MOCKI

Math 114: Integration and Differentiation 1 hour

Answer ALL questions. Please use a separate answer book for each module. There are a total of 100 marks. You may only use the calculator that has been provided.

Some general information about the structure of the real exam paper in Summer:

• There will be one big question about each of the three sections in the lectures, but possibly with different weighting and possibly containing some material from other sections.

• There will be subquestions asking you to formulate/recall definitions or theorems from the lectures.

• There will be subquestions asking you to construct examples with certain properties and to prove them – either your own examples or recalling examples from the lectures.

• “Briefly explain” means to write an easily understandable rough (but correct) explanation in one or two sentences. Not a full proof!

• There will be one subquestion (worth 20 marks) asking you to properly prove something – it will use ideas of proofs from the lecture notes.

• Indicate clearly where you use statements from the lecture notes (you do not have to include the theorem numbers or page numbers, of course).

• Write readably and use mathematical notation carefully and correctly as introduced in the course of MATH110. Do not follow your high school notation and language wherever they differ.

1. • State the definition of the term step function. [10]

   • Provide an explicit example (with domain details etc.) of an increasing step function with 4 nontrivial steps. Draw its graph and compute its integral as a real number.[20]

   • Can a step function be differentiable? Briefly explain.[10]

2. Consider a series ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$.

   • Define what it means for ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$ to be convergent.[10]

   • State a convergence criterion for ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$ that we learnt in the lectures. You should write down the whole criterion and explain the terminology. Any of the the criteria from the lectures is fine, the choice is yours. [10]

   • Construct a nontrivial series which does not fulfill this criterion. No explanation nor proof required. [10]

3. Given a differentiable function $f:[a,b]\to$ which has continuous derivative $f^{\prime }:[a,b]\to$.

   • Let $c,d\in (a,b)$ with . Using results from the lectures which you should clearly reference, prove that the restriction

     $$f^{\prime }{\upharpoonright }_{(c,d)}:(c,d)\to$$

     is uniformly continuous.[20]

   • Provide an explicit example of $f$ as above such that

     $$f^{\prime }:(a,b)\to$$

     is not uniformly continuous. Briefly explain. [10]