Calculus

Calculus is a major part of the A level specifications, but different schools will teach the topics at different times.  If you are asked to differentiate a product or composition of functions and you don't know how to do it (yet!) then just tell your interviewer that you have not met that method.  You are not expected to have covered the whole A level specification, but you will be expected to be able to use what you do know fluently.  See this video for more explanation on what we mean by fluency.

You will often need to differentiate when doing curve sketching questions, but other interview questions might include integration, differential equations or optimisation.

The warm ups for Foundation Assignments 20 to 25 introduce some of the the A level Calculus techniques such as the product rule and the chain rule.  You might also find the STEP 2 Calculus topic notes helpful.

Interview question examples

Here are some questions for you to try.  Each one has a link to a video where we discuss the question and some possible methods which can be used to solve it.

Question 1; Which is bigger, $2^{\frac 1 2}$ or $3^{\frac 1 3}$?  Which is bigger, $\pi^e$ or $e^{\pi}$?

This is a bit of an "old chestnut", but there are a few different ways of approaching it.  The video below discusses some of them.

Question 2: Two points lie on the curve $y=x^2$ in such as way that the two points and the origin form a triangle with a right angle at the origin.  What is the locus of the midpoint of the line segment joining the two points?

The video below shows one of our undergraduate mentors tackling the problem as part of a "mock interview".  It is taken from 1994 STEP 1 Q5.  You can see the whole question, along with a video solution and written solutions from sixth form students on this page.

Question 3: The floor function $\lfloor x \rfloor$ gives the greatest integer which is less than or equal to $x$. For example $\lfloor 4.2 \rfloor = 4$, $\lfloor \pi \rfloor = 3$ and $\lfloor -2.3 \rfloor=-3$.  Can you find $\int_0^5 \lfloor x \rfloor {\rm d}x$?  Can you find $\int_0^{\ln N} \lfloor e^x \rfloor {\rm d}x $?

The video below shows one of our undergraduate mentors tackling the problem as part of a "mock interview".  

More practice questions

Here are some more questions which involve calculus.  For STEP questions use the STEP database to find the question.

• 1991 STEP 1 Question 4.  This feels quite interview-y as you are not given a lot of help with the method, apart from the one hint. This hint probably would not be given initially in an interview, but does make the STEP question a little easier to mark!
• 1991 STEP 1 Question 2. Note that the version that appears on the STEP paper is a little different to the version in Stephen Siklos’s book.
• 1995 STEP 2 Question 5.  This problem also appears on Nrich "To Swim or to Run?", which explains the origin of the "famous film star's" name, and well as showing some student solutions to the problem.
• 1998 STEP 3 Question 2.  This question needs integration by parts.
• 2000 STEP 1 Question 8.  Curve sketching might be helpful!
• You could try the STEP 2 Calculus questions, but these do assume you have covered all of A level single maths.