What you’ll learn8 learning objectivesChoose one objective for a focused lesson, or study the complete topic.—AHL 5.12 (HL)—Continuity, differentiability and first principles• Understand continuity and differentiability informally.• Use convergence/divergence of limits.• Define derivative from first principles for polynomials; use higher-derivative notation.Syllabus objective—AHL 5.13 (HL)—L'Hopital's rule and limits• Evaluate limits of f(x)/g(x) as x approaches a value or infinity.• Use L'Hopital's rule or Maclaurin series for indeterminate forms 0/0 and infinity/infinity.• Repeated use of L'Hopital's rule may be required.Syllabus objective—AHL 5.14 (HL)—Implicit differentiation and related rates• Use implicit differentiation and related rates.• Solve optimization problems, including endpoint cases where appropriate.Syllabus objective—AHL 5.15 (HL)—Further derivatives and integrals• Differentiate tan x, sec x, cosec x, cot x, a^x, log_a x, arcsin x, arccos x and arctan x.• Integrate their derivative forms, including composites with linear functions.• Use partial fractions to rearrange integrands.Syllabus objective—AHL 5.16 (HL)—Advanced integration techniques• Use integration by substitution; substitutions are provided where not directly recognizable.• Use integration by parts, including repeated integration by parts.Syllabus objective—AHL 5.17 (HL)—Areas and volumes of revolution• Find area enclosed by a curve and the y-axis over an interval.• Find volumes of revolution about the x-axis or y-axis.Syllabus objective—AHL 5.18 (HL)—Differential equations• Solve first-order differential equations.• Use Euler's numerical method.• Solve separable, homogeneous and first-order linear equations with integrating factor.Syllabus objective—AHL 5.19 (HL)—Maclaurin series• Use Maclaurin series for e^x, sin x, cos x, ln(1+x) and (1+x)^p.• Obtain other series by substitution, products, integration and differentiation.• Develop Maclaurin series from differential equations.Syllabus objective