MATH40004 Practice Paper 4 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: 8a82b0d5d4108890eb3bd873efd1b97efc6c585c0a4a52fd4fb47d9f2e2adc83
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 At a special point, use the definitions before differentiating the formula for x̸ = 0. Parametric tangents come from numerator/denominator zero conditions.

English passage 2 (74% down the source page) Common errors Using the derivative formula at x = 0; confusing non-continuity of f ′ with non-existence of f ′(0); setting both numerator and denominator to zero without checking.

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 Split an improper integral at a finite point and analyse each endpoint separately. The centroid uses first moments divided by area.

English passage 2 (70% down the source page) Common errors Testing only infinity; claiming oscillation alone guarantees convergence; using

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 Use Taylor coefficients plus a remainder estimate; use oddness for the sign function; differentiate a continuous Fourier series only after identifying its piecewise derivative.

English passage 2 (49% down the source page) Common errors Giving a decimal without a remainder argument; using a cosine series for an odd function; assigning the one- sided derivative value at a jump instead of the midpoint.

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 y′′ = F(y, y′), reduce order by treating u = y′ as a function of y. The delta equation is a standard transform-domain Green function.

English passage 2 (89% down the source page) Common errors Cancelling u without noting the exceptional constant solution; missing the shift phase; saying the origin is the only fixed point when λ = 0.