MATH40004 Practice Paper 5 REVISED 2

English review edition prepared on 4 October 2026 from a preserved source copy. It is a later presentation, not the historical study interface. Source checks and difficulty judgements describe the original material author's own process; they do not indicate Imperial College London endorsement.

Source SHA-256: 317e157493888632a81fbb07bd75c54e82ff9df5c6c06bdd097252893c58f7a5
Source date: 2026-08-06

Source page 1

Source page 2

Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

English passage 1 (13% down the source page) Recognition signal Flat exponentials defeat every algebraic power; the remaining parts are standard integration by parts, Riemann sum and integral test.

English passage 2 (35% down the source page) Common errors Assuming smoothness merely because the pointwise limit exists; claiming the integral diverges from f →0 without rate; using mesh 1/n.

Source page 3

Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

English passage 1 (13% down the source page) Recognition signal Repeated root + switched forcing: solve before and after the switching time and match both y and y′. Fourier moments come from differentiating bf.

English passage 2 (46% down the source page) Common errors Matching only y; forgetting the forcing vanishes after t = 1; losing the factor i in the third transform derivative.

Source page 4

Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

English passage 1 (13% down the source page) Recognition signal A zero eigenvalue means a line of fixed points. In implicit differentiation, remember that z depends on x, y again when taking second derivatives.

English passage 2 (43% down the source page) Common errors Saying every trajectory tends to the origin; labelling a centre asymptotically stable; treating z as constant in the second derivative.

Source page 5

Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.

English passage 1 (13% down the source page) Recognition signal For a monomial integrating factor, equate powers after differentiating. For multivariable classification, keep the gradient equations factored.

English passage 2 (82% down the source page) Common errors Integrating the unmultiplied equation; expanding the gradient too early and losing roots; classifying a point with negative determinant as an extremum.