output:

The input change is $(2+h)-2=h$. For $h\ne0$,

This cancellation removes a nonzero factor multiplying the entire numerator and denominator. It does not delete $h$ from a sum. Dividing $3+h$ by $h$, for example, would instead give $3/h+1$.

The original quotient is undefined at zero. Its simplified formula has a continuous extension there, and the limit is

The derivative is the slope at input 2. The function value $-1$ is the height at input 2. Neither is automatically the intercept of a line. Starting with a point and slope gives

Substitute the point to check: $3(2)-7=-1$. Substitute zero to find the intercept: $-7$. The tempting line $y=3x-1$ has the correct slope but the wrong point, since it gives 5 at input 2. One successful check of slope alone would miss the error.

Why the limit is not merely “set the change to zero”#

The quotient compares two distinct inputs. Its meaning requires a nonzero interval. The derivative describes how those comparisons behave as the interval shrinks. The simplified polynomial $3+h$ makes the limiting behavior easy to evaluate, but it does not retroactively define the original quotient at zero. Keeping these domains separate is mathematical bookkeeping with a purpose: the same distinction arises when an implementation rejects an input that a mathematically equivalent formula would accept after simplification.

For $r(x)=(x^2-9)/(x-3)$, factoring gives $r(x)=x+3$ only for $x\ne3$. The function $s(x)=x+3$ on all real inputs is not the identical function: its domain includes 3. They agree on the original domain. The limit of $r$ at 3 is 6, but $r(3)$ remains undefined unless we explicitly define a new function. An optimizer, database migration, or API rewrite likewise must say whether it preserves outputs on accepted inputs or changes the accepted inputs themselves.

From an algebraic identity to a code specification#

For $q(x)=Ax^2+Bx+C$, expand before memorizing:

Thus the average rate is $2Ax+Ah+B$ for $h\ne0$, while the derivative is $2Ax+B$. The tangent intercept at input $a$ is $q(a)-a(2Aa+B)$, obtained by expanding point-slope form. This derivation supplies a specification and an independent implementation check.

The lab uses exact rational arithmetic so that roundoff does not obscure the substitution lesson. A Fraction represents a rational numerator and nonzero denominator; it is not a symbolic expression or an arbitrary real number. Rejecting floating-point inputs is a deliberate educational contract, not a rule that all quadratic libraries should follow. The exact formula avoids subtracting nearly equal approximate outputs, but real-world measurement inputs would still have uncertainty.

Assignment is a transition, not an equation to solve#

In mathematics, $x=x+1$ has no real solution. In Python, x = x + 1 evaluates the right side using the old binding, then assigns a new value to the name on the left. When several state components must use the same old state, compute them before overwriting it. For a particle,

is one explicit update rule. Updating velocity first and then using it for position defines a different numerical method. Neither should silently be called the same formula. W05 later connects this choice to stability and conservation.

Independent practice#

F foundation C standard course work T transfer S optional stretch L runnable lab or project. Attempt the complete set before opening its solutions.

Complete solutions: W01#

Compare the first step where your work differs. A corrected answer is not yet an independent success.

Open every solution

"""W01: exact quadratic rates. Inputs are int or Fraction, not binary floats."""
from dataclasses import dataclass
from fractions import Fraction


def rational(value):
    if type(value) not in (int,Fraction):
        raise TypeError("use an integer or Fraction for this exact lab")
    return Fraction(value)


@dataclass(frozen=True)
class Quadratic:
    a: Fraction
    b: Fraction
    c: Fraction

    def __post_init__(self):
        for key in ('a','b','c'):
            object.__setattr__(self,key,rational(getattr(self,key)))

    def value(self,x):
        x=rational(x)
        return self.a*x*x+self.b*x+self.c

    def average_rate(self,x,h):
        x,h=rational(x),rational(h)
        if h == 0:
            raise ValueError("average rate needs a nonzero interval")
        return (self.value(x+h)-self.value(x))/h

    def tangent(self,x):
        x=rational(x)
        slope=2*self.a*x+self.b
        intercept=self.value(x)-slope*x
        return slope,intercept


if __name__=='__main__':
    f=Quadratic(3,2,-4)
    print('point height:',f.value(-1))
    print('average rate:',f.average_rate(-1,Fraction(1,10)))
    print('tangent slope and intercept:',f.tangent(-1))

Retention and readiness#

Without notes, explain the differences among $f(x+h)$, $f(x)+h$, $f(a)$, and $f'(a)$. Reconstruct a tangent from a fresh point/slope pair and reject a wrong intercept by substitution. Advance when the object and the calculation can both be explained; repair a sign or state-order gap before increasing complexity.