Calculus

The route

1. 01

   Module 1: The derivative

   What the derivative is a limit of, which rule a function's structure calls for, and what the answer says about the function's shape.

   • 1
     Limits and Continuity

     • Given a function and a point, the learner can evaluate the limit there, by substitution where continuity permits, by algebra where an indeterminate form blocks it, or by the squeeze theorem where neither applies, decide whether the function is continuous at the point, classify any discontinuity, and evaluate limits at infinity.
   • 2
     The Derivative

     • Given a function built from powers, products, quotients and compositions, the learner can compute its derivative by selecting the rule its structure demands, evaluate a derivative from the limit definition when asked, decide where a function fails to be differentiable and say which failure it is, and read the derivative as a slope, a rate and a linear approximation.
   • 3
     L'Hôpital's Rule

     • Given a limit, the learner can decide whether it is of an indeterminate form the rule covers, apply the rule by differentiating numerator and denominator separately, repeat it while the hypotheses continue to hold, rewrite a product, difference or power into a quotient first, and identify cases where the rule is inapplicable, circular or unhelpful.
2. 02

   Module 2: The integral

   Accumulation as a limit of sums, the theorem that makes it computable by antidifferentiation, and two cases where the method gives a wrong answer if its hypotheses are not checked.

   • 1
     The Definite Integral

     • Given a definite or indefinite integral, the learner can approximate it by a Riemann sum, evaluate it exactly by the fundamental theorem, apply the substitution rule with converted limits, and interpret the result as signed area or accumulated change, including recognising when the value is negative and when the theorem does not apply.
   • 2
     Techniques of Integration

     • Given an integral, the learner can recognise which technique its structure calls for, apply integration by parts with a choice of $u$ that makes the remaining integral easier, decompose a proper rational function into partial fractions, and evaluate an improper integral as a limit, reporting convergence with a value or divergence with the reason.
3. 03

   Module 3: Infinite sums

   When an infinite sum has a limit, the tests that decide it, and the series that represent a function to a stated accuracy.

   • 1
     Convergence of Infinite Series

     • Given an infinite series, the learner can select and apply an appropriate convergence test, state the verdict with the test's hypotheses checked, sum a geometric series in closed form when the ratio permits, and distinguish absolute from conditional convergence.
   • 2
     Power Series, Taylor Expansion and the Remainder

     • Given a function, the learner can produce its Taylor polynomial from derivatives at a point, find the radius of convergence of a power series, bound the truncation error by the Lagrange remainder, and distinguish a series that converges from one that represents its function.
4. 04

   Module 4: Several variables

   Rates that depend on direction, the gradient that computes them all, and accumulation over a region rather than an interval.

   • 1
     Partial Derivatives, the Gradient and Critical Points

     • Given a function of two variables, the learner can compute its partial derivatives, assemble the gradient, evaluate a directional derivative along a normalised direction, identify the direction of steepest increase and the level-curve direction, and locate and classify critical points.
   • 2
     Double Integrals and Fubini's Theorem

     • Given a double integral over a rectangle, the learner can evaluate it as an iterated integral in either order, state the hypothesis that guarantees the two agree, and say what changes when the region is not a rectangle.
5. 05

   Module 5: Integrating along a route

   What changes when an integral runs along a path through a plane rather than over an interval: the answer generally depends on the route taken, one pair of derivatives decides whether it does, and a field that passes that test still fails to be conservative when its domain has a hole. Green's theorem trades a closed route for the region it encloses, which among other things computes an area by walking only its boundary.

   • 1
     Line Integrals and Path Parametrisation

     • Given a vector field and a stated path, the learner can parametrise the path, substitute into the line integral, evaluate the resulting single-variable integral, and report the value as a property of the oriented curve.
   • 2
     Conservative Fields, Potentials and Path Independence

     • Given a vector field on a stated region, the learner can compute both cross-partials and say what their agreement does and does not establish, construct a potential by partial integration where one exists, evaluate a conservative integral by endpoints, and recognise that agreement on a region with a hole settles nothing.
   • 3
     Green's Theorem

     • Given a positively oriented simple closed curve, the learner can convert the closed line integral into a double integral over the enclosed region, compute both sides independently as a check, choose a field that makes the region integrand constant so the boundary integral returns an area, and recognise when an enclosed singularity voids the hypothesis.

What finishing means

Finishing this course means you have demonstrated the required skills with the level of support this course currently assesses.

12 required skills. If you reach a lesson without the background it assumes, you are pointed at the prerequisite first, and returned here afterwards.