### Module Details

 The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
 Title Calculus I Code MATH101 Coordinator Dr M Gorbahn Mathematical Sciences Martin.Gorbahn@liverpool.ac.uk Year CATS Level Semester CATS Value Session 2023-24 Level 4 FHEQ First Semester 15

### Aims

1. To introduce the basic ideas of differential and integral calculus, to develop the basic skills required to work with them and to apply these skills to a range of problems.

2. To introduce some of the fundamental concepts and techniques of real analysis, including limits and continuity.

3. To introduce the notions of sequences and series and of their convergence.

### Learning Outcomes

(LO1) Understand the key definitions that underpin real analysis and interpret these in terms of straightforward examples.

(LO2) Apply the methods of calculus and real analysis to solve previously unseen problems (of a similar style to those covered in the course).

(LO3) Understand in interpret proofs in the context of real analysis and apply the theorems developed in the course to straightforward examples.

(LO4) Independently construct proofs of previously unseen mathematical results in real analysis (of a similar style to those demonstrated in the course).

(LO5) Differentiate and integrate a wide range of functions;

(LO6) Sketch graphs and solve problems involving optimisation and mensuration

(LO7) Understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

(S1) Numeracy

### Syllabus

Properties of subsets of the real numbers including their boundedness, supremum and infimum.

Algebraic and trigonometric functions.
Absolute values and inequalities.
Inverse functions.
Sequences.
Limit of a sequence.
Continuity of functions (via sequences).
Derivative.
Differentiation of sums, products and quotients.
Implicit differentiation.
Critical points and extrema.
Optimisation.
L'Hôpital's Theorem.
Indefinite integrals, definite integrals and the Fundamental Theorem of Calculus.
The exponential and logarithm functions, hyperbolic functions.
Techniques of integration.
Series.
Convergence of a series.
Tests for convergence.
Alternating series and absolute convergence.

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Final Assessment on campus There is a resit opportunity. This is an anonymous assessment. One hour time on task  60    50
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
class test on campus This is an anonymous assessment.  40    25
Homework Standard UoL penalty applies for late submission. This is not an anonymous assessment.    25