Calculus

The slope of a curve at a single point and the area under it, why every algebraic method stalls on both, and the hint that the two problems are one problem run in opposite directions.

What "approaches" means precisely, the laws that let most limits be computed by substitution, and the indeterminate forms where the whole content of calculus hides.

The functions whose limits are their values, the three ways that fails, root finding by bisection, and the existence theorems that the rest of the course stands on.

The difference quotient and its limit, worked from scratch for powers and roots, what differentiability rules out, and the two notations and what each is good for.

Sum, product, quotient and chain, each derived from the definition rather than quoted, so that any algebraic function can be differentiated without returning to a limit.

Two limits the algebraic rules cannot reach: the squeeze that gives the derivative of sine, and the base for which an exponential is its own derivative, which is where the number $e$ comes from.

Differentiating a curve that is not a function, getting the derivative of every inverse for free, and reading one rate of change off another in a moving system.

Fermat's condition for a maximum, the closed interval method, the second derivative as curvature, and real optimisation problems including the one that yields Snell's law.

Rolle's theorem and the tangent that matches the average slope, the corollaries that justify reading a graph off its derivative, and the fact that a derivative determines its function up to one constant.

Area defined as a limit of sums rather than assumed, computed by hand for $x^2$, with the properties that follow from the definition and the theorem that continuous functions are integrable.

Both halves, proved: the area accumulated up to $x$ has the integrand as its derivative, and any antiderivative evaluates a definite integral in two subtractions.

The chain rule and the product rule run backwards, partial fractions, and an honest account of why most integrands have no elementary antiderivative and what to do instead.

Anything accumulated: areas between curves, volumes of revolution, work against a varying force, escape velocity from an infinite domain, averages and centres of mass.

Replacing a function by the polynomial that matches all its derivatives at a point, with an error bound, which is where the approximations in every physics text come from and where they stop being valid.