Please find the solutions to the Summer exam here. Note that these also include the marking scheme — numbers in bold brackets indicate marks, i.e. [3] implies three marks.
You will need to do exercises to prepare for your repeat exam. All of the following are of exam grade. If you have any questions please do not hesitate to use the comment function on the bottom:

Exercise Sheets

Qns. 2,4,7 — 9

Qns. 1, 5, 6 (but not the questions on singularities), 8, 9

Qns. 3 (especially the part on where they are differentiable), 5, 9, 10, 11, 14

Qns. 10 (also consider vertical asymptotes (well, vertical asymptotes are singularities), the domain (where the function is defined), roots, y-intercepts. Examine the local maxima & minima using both the second and first derivative tests. Examine concavity by using the method of split points.), 11 — 17

Past Papers

— except Q. 1(d)

— except Q. 1(d)

— except Q. 1(d)

— except Q. 1(d)

— except Q. 2

— except Q. 1(d)

— except Q. 1(d),(e)

For these older papers the layout is different to ours:

— except Q. 6(c)

— except Q. 2(b)

— except Q. 6(a)

New Material

The material that was additional to previous years was:

  1. Closed Interval Method (note that Wills does define Critical points – however we define critical points on closed intervals and include the endpoints)
  2. First Derivative Test
  3. Asymptotes
  4. Concavity

Past paper questions on maxima and minima on closed intervals can (in general) be answered using the Closed Interval Method.

Past paper questions on maxima and minima on more general sets (including the entire real line) can (in general) be answered using the First Derivative Test.

Finally we use asymptotes to help in curve sketching.

Section 4 from Problems has examples of questions on this new material.

In particular, questions 4 and 5 are of exam grade.