Subject
Calculus
The mathematics of change and accumulation. Begins with the limit, which makes both precise, then the derivative as a rate and the integral as a total, the fundamental theorem relating them, infinite series, functions of several variables, and line integrals in the plane.
Learning paths
Calculus
Limits, differentiation, integration, infinite series, functions of several variables, and line integrals in the plane. The derivative is built as a limit of difference quotients and the integral as a limit of Riemann sums, so the differentiation rules and the fundamental theorem are derived rather than asserted, and the cases where a derivative fails to exist can be stated.
What this subject develops
Differentiate a function and read what the derivative says
Compute derivatives by choosing the rule a function's structure demands, and interpret the result as a rate, a slope and a linear approximation. This is the computational and interpretive core of the first half of calculus: a learner may apply the power rule fluently and still be unable to say why the product rule is not a product of derivatives, or what a derivative of zero establishes about a point, which are the judgements this competency claims.
Evaluate an integral and read what it accumulates
Evaluate definite and indefinite integrals by the fundamental theorem and by substitution, and interpret the result as signed area or accumulated change.
Decide convergence and expand a function as a series
Apply convergence tests to infinite series and represent functions as power series with a stated error bound.
Differentiate and integrate a function of several variables
Compute partial derivatives and gradients, evaluate directional derivatives and double integrals, and classify critical points as minima, maxima or saddles.
Evaluate line integrals and determine path independence
Evaluate the integral of a vector field along a curve, determine whether the value depends on the path, and convert between a closed line integral and an integral over the enclosed region.
See detailed outcomes
Differentiate a function and read what the derivative says
- 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.
- 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.
- 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.
Evaluate an integral and read what it accumulates
- 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.
- Given an integral, the learner can recognise which technique its structure calls for, apply integration by parts with a choice of
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.
Decide convergence and expand a function as a 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.
- 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.
Differentiate and integrate a function of several variables
- 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.
- 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.
Evaluate line integrals and determine path independence
- 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.
- 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.
- 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.