Course

Every lecture has two pages. Theory follows the lecture notes: the same Definitions, Facts, Methods and Watch outs, with the same numbers. Around them is what a page without a lecturer needs: a question to answer before you read on, figures, animations, and worked examples that open one step at a time. Problems has problems that are checked instantly, problems with a solution in steps and, last, exam problems with points and a marking guide.

A Fact is shown with its statement. The reason is behind the button Why is this true?. All Facts are collected under Facts, and all concepts under Concepts.

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Placeholder. Fifteen lectures, four a week for four weeks, each followed by an exercise session. That is fast: one lecture is one day's reading, and a lecture you fall behind is hard to recover. The schedule is in TimeEdit.

Placeholder, as in the latest course round. A written exam with six or seven problems. The grade is based on the exam alone: 50 % of the points guarantees E, 60 % D, 70 % C, 80 % B and 90 % A.

The problem pages end with problems in the exam's form, with full solutions and a marking guide.

Eric Ahlqvist: eric.ahlqvist@math.su.se

Administrative questions (registration, exam sign-up): studentexpeditionen@math.su.se

hub = steps, exam = chains + argument. The practice hub and the Problems pages train one step at a time. An exam problem joins three or four steps into a chain, and every step must be argued in words.

The written exam has seven problems of 10 points each, five hours, no aids but a non-graphical calculator. Grades: E 35, D 42, C 49, B 56, A 63 of 70 points. Problem k is always about the same topic, so you can practise topic by topic: below, for each of the seven problems, the lectures it rests on, the exam problems at the end of those lectures' Problems pages, the same problem on each mock exam, and what a solution must contain to get full points (the same things the marking guides ask for).

A good order: the lectures' exam problems first, topic by topic; then the mock exams, each in five hours on paper, marked with the guide under each solution.

Problem 1 · Limits

Lectures: Lecture 5.

On the exam: (a) 5 p parameter from a 0/0 limit; (b) 5 p root expression at ∞.

Exam problems in the lectures:

• Lecture 5: A parameter from a 0/0 limit, and a root at infinity · 10 p
• Lecture 5: l'Hôpital twice with a parameter, and two differences at infinity · 10 p
• Lecture 5: Quotients at infinity: the leading terms decide · 5 p

The same problem on the mock exams: Practice exam 1 · Practice exam 2 · Exam 28 Sept 2026.

Before you hand in, check: Name the form (0/0) before using l'Hôpital; say that x > 0 when √(x²) is replaced by x; give every value of the parameter.

Problem 2 · Integrals

Lectures: Lecture 7, Lecture 8; steps from 1.

On the exam: (a) 5 p division then ln, or by parts with ln; (b) 5 p improper integral by substitution.

Exam problems in the lectures:

• Lecture 1: A polynomial plus a remainder over x − 2 · 4 p
• Lecture 7: Divide first, then integrate · 5 p
• Lecture 7: By parts: a logarithm, and the base 3 · 5 p
• Lecture 8: Two improper integrals by substitution · 10 p

The same problem on the mock exams: Practice exam 1 · Practice exam 2 · Exam 28 Sept 2026.

Before you hand in, check: The check of the division by multiplying back; the constant C; the limits in u, not in x, after the substitution; the improper integral written as a limit, not with ∞ substituted.

Problem 3 · Implicit function

Lectures: Lecture 4; steps from 1, 3, 6, 9.

On the exam: (a) 3 p all values of y(x₀); (b) 3–5 p tangent or Taylor polynomial of degree 2; (c) 2–4 p shape.

Exam problems in the lectures:

• Lecture 1: Two cubics: all real roots, and why there are no more · 5 p
• Lecture 3: First and second derivative of a composition, and the shape of the graph · 5 p
• Lecture 4: An implicit function: its value, its tangent, and which way it goes · 10 p
• Lecture 4: An implicit function: Taylor polynomial of degree 2, and its shape · 10 p
• Lecture 4: An implicit function with an exponential, in one line · 8 p
• Lecture 4: Taylor polynomials of a logarithm and of a root, and an estimate · 5 p
• Lecture 6: Exactly one solution: the argument behind “the only possible value” · 5 p
• Lecture 9: Where a level curve meets a line, and its tangents there · 6 p
• Lecture 9: An implicit function: its value, its slope by partial derivatives, its tangent · 6 p

The same problem on the mock exams: Practice exam 1 · Practice exam 2 · Exam 28 Sept 2026.

Before you hand in, check: Why the value is the only one (negative discriminant, or a strictly increasing function); the differentiated equation written out before numbers go in; the verdict tied to a sign.

Problem 4 · One-variable function with parameters

Lectures: Lecture 6, Lecture 7; steps from 1, 2, 3.

On the exam: (a) 3 p a and b from two conditions; (b) 4 p sign table, local extrema; (c) 3 p convexity or global extrema.

Exam problems in the lectures:

• Lecture 1: A parabola from two conditions · 5 p
• Lecture 2: A cubic: where it increases and decreases, and a tangent · 5 p
• Lecture 3: A rational function: derivative, sign table, increase and decrease · 6 p
• Lecture 3: Differentiating with the letters left in · 4 p
• Lecture 6: A rational function with two parameters: conditions, sign table, extreme points · 10 p
• Lecture 6: A cubic with two parameters, then its extremes on a closed interval · 8 p
• Lecture 7: A polynomial times $e^{-x}$: parameters, extrema, and global extremes · 10 p
• Lecture 7: Where a rational function and a cubic are convex · 6 p

The same problem on the mock exams: Practice exam 1 · Practice exam 2 · Exam 28 Sept 2026.

Before you hand in, check: The sign table itself; intervals stated separately on each side of the pole; local extreme points given with their values; 'no global maximum because f → +∞ as …'.

Problem 5 · Two variables

Lectures: Lecture 10, Lecture 11; steps from 9.

On the exam: (a) 2–4 p stationary points, classification, global?; (b) 3–5 p the segment; (c) 3 p λ and its meaning.

Exam problems in the lectures:

• Lecture 9: Profit on the move: the chain rule, and in growth rates · 5 p
• Lecture 10: A quadratic: the global minimum, and the extremes on a segment · 10 p
• Lecture 10: Profit from two goods, with and without a capacity limit · 8 p
• Lecture 10: All the stationary points, by cases · 6 p
• Lecture 11: Three stationary points, found by cases · 10 p
• Lecture 11: Largest and smallest value on a triangle · 10 p
• Lecture 11: Lagrange on a segment, and what λ predicts · 10 p
• Lecture 11: A consumer on a budget line · 10 p

The same problem on the mock exams: Practice exam 1 · Practice exam 2 · Exam 28 Sept 2026.

Before you hand in, check: That the segment is closed and bounded and f continuous, so the extremes exist; the endpoints evaluated, not forgotten; λ with its meaning in words, and the units when the problem has them.

Problem 6 · Series

Lectures: Lecture 8; steps from 2.

On the exam: (a) 5 p convergence set and sum = c; (b) 5 p tail, savings plan, or perpetuity and annuity.

Exam problems in the lectures:

• Lecture 2: Growth by a fixed factor, and the first year past a threshold · 5 p
• Lecture 2: The smallest n: two inequalities with n in the exponent · 4 p
• Lecture 8: A series with a root, and the tail of another · 10 p
• Lecture 8: A savings plan, a perpetuity and an annuity · 10 p
• Lecture 8: Convergence on two rays, and a logarithm as the ratio · 6 p

The same problem on the mock exams: Practice exam 1 · Practice exam 2 · Exam 28 Sept 2026.

Before you hand in, check: The first term of the series named (n starts at 0 or at 1); the endpoint that the root allows and the one that convergence forbids; the direction of the inequality when taking logarithms; a check of the n found against n − 1.

Problem 7 · Linear algebra

Lectures: Lecture 13, Lecture 14, Lecture 15; steps from 12.

On the exam: (a) 3 p determinant; (b) 4 p the remaining values, all solutions (or Gauss); (c) 3 p Gauss for one value (or a singular matrix).

Exam problems in the lectures:

• Lecture 12: A system in matrix form, checked by a product · 4 p
• Lecture 12: Which products exist, and a matrix expression · 5 p
• Lecture 12: Two lines with a parameter · 5 p
• Lecture 13: Matrix form and Gaussian elimination to the only solution · 6 p
• Lecture 13: A parameter decided by the last row · 10 p
• Lecture 13: Four equations in three unknowns, and two · 6 p
• Lecture 14: A cubic determinant, and the values it leaves · 10 p
• Lecture 14: Matrix form, a determinant, elimination — and a singular matrix · 10 p
• Lecture 14: A matrix equation, solved from the right · 8 p
• Lecture 15: A determinant with a parameter, and every value decided · 10 p
• Lecture 15: An inverse by row operations, used twice, and a matrix without one · 6 p
• Lecture 15: One matrix, four questions · 8 p

The same problem on the mock exams: Practice exam 1 · Practice exam 2 · Exam 28 Sept 2026.

Before you hand in, check: Every row operation written next to its arrow; the solution set as (x, y, z) = … with t ∈ R; a check by substitution; 'determinant zero' never read as 'no solution' (Watch out 15.8).