MATH2089 Chap.5 Differentiation and Quadrature
Differentiation and Quadrature
Taylor expansion derives finite difference and quadrature formulas and exposes their leading truncation errors. Numerical differentiation must balance decreasing truncation against increasing roundoff at very small steps. Numerical integration compares trapezoidal, Simpson and Gauss Legendre rules through nodes, weights, degree of precision, smoothness and refinement behaviour.
Reliable answers show the rule, the expected order and an observed or geometric error check. A finite difference replaces a derivative by nearby function values. Forward and backward differences are useful at boundaries and usually have lower order than a centred formula using comparable spacing. Taylor expansions show which terms cancel and identify the leading truncation error.
The derivation matters because it explains both the coefficient pattern and the power of h expected under refinement. Making h smaller reduces truncation only while arithmetic remains trustworthy. Difference formulas subtract nearby function values and divide by h, so very small steps can magnify rounding and cancellation.
A useful experiment evaluates several step sizes, compares against a reference when available, and looks for an initial error decrease followed by a roundoff-dominated plateau or rise. The smallest computed difference is not automatically the best derivative. Quadrature replaces an integral by weighted samples.
Composite trapezoidal panels approximate the curve by straight segments and give second-order behaviour for a sufficiently smooth integrand. Composite Simpson uses a specific alternating weight pattern, requires an even number of equal subintervals, and has fourth-order behaviour under stronger smoothness. A correct-looking total can still come from the wrong number of panels or a broken weight sequence.
Gauss-Legendre rules choose nodes and weights to maximise polynomial exactness for a fixed number of evaluations on a standard interval. Mapping another interval to the standard one requires the coordinate change and its scale factor. Degree of precision states the highest polynomial degree integrated exactly. Convergence order states how leading error changes for a general smooth function as the discretisation is refined.
These ideas are connected but not interchangeable. A particular integrand may be integrated exactly even when the rule has finite general order. Smooth-error formulas can fail around corners, singular derivatives, discontinuities, rapid oscillation or narrow peaks. Split the interval, transform the variable or refine locally when the function structure demands it.
Show results at related mesh sizes and compare the observed change with the predicted order. The final answer should name the formula, eligibility conditions, step or panel count, expected rate and evidence that the numerical value is stable enough for the intended use.
What this chapter covers
- 01
Taylor polynomial and remainder
- 02
Forward and centred differences
- 03
Roundoff versus truncation
- 04
Composite trapezoidal rule
- 05
Simpson's rule
- 06
Gauss Legendre nodes and weights
Worked example · free
Two trapezoidal panels
- +1The step size is one with nodes zero, one and two.
- +1Apply endpoint half weights and the full interior weight.
- +1The trapezoidal approximation is 3.
- +1The exact integral is 8/3, so convexity correctly predicts a positive error of 1/3.
Key terms
- Truncation order
- The power governing the leading approximation error as the step tends to zero.
- Degree of precision
- The highest polynomial degree integrated exactly by a quadrature rule.
- Gauss Legendre rule
- A quadrature rule using optimally selected nodes and weights on a standard interval.
Differentiation and Quadrature FAQ
Why can differentiation worsen with smaller h?
A smaller h reduces truncation initially, but nearby function values become almost equal. Their subtraction loses significant digits and division by h magnifies the roundoff.
When is Simpson's rule exact?
Its degree of precision includes cubic polynomials, so it exactly integrates them under exact arithmetic. General smooth functions still have a fourth order error term.
How should convergence order be checked?
Compare results after systematic refinement at the same target point or interval. The difference ratio should approach the predicted power while truncation dominates and assumptions hold.
Exam move
Write one card for each rule with nodes, weights, eligibility and expected order. Recalculate the trapezoidal example using Simpson's rule and explain the exactness. For differentiation, tabulate errors across several h values so the transition from truncation dominated to roundoff dominated behaviour becomes visible. Derive one forward and one centred difference from Taylor expansions, crossing out the terms that cancel.
Transform a nonstandard interval for Gauss-Legendre and highlight the scale factor. Test degree of precision using simple monomials, then keep that result separate from a refinement-order claim for a general smooth function. Add a nonsmooth or sharply peaked integrand and decide whether splitting the interval is more sensible than raising nominal order.
In every worked answer, print the step or panel count, eligibility condition and observed refinement ratio before the final approximation.
Working through Differentiation and Quadrature in MATH2089? Sia is AskSia’s AI Mathematics tutor — ask any MATH2089 Differentiation and Quadrature question and get a clear, step-by-step explanation grounded in how MATH2089 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.