MATH40002 Set5 REVISED
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: d0da8a68f789fdd7040c3a430317aa4aa6c5f00dbe9c14889f68eebd3201abfcSource date: 2026-08-06
Source page 1
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (19% down the source page)
The main solutions prioritise methods actually used in the official answers accessible during preparation.
English passage 2 (21% down the source page)
No lecture notes, problem sheets, tutorials or marked scripts from Tony were accessible. This booklet does not invent their contents, and advanced alternative methods are not the sole standard for solutions.
English passage 3 (38% down the source page)
Recognition signal
English passage 4 (40% down the source page)
For rearrangements return to partial sums; for a tail r first write a = r − r ; a square-root denominator suggests rationalisation and telescoping; for a /r
use blocks in the Cauchy criterion.
English passage 5 (46% down the source page)
First key step
English passage 6 (48% down the source page)
Begin the definition with “all sufficiently late partial sums”; for a tail, first write the difference identity.
English passage 7 (88% down the source page)
Definition: 3 marks; rearrangement: 7; square-root tail: 4; Cauchy divergence: 6.
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 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Treating an infinite sum as a number whose terms may be rearranged freely; reversing the inequality between negative partial sums and their limit; trying termwise comparison with the harmonic series in (d).
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
r decreases; any finite sum of negative terms is at least the sum of all negative terms.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (25% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (26% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (43% down the source page)
Recognition signal
English passage 9 (45% down the source page)
Use neighbourhoods and sequences for open/closed sets. For openness of the image of a strictly monotone continuous function, use local endpoints and IVT. Subdivide and add estimates to control long intervals using uniform continuity. Disprove uniform continuity using two sequences.
English passage 10 (51% down the source page)
First key step
English passage 11 (53% down the source page)
For openness of the image, fix y = f (x); for a growth bound, first take ε = 1; for oscillation, choose a phase difference of π.
English passage 12 (85% down the source page)
Set classification: 5 marks; open mapping: 5; linear growth bound: 5; two-sequence method: 5.
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 (9% down the source page)
Common errors
English passage 2 (12% down the source page)
Assuming every continuous function is an open map; incorrectly making the number of subintervals independent of x; choosing two sequences with fixed phase difference but whose distance does not tend to zero.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
Use both strict monotonicity and IVT; simplify x − y using a difference of squares.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (25% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 7 (26% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (43% down the source page)
Recognition signal
English passage 9 (45% down the source page)
Use a standard collection of examples; repeatedly apply Rolle to equal values; first construct a difference function for MVT; read Taylor coefficients from the power series, and use the second-derivative sign for convexity.
English passage 10 (51% down the source page)
First key step
English passage 11 (53% down the source page)
Start with the simplest example; track the decreasing number of zeros under Rolle; rewrite two solutions of an equation as two zeros of a difference function.
English passage 12 (81% down the source page)
Examples: 6 marks; repeated Rolle: 4; three MVT applications: 6; power series + convexity: 4.
English passage 13 (83% down the source page)
Common errors
English passage 14 (86% down the source page)
Common mistakes: use a constant function as an example with no local extrema; apply Rolle once too few; reverse the arctan inequality; reverse the midpoint-convexity inequality.
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 (9% down the source page)
Final self-check
English passage 2 (12% down the source page)
F should be convex on (0, 1), so its midpoint value does not exceed the average of the endpoint values.
English passage 3 (16% down the source page)
Tony study template card
English passage 4 (18% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 5 (20% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 6 (38% down the source page)
Recognition signal
English passage 7 (41% down the source page)
For an ODE with zero endpoints, use a positive maximum/negative minimum; if the integral vanishes for all test functions, use a local tent function; for nested integral identities,
English passage 8 (43% down the source page)
compare derivatives and initial values; isolate a single-point spike with a narrow partition interval.
English passage 9 (46% down the source page)
First key step
English passage 10 (49% down the source page)
Assume a positive value exists and take a global maximum; explicitly choose a nonnegative test function that is locally positive; name both sides of the identity.
English passage 11 (74% down the source page)
Maximum principle: 5 marks; test functions: 6; integral identity: 5; integrability of a spike: 4.
English passage 12 (77% down the source page)
Common errors
English passage 13 (80% down the source page)
Forgetting the zero endpoints when considering where the maximum occurs; choosing a test function that fails the endpoint conditions; exchanging integration order without stating the conditions;
English passage 14 (82% down the source page)
Incorrectly calculate the spike upper sum as 1.
English passage 15 (85% down the source page)
Final self-check
English passage 16 (88% down the source page)
Repeat the argument for −f to obtain equality to 0; both sides of the identity vanish at x = 0.
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 (9% down the source page)
Tony study template card
English passage 2 (12% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 3 (14% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.