Calculus 1 tutoring, online and built around your exam.
Limits that do not add up, integrals that stall halfway, series whose test is never clear: Calculus 1 is the exam students resit most often. We work in 1-to-1 online lessons on your syllabus and your exercises, starting from where you are stuck now rather than from chapter one.
Let’s start from where you are stuck now.
Who asks us for help with Calculus 1.
- The first sitting went badly: the mark does not match the hours spent on the textbook and you need to see what is really going wrong.
- An exam to resit for the second or third time, with a date already fixed and little time to restart.
- A gap in the basics of algebra, trigonometry or elementary functions that only shows up when an exercise gets long.
- Solid theory but exercises that stall, or the opposite: fast calculations but no idea when a theorem can be used.
How we work on Calculus 1.
- We start from your paper: you bring the last exam or the exercises that fail, and we look at the reasoning, not only the result.
- Shared whiteboard: every step stays written down, so the lesson becomes revision material for the days before the exam.
- Timed practice with a practice test and a guided review of the mistakes, to reach the exam without surprises.
What Calculus 1 covers.
Contents change from one course and one lecturer to another: this is the structure we meet most often. Your course syllabus has the last word: in the first lesson we read it together and work on that.
- Limits and continuity
- Indeterminate forms and standard limits
- Comparing infinities and infinitesimals
- Continuity and the theorems on continuous functions
- Derivatives and curve sketching
- Differentiation rules and the chain rule
- Maxima, minima, convexity and asymptotes
- Rolle's, Lagrange's and de l'Hôpital's theorems
- Integrals
- Integration by substitution and by parts
- The definite integral and area
- Improper integrals and convergence tests
- Sequences and series
- Limits of sequences
- Series of positive terms and alternating series
- Taylor and Maclaurin expansions with the remainder
- Differential equations
- First order: separable and linear
- Second order with constant coefficients, where the course includes them
How the exam is usually structured.
In most courses the exam is a written paper of two or three hours with four or five exercises, followed, depending on the lecturer, by an oral. The exact format changes, and the first useful thing to do is to get the papers of the last few sittings of your course: they are the most faithful source of what will be asked.
The written paper usually tests technique: a limit or a curve study, an integral, often improper, a series with a parameter, a differential equation. The oral tests the hypotheses instead: stating a theorem, saying when it can be applied and when it cannot, sometimes a short proof. Preparing only the exercises is not enough, and neither is preparing only the theory.
The real opponent is time: five exercises in a few hours leave no room to restart from zero on a single step. We train that too, with timed exercises and by choosing the order in which to tackle the parts.
Three mistakes that cost marks.
- Truncating Taylor at the wrong order
To compute a limit with an expansion you have to reach the first order at which something survives. Stopping one term too early leads to 0 or to a result that is plausible but false. Practical rule: expand until a nonzero term remains, then compare the infinitesimals.
- Applying a test without checking its hypotheses
The asymptotic comparison test holds for positive terms; the alternating series test needs a term that decreases to zero. A series with mixed signs treated as positive gives the wrong conclusion, and it is the oral question that makes candidates fail. Write the hypotheses next to the series before choosing the test.
- Losing marks on the algebra
The limit is right and so is the integral, but a sign or a denominator wrong in one step (rationalising, factorising, completing the square) throws everything off. It is not a calculus mistake: it is a gap in the basics that a long exercise exposes, and it is fixed with targeted exercises, not by rereading the chapter.
An exam-style exercise, step by step.
A limit solved with Taylor
Let us compute the limit
Numerator and denominator both tend to : an indeterminate form. The denominator has order , so we expand sine and cosine up to that order (for the cosine up to order is enough).
Subtracting, the terms of order cancel and what is left is
Dividing by and taking the limit:
Stopping at order would have left with no information: the term that survives is exactly the one of order . It is the first mistake in the list above.
Timed Calculus 1 practice test.
A test that mixes the topics of the syllabus the way a real exam sitting does, with marking and weak points. It is free and you do not need an account to try it.
Frequently asked questions about Calculus 1.
Do I need to have attended the course to get Calculus 1 tutoring?
No. If the course is under way we work alongside the lectures; if you have already sat the exam and must resit it we start from the mistakes of the last paper. In both cases the first lesson is used to understand your real syllabus.
Is it better to work on theory or on exercises?
On both, but in the right order: first the exercise, because it shows which statements you are missing, then the theory that exercise needs. A proof memorised without ever using it is forgotten; one used ten times stays.
How many lessons are needed to pass Calculus 1?
It depends on how long is left before the exam and where you stand. We do not promise a number: after the first lesson we tell you honestly whether the time you have is enough and which topics come first.
Can we use the papers of my past exam sittings?
Yes, it is the best way: bring the papers of the last sittings and your solutions. We correct together where you lose marks and build the practice exercises in the same style.
The other exams we cover.
- Calculus 2
Several variables, multiple integrals, fields and surfaces: the sketch comes before the calculation.
See the page → - Linear algebra
Matrices, vector spaces, eigenvalues and quadratic forms: the vocabulary matters as much as the calculations.
See the page → - Physics 1
Mechanics and thermodynamics: the method for setting up the problem, symbols before numbers.
See the page →
Want to review the concepts before the lesson? Read limits explained well, integrals explained well and the guide to series convergence tests.
For the full picture go to university tutoring and the maths syllabus.
Your next Calculus 1 exam.
Message me on WhatsApp with the exam date and the result of your last paper: I will tell you where to start.