Part V: Calculus II
Integration strategy, honest numerical error, differential equations, series, and curves in the plane.
- Module 20Integration by parts and structural choices8 exercisesRecognize when a product suggests integration by parts, choose useful components, repeat the method when necessary, and solve a cyclic integral.
- Module 21Trigonometric integrals and trigonometric substitution8 exercisesChoose a parity-based trig integration strategy and use a trig substitution while preserving its interval and square-root sign.
- Module 22Partial fractions and a mixed integration strategy8 exercisesDecompose rational functions including repeated and irreducible factors; select methods for an unlabeled integral; and preserve domain intervals in a primitive.
- Module 23Numerical integration and honest error bounds8 exercisesCompute composite midpoint, trapezoidal, and Simpson approximations; distinguish actual error from a bound; and select a mesh meeting a stated guarantee.
- Module 24Improper integrals and probability density8 exercisesDefine improper integrals by separate limits, decide convergence, distinguish convergence from value, and normalize a continuous probability density.
- Module 25Introductory differential equations and dynamical models8 exercisesRecognize separable and first-order linear equations, preserve equilibrium solutions, verify an initial-value solution, and interpret Euler approximations.
- Mixed review R2: choosing a rate or accumulation method8 exercisesUse this review after Module 25. It deliberately combines earlier derivative ideas with integration and approximation. R2.07 is an introductory differential-equation extension; complete it when that topic is part of your study…
- Module 26Sequences, series, and the meaning of convergence8 exercisesDistinguish a sequence of terms from a sequence of partial sums, evaluate geometric and telescoping series, and use the term test only in its valid direction.
- Module 27Positive series: comparison, integral, ratio, and root tests8 exercisesSelect a convergence test based on term structure, verify its hypotheses, and distinguish a decisive result from an inconclusive one.
- Module 28Alternating series, absolute convergence, and controlled error8 exercisesDistinguish absolute from conditional convergence, verify the alternating-series hypotheses, and bound an approximation with correct next-term indexing.
- Module 29Power series and endpoint testing8 exercisesFind a radius and interval of convergence, test endpoints separately, and differentiate or integrate a power series only where justified.
- Module 30Taylor approximation, series, and remainder control8 exercisesConstruct Taylor polynomials, distinguish approximation from equality to an infinite series, and choose a justified remainder bound.
- Module 31Parametric curves and Cartesian conics8 exercisesRead a curve’s geometry and orientation from a parameter, compute slopes and lengths, and identify basic conics without losing parameter restrictions.
- Module 32Polar coordinates, polar calculus, and conics8 exercisesInterpret negative radius and equivalent coordinates, compute polar slopes and areas, choose intervals that trace a region once, and recognize focus-directrix conic forms.
- Cumulative assessment: Calculus II12 exercisesUse this assessment after Module 32, retaining Calculus I prerequisites. The method is not named in most statements. A convergence classification is not automatically an evaluation of a sum. A numerical approximation needs its…
- chapter
- workshop: connects two subjects
- checkpoint, review or assessment