About this topic
Before you begin
Choose valid local expansions, derive composite and implicit coefficients, and check the accuracy of truncated approximations.
Learning goals
- Expand by substitution into standard series and state where the standard expansions are valid
- Use small-angle approximations to give a quadratic approximation of a trigonometric expression
- Use and derive bounded Maclaurin approximations
- Find leading Maclaurin terms of products, quotients and compositions of standard series
- Find leading Maclaurin terms of an implicitly defined function from its relation and value at zero
Syllabus statements covered
- Include: standard expansions of (1 + x)^n for rational n, e^x, sin x, cos x and ln(1 + x); first few terms by repeated differentiation (e.g. sec x), repeated implicit differentiation (e.g. y³ + y² + y = x² − 2x), or standard series (e.g. e^x cos 2x, ln((1 + x)/(1 − x))); convergence ranges; approximation of a function; small-angle approximations sin x ≈ x, cos x ≈ 1 − x²/2, tan x ≈ x. Exclude derivation of a general term.
- Include: standard expansions of (1 + x)^n for rational n, e^x, sin x, cos x and ln(1 + x); first few terms by repeated differentiation (e.g. sec x), repeated implicit differentiation (e.g. y³ + y² + y = x² − 2x), or standard series (e.g. e^x cos 2x, ln((1 + x)/(1 − x))); convergence ranges; approximation of a function; small-angle approximations sin x ≈ x, cos x ≈ 1 − x²/2, tan x ≈ x. Exclude derivation of a general term.
Choose a lesson
Work through them in order.
- Not startedStandard series and convergenceNextChoose standard expansions with their transformed intervals, then distinguish convergence, domain and useful numerical accuracy.
- Not startedComposite and implicit seriesNextCombine series through a chosen order, derive local coefficients by repeated differentiation, and verify implicit approximations.